id sid tid token lemma pos iajs-2288 1 1 117 117 NUM iajs-2288 1 2 ibn ibn PROPN iajs-2288 1 3 al al PROPN iajs-2288 1 4 - - PUNCT iajs-2288 1 5 haitham haitham PROPN iajs-2288 1 6 jour jour X iajs-2288 1 7 . . PROPN iajs-2288 1 8 for for ADP iajs-2288 1 9 pure pure ADJ iajs-2288 1 10 & & CCONJ iajs-2288 1 11 appl appl PROPN iajs-2288 1 12 . . PUNCT iajs-2288 2 1 sci sci PROPN iajs-2288 2 2 . . PROPN iajs-2288 2 3 32 32 NUM iajs-2288 2 4 ( ( PUNCT iajs-2288 2 5 3 3 NUM iajs-2288 2 6 ) ) PUNCT iajs-2288 2 7 2019 2019 NUM iajs-2288 2 8 ali ali PROPN iajs-2288 2 9 sh sh PROPN iajs-2288 2 10 . . PROPN iajs-2288 2 11 ajeel ajeel PROPN iajs-2288 2 12 haibat haibat PROPN iajs-2288 2 13 k. k. PROPN iajs-2288 2 14 mohammadali mohammadali PROPN iajs-2288 3 1 abstract abstract ADJ iajs-2288 3 2 we we PRON iajs-2288 3 3 introduce introduce VERB iajs-2288 3 4 in in ADP iajs-2288 3 5 this this DET iajs-2288 3 6 paper paper NOUN iajs-2288 3 7 the the DET iajs-2288 3 8 concept concept NOUN iajs-2288 3 9 of of ADP iajs-2288 3 10 approximaitly approximaitly ADV iajs-2288 3 11 semi semi ADJ iajs-2288 3 12 - - ADJ iajs-2288 3 13 prime prime ADJ iajs-2288 3 14 submodules submodule NOUN iajs-2288 3 15 of of ADP iajs-2288 3 16 unitary unitary ADJ iajs-2288 3 17 left leave VERB iajs-2288 3 18 𝑅-module 𝑅-module PROPN iajs-2288 3 19 𝑇 𝑇 PROPN iajs-2288 3 20 over over ADP iajs-2288 3 21 a a DET iajs-2288 3 22 commutative commutative ADJ iajs-2288 3 23 ring ring NOUN iajs-2288 3 24 𝑅 𝑅 PROPN iajs-2288 3 25 with with ADP iajs-2288 3 26 identity identity NOUN iajs-2288 3 27 as as ADP iajs-2288 3 28 a a DET iajs-2288 3 29 generalization generalization NOUN iajs-2288 3 30 of of ADP iajs-2288 3 31 a a DET iajs-2288 3 32 prime prime ADJ iajs-2288 3 33 submodules submodule NOUN iajs-2288 3 34 and and CCONJ iajs-2288 3 35 semi semi ADJ iajs-2288 3 36 - - ADJ iajs-2288 3 37 prime prime ADJ iajs-2288 3 38 submodules submodule NOUN iajs-2288 3 39 , , PUNCT iajs-2288 3 40 also also ADV iajs-2288 3 41 generalization generalization NOUN iajs-2288 3 42 of of ADP iajs-2288 3 43 quasi quasi ADJ iajs-2288 3 44 - - ADJ iajs-2288 3 45 prime prime ADJ iajs-2288 3 46 submodules submodule NOUN iajs-2288 3 47 and and CCONJ iajs-2288 3 48 approximaitly approximaitly ADV iajs-2288 3 49 prime prime ADJ iajs-2288 3 50 submodules submodule NOUN iajs-2288 3 51 . . PUNCT iajs-2288 4 1 various various ADJ iajs-2288 4 2 basic basic ADJ iajs-2288 4 3 properties property NOUN iajs-2288 4 4 of of ADP iajs-2288 4 5 an an DET iajs-2288 4 6 approximaitly approximaitly ADV iajs-2288 4 7 semi semi ADJ iajs-2288 4 8 - - ADJ iajs-2288 4 9 prime prime ADJ iajs-2288 4 10 submodules submodule NOUN iajs-2288 4 11 are be AUX iajs-2288 4 12 discussed discuss VERB iajs-2288 4 13 , , PUNCT iajs-2288 4 14 where where SCONJ iajs-2288 4 15 a a DET iajs-2288 4 16 proper proper ADJ iajs-2288 4 17 submodule submodule NOUN iajs-2288 4 18 𝐿 𝐿 PROPN iajs-2288 4 19 of of ADP iajs-2288 4 20 an an DET iajs-2288 4 21 𝑅-module 𝑅-module PROPN iajs-2288 4 22 𝑇 𝑇 PROPN iajs-2288 4 23 is be AUX iajs-2288 4 24 called call VERB iajs-2288 4 25 an an DET iajs-2288 4 26 approximaitly approximaitly ADV iajs-2288 4 27 semi semi ADJ iajs-2288 4 28 - - ADJ iajs-2288 4 29 prime prime ADJ iajs-2288 4 30 submodule submodule NOUN iajs-2288 4 31 of of ADP iajs-2288 4 32 , , PUNCT iajs-2288 4 33 if if SCONJ iajs-2288 4 34 whenever whenever SCONJ iajs-2288 4 35 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 4 36 ∈ ∈ PROPN iajs-2288 4 37 𝐿 𝐿 PROPN iajs-2288 4 38 , , PUNCT iajs-2288 4 39 where where SCONJ iajs-2288 4 40 π‘Ž π‘Ž PROPN iajs-2288 4 41 ∈ ∈ PROPN iajs-2288 4 42 𝑅 𝑅 PROPN iajs-2288 4 43 , , PUNCT iajs-2288 4 44 𝑑 𝑑 PROPN iajs-2288 4 45 ∈ ∈ PROPN iajs-2288 4 46 𝑇 𝑇 PROPN iajs-2288 4 47 and and CCONJ iajs-2288 4 48 𝑛 𝑛 DET iajs-2288 4 49 ∈ ∈ PROPN iajs-2288 4 50 𝑍+ 𝑍+ NOUN iajs-2288 4 51 , , PUNCT iajs-2288 4 52 implies imply VERB iajs-2288 5 1 that that SCONJ iajs-2288 5 2 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 5 3 ∈ ∈ PROPN iajs-2288 5 4 𝐿 𝐿 PROPN iajs-2288 5 5 + + NOUN iajs-2288 5 6 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 5 7 ) ) PUNCT iajs-2288 5 8 . . PUNCT iajs-2288 6 1 furthermore furthermore ADV iajs-2288 6 2 the the DET iajs-2288 6 3 behaviors behavior NOUN iajs-2288 6 4 of of ADP iajs-2288 6 5 approximaitly approximaitly ADV iajs-2288 6 6 semiprime semiprime NOUN iajs-2288 6 7 submodule submodule NOUN iajs-2288 6 8 in in ADP iajs-2288 6 9 some some DET iajs-2288 6 10 classes class NOUN iajs-2288 6 11 of of ADP iajs-2288 6 12 modules module NOUN iajs-2288 6 13 are be AUX iajs-2288 6 14 studied study VERB iajs-2288 6 15 . . PUNCT iajs-2288 7 1 on on ADP iajs-2288 7 2 the the DET iajs-2288 7 3 other other ADJ iajs-2288 7 4 hand hand NOUN iajs-2288 7 5 several several ADJ iajs-2288 7 6 characterizations characterization NOUN iajs-2288 7 7 of of ADP iajs-2288 7 8 this this DET iajs-2288 7 9 concept concept NOUN iajs-2288 7 10 are be AUX iajs-2288 7 11 introduced introduce VERB iajs-2288 7 12 . . PUNCT iajs-2288 8 1 keywords keyword NOUN iajs-2288 8 2 : : PUNCT iajs-2288 8 3 prime prime ADJ iajs-2288 8 4 submodules submodule NOUN iajs-2288 8 5 , , PUNCT iajs-2288 8 6 semi semi ADJ iajs-2288 8 7 - - ADJ iajs-2288 8 8 prime prime ADJ iajs-2288 8 9 submodules submodule NOUN iajs-2288 8 10 , , PUNCT iajs-2288 8 11 quasi quasi ADJ iajs-2288 8 12 - - ADJ iajs-2288 8 13 prime prime ADJ iajs-2288 8 14 submodules submodule NOUN iajs-2288 8 15 , , PUNCT iajs-2288 8 16 approximaitly approximaitly ADV iajs-2288 8 17 prime prime ADJ iajs-2288 8 18 submodules submodule NOUN iajs-2288 8 19 , , PUNCT iajs-2288 8 20 approximaitly approximaitly ADV iajs-2288 8 21 semi semi ADJ iajs-2288 8 22 - - ADJ iajs-2288 8 23 prime prime ADJ iajs-2288 8 24 submodules submodule NOUN iajs-2288 8 25 , , PUNCT iajs-2288 8 26 multiplication multiplication NOUN iajs-2288 8 27 module module NOUN iajs-2288 8 28 , , PUNCT iajs-2288 8 29 socle socle NOUN iajs-2288 8 30 of of ADP iajs-2288 8 31 modules module NOUN iajs-2288 8 32 . . PUNCT iajs-2288 9 1 1 1 X iajs-2288 9 2 . . X iajs-2288 9 3 introduction introduction NOUN iajs-2288 9 4 throughout throughout ADP iajs-2288 9 5 this this DET iajs-2288 9 6 article article NOUN iajs-2288 9 7 , , PUNCT iajs-2288 9 8 all all DET iajs-2288 9 9 rings ring NOUN iajs-2288 9 10 are be AUX iajs-2288 9 11 commutative commutative ADJ iajs-2288 9 12 rings ring NOUN iajs-2288 9 13 with with ADP iajs-2288 9 14 identity identity NOUN iajs-2288 9 15 and and CCONJ iajs-2288 9 16 all all DET iajs-2288 9 17 modules module NOUN iajs-2288 9 18 are be AUX iajs-2288 9 19 unitary unitary ADJ iajs-2288 9 20 . . PUNCT iajs-2288 10 1 prime prime ADJ iajs-2288 10 2 submodules submodule NOUN iajs-2288 10 3 play play VERB iajs-2288 10 4 an an DET iajs-2288 10 5 important important ADJ iajs-2288 10 6 role role NOUN iajs-2288 10 7 in in ADP iajs-2288 10 8 module module NOUN iajs-2288 10 9 theory theory NOUN iajs-2288 10 10 over over ADP iajs-2288 10 11 commutative commutative ADJ iajs-2288 10 12 ring ring NOUN iajs-2288 10 13 with with ADP iajs-2288 10 14 identity identity NOUN iajs-2288 10 15 , , PUNCT iajs-2288 10 16 where where SCONJ iajs-2288 10 17 a a DET iajs-2288 10 18 proper proper ADJ iajs-2288 10 19 submodule submodule NOUN iajs-2288 10 20 𝐿 𝐿 PROPN iajs-2288 10 21 of of ADP iajs-2288 10 22 an an DET iajs-2288 10 23 𝑅module 𝑅module PROPN iajs-2288 10 24 𝑇 𝑇 PROPN iajs-2288 10 25 is be AUX iajs-2288 10 26 called call VERB iajs-2288 10 27 a a DET iajs-2288 10 28 prime prime NOUN iajs-2288 10 29 , , PUNCT iajs-2288 10 30 if if SCONJ iajs-2288 10 31 whenever whenever SCONJ iajs-2288 10 32 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 10 33 ∈ ∈ PROPN iajs-2288 10 34 𝐿 𝐿 PROPN iajs-2288 10 35 , , PUNCT iajs-2288 10 36 with with ADP iajs-2288 10 37 π‘Ž π‘Ž PROPN iajs-2288 10 38 ∈ ∈ PROPN iajs-2288 10 39 𝑅 𝑅 PROPN iajs-2288 10 40 , , PUNCT iajs-2288 10 41 𝑑 𝑑 PROPN iajs-2288 10 42 ∈ ∈ PROPN iajs-2288 10 43 𝑇 𝑇 PROPN iajs-2288 10 44 , , PUNCT iajs-2288 10 45 implies imply VERB iajs-2288 10 46 that that SCONJ iajs-2288 10 47 either either CCONJ iajs-2288 10 48 𝑑 𝑑 PROPN iajs-2288 10 49 ∈ ∈ PROPN iajs-2288 10 50 𝐿 𝐿 PROPN iajs-2288 10 51 or or CCONJ iajs-2288 10 52 π‘Ž π‘Ž PROPN iajs-2288 10 53 ∈ ∈ NOUN iajs-2288 10 54 [ [ X iajs-2288 10 55 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 10 56 𝑇 𝑇 PROPN iajs-2288 10 57 ] ] PUNCT iajs-2288 10 58 where where SCONJ iajs-2288 10 59 [ [ X iajs-2288 10 60 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 10 61 𝑇 𝑇 PROPN iajs-2288 10 62 ] ] X iajs-2288 10 63 = = SYM iajs-2288 10 64 { { PUNCT iajs-2288 10 65 π‘Ÿ π‘Ÿ X iajs-2288 10 66 ∈ ∈ PROPN iajs-2288 10 67 𝑅 𝑅 NOUN iajs-2288 10 68 : : PUNCT iajs-2288 10 69 π‘Ÿπ‘‡ π‘Ÿπ‘‡ NOUN iajs-2288 10 70 βŠ† βŠ† NUM iajs-2288 10 71 𝐿}[1 𝐿}[1 NOUN iajs-2288 10 72 ] ] PUNCT iajs-2288 10 73 . . PUNCT iajs-2288 11 1 recently recently ADV iajs-2288 11 2 several several ADJ iajs-2288 11 3 generalization generalization NOUN iajs-2288 11 4 of of ADP iajs-2288 11 5 the the DET iajs-2288 11 6 concept concept NOUN iajs-2288 11 7 of of ADP iajs-2288 11 8 prime prime ADJ iajs-2288 11 9 submodules submodule NOUN iajs-2288 11 10 are be AUX iajs-2288 11 11 studied study VERB iajs-2288 11 12 in in ADP iajs-2288 11 13 [ [ PUNCT iajs-2288 11 14 2 2 NUM iajs-2288 11 15 - - SYM iajs-2288 11 16 5 5 NUM iajs-2288 11 17 ] ] PUNCT iajs-2288 11 18 . . PUNCT iajs-2288 12 1 the the DET iajs-2288 12 2 concept concept NOUN iajs-2288 12 3 semi semi ADJ iajs-2288 12 4 - - ADJ iajs-2288 12 5 prime prime ADJ iajs-2288 12 6 submodule submodule NOUN iajs-2288 12 7 which which PRON iajs-2288 12 8 was be AUX iajs-2288 12 9 first first ADV iajs-2288 12 10 introduced introduce VERB iajs-2288 12 11 in in ADP iajs-2288 12 12 [ [ X iajs-2288 12 13 6 6 NUM iajs-2288 12 14 ] ] PUNCT iajs-2288 12 15 . . PUNCT iajs-2288 13 1 and and CCONJ iajs-2288 13 2 extensively extensively ADV iajs-2288 13 3 studied study VERB iajs-2288 13 4 in in ADP iajs-2288 13 5 [ [ X iajs-2288 13 6 7 7 NUM iajs-2288 13 7 ] ] PUNCT iajs-2288 13 8 . . PUNCT iajs-2288 14 1 is be AUX iajs-2288 14 2 given give VERB iajs-2288 14 3 as as SCONJ iajs-2288 14 4 a a DET iajs-2288 14 5 proper proper ADJ iajs-2288 14 6 submodule submodule NOUN iajs-2288 14 7 𝐿 𝐿 PROPN iajs-2288 14 8 of of ADP iajs-2288 14 9 an an DET iajs-2288 14 10 𝑅-module 𝑅-module PROPN iajs-2288 14 11 𝑇 𝑇 PROPN iajs-2288 14 12 is be AUX iajs-2288 14 13 called call VERB iajs-2288 14 14 semiprime semiprime NOUN iajs-2288 14 15 submodule submodule NOUN iajs-2288 14 16 , , PUNCT iajs-2288 14 17 if if SCONJ iajs-2288 14 18 wheneverπ‘Žπ‘›π‘‘ wheneverπ‘Žπ‘›π‘‘ ADP iajs-2288 14 19 ∈ ∈ PROPN iajs-2288 14 20 𝐿 𝐿 PROPN iajs-2288 14 21 , , PUNCT iajs-2288 14 22 where where SCONJ iajs-2288 14 23 π‘Ž π‘Ž PROPN iajs-2288 14 24 ∈ ∈ PROPN iajs-2288 14 25 𝑅 𝑅 PROPN iajs-2288 14 26 , , PUNCT iajs-2288 14 27 𝑑 𝑑 PROPN iajs-2288 14 28 ∈ ∈ PROPN iajs-2288 14 29 𝑇 𝑇 PROPN iajs-2288 14 30 and and CCONJ iajs-2288 14 31 𝑛 𝑛 DET iajs-2288 14 32 ∈ ∈ PROPN iajs-2288 14 33 𝑍+ 𝑍+ NOUN iajs-2288 14 34 , , PUNCT iajs-2288 14 35 implies imply VERB iajs-2288 14 36 that that SCONJ iajs-2288 14 37 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 14 38 ∈ ∈ PROPN iajs-2288 14 39 𝐿. 𝐿. VERB iajs-2288 15 1 [ [ X iajs-2288 15 2 7 7 NUM iajs-2288 15 3 ] ] PUNCT iajs-2288 15 4 . . PUNCT iajs-2288 16 1 characterized characterize VERB iajs-2288 16 2 semi semi ADJ iajs-2288 16 3 - - ADJ iajs-2288 16 4 prime prime ADJ iajs-2288 16 5 submodules submodule NOUN iajs-2288 16 6 as as SCONJ iajs-2288 16 7 follows follow VERB iajs-2288 16 8 : : PUNCT iajs-2288 16 9 a a DET iajs-2288 16 10 proper proper ADJ iajs-2288 16 11 submodule submodule NOUN iajs-2288 16 12 𝐿 𝐿 PROPN iajs-2288 16 13 of of ADP iajs-2288 16 14 an an DET iajs-2288 16 15 𝑅-module 𝑅-module PROPN iajs-2288 16 16 𝑇 𝑇 PROPN iajs-2288 16 17 is be AUX iajs-2288 16 18 semi semi ADJ iajs-2288 16 19 - - ADJ iajs-2288 16 20 prime prime ADJ iajs-2288 16 21 if if SCONJ iajs-2288 17 1 and and CCONJ iajs-2288 17 2 only only ADV iajs-2288 17 3 if if SCONJ iajs-2288 17 4 whenever whenever SCONJ iajs-2288 17 5 π‘Ž2𝑑 π‘Ž2𝑑 PROPN iajs-2288 17 6 ∈ ∈ PROPN iajs-2288 17 7 𝐿 𝐿 PROPN iajs-2288 17 8 , , PUNCT iajs-2288 17 9 where where SCONJ iajs-2288 17 10 π‘Ž π‘Ž PROPN iajs-2288 17 11 ∈ ∈ PROPN iajs-2288 17 12 𝑅 𝑅 PROPN iajs-2288 17 13 , , PUNCT iajs-2288 17 14 𝑑 𝑑 PROPN iajs-2288 17 15 ∈ ∈ PROPN iajs-2288 17 16 𝑇 𝑇 PROPN iajs-2288 17 17 , , PUNCT iajs-2288 17 18 implies imply VERB iajs-2288 17 19 that that SCONJ iajs-2288 17 20 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 17 21 ∈ ∈ PROPN iajs-2288 17 22 𝐿. 𝐿. VERB iajs-2288 17 23 the the DET iajs-2288 17 24 concept concept NOUN iajs-2288 17 25 quasi quasi ADJ iajs-2288 17 26 - - ADJ iajs-2288 17 27 prime prime ADJ iajs-2288 17 28 submodule submodule NOUN iajs-2288 17 29 which which PRON iajs-2288 17 30 introduced introduce VERB iajs-2288 17 31 and and CCONJ iajs-2288 17 32 studied study VERB iajs-2288 17 33 in in ADP iajs-2288 17 34 [ [ X iajs-2288 17 35 8 8 NUM iajs-2288 17 36 ] ] PUNCT iajs-2288 17 37 . . PUNCT iajs-2288 18 1 is be AUX iajs-2288 18 2 a a DET iajs-2288 18 3 strong strong ADJ iajs-2288 18 4 form form NOUN iajs-2288 18 5 of of ADP iajs-2288 18 6 a a DET iajs-2288 18 7 semi semi ADJ iajs-2288 18 8 - - ADJ iajs-2288 18 9 prime prime ADJ iajs-2288 18 10 submodule submodule NOUN iajs-2288 18 11 , , PUNCT iajs-2288 18 12 where where SCONJ iajs-2288 18 13 a a DET iajs-2288 18 14 proper proper ADJ iajs-2288 18 15 submodule submodule NOUN iajs-2288 18 16 𝐿 𝐿 PROPN iajs-2288 18 17 of of ADP iajs-2288 18 18 an an DET iajs-2288 18 19 𝑅module 𝑅module PROPN iajs-2288 18 20 𝑇 𝑇 PROPN iajs-2288 18 21 is be AUX iajs-2288 18 22 called call VERB iajs-2288 18 23 a a DET iajs-2288 18 24 quasiprime quasiprime ADJ iajs-2288 18 25 , , PUNCT iajs-2288 18 26 if if SCONJ iajs-2288 18 27 whenever whenever SCONJ iajs-2288 18 28 π‘Žπ‘π‘‘ π‘Žπ‘π‘‘ PROPN iajs-2288 18 29 ∈ ∈ PROPN iajs-2288 18 30 𝐿 𝐿 PROPN iajs-2288 18 31 , , PUNCT iajs-2288 18 32 with with ADP iajs-2288 18 33 π‘Ž π‘Ž PROPN iajs-2288 18 34 , , PUNCT iajs-2288 18 35 𝑏 𝑏 PROPN iajs-2288 18 36 ∈ ∈ PROPN iajs-2288 18 37 𝑅 𝑅 PROPN iajs-2288 18 38 , , PUNCT iajs-2288 18 39 𝑑 𝑑 PROPN iajs-2288 18 40 ∈ ∈ PROPN iajs-2288 18 41 𝑇 𝑇 PROPN iajs-2288 18 42 , , PUNCT iajs-2288 18 43 implies imply VERB iajs-2288 18 44 that that SCONJ iajs-2288 18 45 either either CCONJ iajs-2288 18 46 π‘Žπ‘‘ π‘Žπ‘‘ DET iajs-2288 18 47 ∈ ∈ PROPN iajs-2288 18 48 𝐿 𝐿 PROPN iajs-2288 18 49 or or CCONJ iajs-2288 18 50 𝑏𝑑 𝑏𝑑 NOUN iajs-2288 18 51 ∈ ∈ PROPN iajs-2288 18 52 𝐿. 𝐿. PROPN iajs-2288 18 53 ibn ibn NOUN iajs-2288 18 54 al al PROPN iajs-2288 18 55 haitham haitham PROPN iajs-2288 18 56 journal journal PROPN iajs-2288 18 57 for for ADP iajs-2288 18 58 pure pure ADJ iajs-2288 18 59 and and CCONJ iajs-2288 18 60 applied apply VERB iajs-2288 18 61 science science NOUN iajs-2288 18 62 journal journal PROPN iajs-2288 18 63 homepage homepage NOUN iajs-2288 18 64 : : PUNCT iajs-2288 18 65 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2288 18 66 doi:10.30526/32.3.2288 doi:10.30526/32.3.2288 VERB iajs-2288 18 67 approximaitly approximaitly ADV iajs-2288 18 68 semi semi ADJ iajs-2288 18 69 - - ADJ iajs-2288 18 70 prime prime ADJ iajs-2288 18 71 submodules submodule NOUN iajs-2288 18 72 and and CCONJ iajs-2288 18 73 some some DET iajs-2288 18 74 related relate VERB iajs-2288 18 75 concepts concept NOUN iajs-2288 18 76 department department PROPN iajs-2288 18 77 of of ADP iajs-2288 18 78 mathematics mathematics PROPN iajs-2288 18 79 , , PUNCT iajs-2288 18 80 college college NOUN iajs-2288 18 81 of of ADP iajs-2288 18 82 computer computer NOUN iajs-2288 18 83 science science NOUN iajs-2288 18 84 and and CCONJ iajs-2288 18 85 mathematics mathematic NOUN iajs-2288 18 86 , , PUNCT iajs-2288 18 87 university university NOUN iajs-2288 18 88 of of ADP iajs-2288 18 89 tikrit tikrit NOUN iajs-2288 18 90 , , PUNCT iajs-2288 18 91 iraq iraq PROPN iajs-2288 18 92 . . PUNCT iajs-2288 19 1 ali.shebl@st.tu.edu.iq ali.shebl@st.tu.edu.iq PROPN iajs-2288 19 2 dr.mohammadali2013@gmail.com dr.mohammadali2013@gmail.com PROPN iajs-2288 19 3 article article NOUN iajs-2288 19 4 history history NOUN iajs-2288 19 5 : : PUNCT iajs-2288 19 6 received receive VERB iajs-2288 19 7 19 19 NUM iajs-2288 19 8 february february NOUN iajs-2288 19 9 2019 2019 NUM iajs-2288 19 10 , , PUNCT iajs-2288 19 11 accepted accept VERB iajs-2288 19 12 4 4 NUM iajs-2288 19 13 march march NOUN iajs-2288 19 14 2019,publish 2019,publish NUM iajs-2288 19 15 september september PROPN iajs-2288 19 16 2019 2019 NUM iajs-2288 19 17 . . PUNCT iajs-2288 20 1 mailto:ali.shebl@st.tu.edu.iq mailto:ali.shebl@st.tu.edu.iq PROPN iajs-2288 20 2 mailto:dr.mohammadali2013@gmail.com mailto:dr.mohammadali2013@gmail.com X iajs-2288 20 3 118 118 NUM iajs-2288 20 4 ibn ibn PROPN iajs-2288 20 5 al al PROPN iajs-2288 20 6 - - PUNCT iajs-2288 20 7 haitham haitham PROPN iajs-2288 20 8 jour jour X iajs-2288 20 9 . . PROPN iajs-2288 21 1 for for ADP iajs-2288 21 2 pure pure ADJ iajs-2288 21 3 & & CCONJ iajs-2288 21 4 appl appl PROPN iajs-2288 21 5 . . PUNCT iajs-2288 22 1 sci sci PROPN iajs-2288 22 2 . . PROPN iajs-2288 22 3 32 32 NUM iajs-2288 22 4 ( ( PUNCT iajs-2288 22 5 3 3 NUM iajs-2288 22 6 ) ) PUNCT iajs-2288 22 7 2019 2019 NUM iajs-2288 22 8 recently recently ADV iajs-2288 22 9 extensive extensive ADJ iajs-2288 22 10 research research NOUN iajs-2288 22 11 has have AUX iajs-2288 22 12 been be AUX iajs-2288 22 13 done do VERB iajs-2288 22 14 on on ADP iajs-2288 22 15 generalizations generalization NOUN iajs-2288 22 16 of of ADP iajs-2288 22 17 semi semi ADJ iajs-2288 22 18 - - ADJ iajs-2288 22 19 prime prime ADJ iajs-2288 22 20 submodules submodule NOUN iajs-2288 22 21 see see VERB iajs-2288 22 22 for for ADP iajs-2288 22 23 example example NOUN iajs-2288 23 1 [ [ X iajs-2288 23 2 9 9 NUM iajs-2288 23 3 , , PUNCT iajs-2288 23 4 10 10 NUM iajs-2288 23 5 ] ] PUNCT iajs-2288 23 6 . . PUNCT iajs-2288 24 1 the the DET iajs-2288 24 2 socle socle NOUN iajs-2288 24 3 of of ADP iajs-2288 24 4 an an DET iajs-2288 24 5 𝑅-module 𝑅-module PROPN iajs-2288 24 6 𝑇 𝑇 PROPN iajs-2288 24 7 ( ( PUNCT iajs-2288 24 8 for for ADP iajs-2288 24 9 short short ADJ iajs-2288 24 10 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 24 11 ) ) PUNCT iajs-2288 24 12 ) ) PUNCT iajs-2288 24 13 is be AUX iajs-2288 24 14 defined define VERB iajs-2288 24 15 by by ADP iajs-2288 24 16 the the DET iajs-2288 24 17 intersection intersection NOUN iajs-2288 24 18 of of ADP iajs-2288 24 19 all all DET iajs-2288 24 20 essential essential ADJ iajs-2288 24 21 submodules submodule NOUN iajs-2288 24 22 of of ADP iajs-2288 24 23 𝑇 𝑇 PROPN iajs-2288 25 1 [ [ X iajs-2288 25 2 11 11 NUM iajs-2288 25 3 ] ] PUNCT iajs-2288 25 4 . . PUNCT iajs-2288 26 1 where where SCONJ iajs-2288 26 2 a a DET iajs-2288 26 3 non non ADJ iajs-2288 26 4 - - ADJ iajs-2288 26 5 zero zero NUM iajs-2288 26 6 submodule submodule NOUN iajs-2288 26 7 𝑁 𝑁 PROPN iajs-2288 26 8 of of ADP iajs-2288 26 9 an an DET iajs-2288 26 10 𝑅module 𝑅module PROPN iajs-2288 26 11 𝑇 𝑇 PROPN iajs-2288 26 12 is be AUX iajs-2288 26 13 called call VERB iajs-2288 26 14 essential essential ADJ iajs-2288 26 15 if if SCONJ iajs-2288 26 16 𝑁 𝑁 PROPN iajs-2288 26 17 ∩ ∩ ADJ iajs-2288 26 18 𝐾 𝐾 PROPN iajs-2288 26 19 β‰  β‰  PROPN iajs-2288 26 20 ( ( PUNCT iajs-2288 26 21 0 0 NUM iajs-2288 26 22 ) ) PUNCT iajs-2288 26 23 for for ADP iajs-2288 26 24 all all DET iajs-2288 26 25 non non ADJ iajs-2288 26 26 - - ADJ iajs-2288 26 27 zero zero NUM iajs-2288 26 28 submodule submodule NOUN iajs-2288 26 29 𝐾 𝐾 PROPN iajs-2288 26 30 of of ADP iajs-2288 26 31 𝑇 𝑇 PROPN iajs-2288 26 32 [ [ X iajs-2288 26 33 12 12 NUM iajs-2288 26 34 ] ] PUNCT iajs-2288 26 35 . . PUNCT iajs-2288 27 1 recall recall VERB iajs-2288 27 2 that that SCONJ iajs-2288 27 3 an an DET iajs-2288 27 4 ideal ideal ADJ iajs-2288 27 5 𝐼 𝐼 NOUN iajs-2288 27 6 of of ADP iajs-2288 27 7 a a DET iajs-2288 27 8 ring ring NOUN iajs-2288 27 9 𝑅 𝑅 PROPN iajs-2288 27 10 is be AUX iajs-2288 27 11 a a DET iajs-2288 27 12 semi semi ADJ iajs-2288 27 13 - - ADJ iajs-2288 27 14 prime prime ADJ iajs-2288 27 15 ideal ideal NOUN iajs-2288 27 16 of of ADP iajs-2288 27 17 𝑅 𝑅 PROPN iajs-2288 27 18 if if SCONJ iajs-2288 27 19 π‘Ž2 π‘Ž2 PROPN iajs-2288 27 20 ∈ ∈ PROPN iajs-2288 27 21 𝐼 𝐼 PROPN iajs-2288 27 22 , , PUNCT iajs-2288 27 23 implies imply VERB iajs-2288 27 24 that that SCONJ iajs-2288 27 25 π‘Ž π‘Ž PROPN iajs-2288 27 26 ∈ ∈ PROPN iajs-2288 27 27 𝐼. 𝐼. NOUN iajs-2288 27 28 equivalent equivalent NOUN iajs-2288 27 29 𝐼 𝐼 PROPN iajs-2288 27 30 = = PUNCT iajs-2288 27 31 √𝐼 √𝐼 X iajs-2288 28 1 = = PUNCT iajs-2288 28 2 { { PUNCT iajs-2288 28 3 π‘Ž π‘Ž PROPN iajs-2288 28 4 ∈ ∈ PROPN iajs-2288 28 5 𝑅 𝑅 NOUN iajs-2288 28 6 : : PUNCT iajs-2288 28 7 π‘Žπ‘› π‘Žπ‘› PROPN iajs-2288 28 8 ∈ ∈ NOUN iajs-2288 28 9 𝐼 𝐼 ADP iajs-2288 28 10 for for ADP iajs-2288 28 11 some some PRON iajs-2288 28 12 𝑛 𝑛 PRON iajs-2288 28 13 ∈ ∈ PROPN iajs-2288 28 14 𝑍+ 𝑍+ NOUN iajs-2288 28 15 } } PUNCT iajs-2288 28 16 [ [ X iajs-2288 28 17 7 7 NUM iajs-2288 28 18 ] ] PUNCT iajs-2288 28 19 . . PUNCT iajs-2288 29 1 recall recall VERB iajs-2288 29 2 that that SCONJ iajs-2288 29 3 a a DET iajs-2288 29 4 proper proper ADJ iajs-2288 29 5 submodule submodule NOUN iajs-2288 29 6 𝐿 𝐿 PROPN iajs-2288 29 7 of of ADP iajs-2288 29 8 an an DET iajs-2288 29 9 𝑅module 𝑅module PROPN iajs-2288 29 10 𝑇 𝑇 PROPN iajs-2288 29 11 is be AUX iajs-2288 29 12 called call VERB iajs-2288 29 13 an an DET iajs-2288 29 14 approximaitly approximaitly ADV iajs-2288 29 15 prime prime ADJ iajs-2288 29 16 submodule submodule NOUN iajs-2288 29 17 of of ADP iajs-2288 29 18 𝑇 𝑇 PROPN iajs-2288 29 19 , , PUNCT iajs-2288 29 20 if if SCONJ iajs-2288 29 21 whenever whenever SCONJ iajs-2288 29 22 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 29 23 ∈ ∈ PROPN iajs-2288 29 24 𝐿 𝐿 PROPN iajs-2288 29 25 , , PUNCT iajs-2288 29 26 where where SCONJ iajs-2288 29 27 π‘Ž π‘Ž PROPN iajs-2288 29 28 ∈ ∈ PROPN iajs-2288 29 29 𝑅 𝑅 PROPN iajs-2288 29 30 , , PUNCT iajs-2288 29 31 𝑑 𝑑 PROPN iajs-2288 29 32 ∈ ∈ PROPN iajs-2288 29 33 𝑇 𝑇 PROPN iajs-2288 29 34 , , PUNCT iajs-2288 29 35 implies imply VERB iajs-2288 29 36 that that SCONJ iajs-2288 29 37 either either CCONJ iajs-2288 29 38 𝑑 𝑑 PROPN iajs-2288 29 39 ∈ ∈ PROPN iajs-2288 29 40 𝐿 𝐿 PROPN iajs-2288 29 41 + + NOUN iajs-2288 29 42 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 29 43 ) ) PUNCT iajs-2288 29 44 or or CCONJ iajs-2288 29 45 π‘Ž π‘Ž PRON iajs-2288 29 46 ∈ ∈ NOUN iajs-2288 29 47 [ [ X iajs-2288 29 48 𝐿 𝐿 PROPN iajs-2288 29 49 + + NOUN iajs-2288 29 50 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 29 51 ): ): PUNCT iajs-2288 29 52 𝑇][2 𝑇][2 PROPN iajs-2288 29 53 ] ] PUNCT iajs-2288 29 54 . . PUNCT iajs-2288 30 1 2 2 X iajs-2288 30 2 . . X iajs-2288 30 3 approximaitly approximaitly ADV iajs-2288 30 4 semi semi ADJ iajs-2288 30 5 - - ADJ iajs-2288 30 6 prime prime ADJ iajs-2288 30 7 submodules submodule NOUN iajs-2288 30 8 in in ADP iajs-2288 30 9 this this DET iajs-2288 30 10 section section NOUN iajs-2288 30 11 , , PUNCT iajs-2288 30 12 we we PRON iajs-2288 30 13 introduce introduce VERB iajs-2288 30 14 the the DET iajs-2288 30 15 definition definition NOUN iajs-2288 30 16 of of ADP iajs-2288 30 17 approximaitly approximaitly ADV iajs-2288 30 18 semi semi ADJ iajs-2288 30 19 - - ADJ iajs-2288 30 20 prime prime ADJ iajs-2288 30 21 submodule submodule NOUN iajs-2288 30 22 and and CCONJ iajs-2288 30 23 give give VERB iajs-2288 30 24 it it PRON iajs-2288 30 25 is be AUX iajs-2288 30 26 basic basic ADJ iajs-2288 30 27 properties property NOUN iajs-2288 30 28 , , PUNCT iajs-2288 30 29 examples example NOUN iajs-2288 30 30 and and CCONJ iajs-2288 30 31 characterizations characterization NOUN iajs-2288 30 32 . . PUNCT iajs-2288 31 1 definition definition NOUN iajs-2288 31 2 ( ( PUNCT iajs-2288 31 3 1 1 X iajs-2288 31 4 ) ) PUNCT iajs-2288 31 5 a a DET iajs-2288 31 6 proper proper ADJ iajs-2288 31 7 submodule submodule NOUN iajs-2288 31 8 𝐿 𝐿 PROPN iajs-2288 31 9 of of ADP iajs-2288 31 10 an an DET iajs-2288 31 11 𝑅-module 𝑅-module PROPN iajs-2288 31 12 𝑇 𝑇 PROPN iajs-2288 31 13 is be AUX iajs-2288 31 14 called call VERB iajs-2288 31 15 an an DET iajs-2288 31 16 approximaitly approximaitly ADV iajs-2288 31 17 semi semi ADJ iajs-2288 31 18 - - ADJ iajs-2288 31 19 prime prime ADJ iajs-2288 31 20 ( ( PUNCT iajs-2288 31 21 for for ADP iajs-2288 31 22 short short ADJ iajs-2288 31 23 app app ADJ iajs-2288 31 24 - - PUNCT iajs-2288 31 25 semi semi ADJ iajs-2288 31 26 - - ADJ iajs-2288 31 27 prime prime ADJ iajs-2288 31 28 ) ) PUNCT iajs-2288 31 29 submodule submodule NOUN iajs-2288 31 30 of of ADP iajs-2288 31 31 𝑇 𝑇 PROPN iajs-2288 31 32 , , PUNCT iajs-2288 31 33 if if SCONJ iajs-2288 31 34 whenever whenever SCONJ iajs-2288 31 35 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 31 36 ∈ ∈ PROPN iajs-2288 31 37 𝐿 𝐿 PROPN iajs-2288 31 38 , , PUNCT iajs-2288 31 39 where where SCONJ iajs-2288 31 40 π‘Ž π‘Ž PROPN iajs-2288 31 41 ∈ ∈ PROPN iajs-2288 31 42 𝑅 𝑅 PROPN iajs-2288 31 43 , , PUNCT iajs-2288 31 44 𝑑 𝑑 PROPN iajs-2288 31 45 ∈ ∈ PROPN iajs-2288 31 46 𝑇 𝑇 PROPN iajs-2288 31 47 and and CCONJ iajs-2288 31 48 𝑛 𝑛 DET iajs-2288 31 49 ∈ ∈ PROPN iajs-2288 31 50 𝑍+ 𝑍+ NOUN iajs-2288 31 51 , , PUNCT iajs-2288 31 52 implies imply VERB iajs-2288 31 53 that that SCONJ iajs-2288 32 1 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 32 2 ∈ ∈ PROPN iajs-2288 32 3 𝐿 𝐿 PROPN iajs-2288 32 4 + + NOUN iajs-2288 32 5 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 32 6 ) ) PUNCT iajs-2288 32 7 . . PUNCT iajs-2288 33 1 an an DET iajs-2288 33 2 ideal ideal ADJ iajs-2288 33 3 𝐼 𝐼 NOUN iajs-2288 33 4 of of ADP iajs-2288 33 5 a a DET iajs-2288 33 6 ring ring NOUN iajs-2288 33 7 𝑅 𝑅 PROPN iajs-2288 33 8 is be AUX iajs-2288 33 9 called call VERB iajs-2288 33 10 an an DET iajs-2288 33 11 approximaitly approximaitly ADV iajs-2288 33 12 semi semi ADJ iajs-2288 33 13 - - ADJ iajs-2288 33 14 prime prime ADJ iajs-2288 33 15 ideal ideal NOUN iajs-2288 33 16 of of ADP iajs-2288 33 17 𝑅 𝑅 PROPN iajs-2288 33 18 if if SCONJ iajs-2288 33 19 𝐼 𝐼 PROPN iajs-2288 33 20 is be AUX iajs-2288 33 21 an an DET iajs-2288 33 22 approximaitly approximaitly ADV iajs-2288 33 23 semi semi ADJ iajs-2288 33 24 - - ADJ iajs-2288 33 25 prime prime ADJ iajs-2288 33 26 submodule submodule NOUN iajs-2288 33 27 of of ADP iajs-2288 33 28 𝑅-module 𝑅-module PROPN iajs-2288 33 29 𝑅. 𝑅. NOUN iajs-2288 33 30 remarks remark NOUN iajs-2288 33 31 and and CCONJ iajs-2288 33 32 examples example NOUN iajs-2288 33 33 ( ( PUNCT iajs-2288 33 34 2 2 NUM iajs-2288 33 35 ) ) PUNCT iajs-2288 33 36 1 1 NUM iajs-2288 33 37 ) ) PUNCT iajs-2288 33 38 it it PRON iajs-2288 33 39 is be AUX iajs-2288 33 40 clear clear ADJ iajs-2288 33 41 that that SCONJ iajs-2288 33 42 every every DET iajs-2288 33 43 semi semi ADJ iajs-2288 33 44 - - ADJ iajs-2288 33 45 prime prime ADJ iajs-2288 33 46 submodule submodule NOUN iajs-2288 33 47 of of ADP iajs-2288 33 48 an an DET iajs-2288 33 49 𝑅-module 𝑅-module PROPN iajs-2288 33 50 𝑇 𝑇 PROPN iajs-2288 33 51 is be AUX iajs-2288 33 52 an an DET iajs-2288 33 53 app app ADJ iajs-2288 33 54 - - PUNCT iajs-2288 33 55 semi semi ADJ iajs-2288 33 56 - - ADJ iajs-2288 33 57 prime prime ADJ iajs-2288 33 58 submodule submodule NOUN iajs-2288 33 59 while while SCONJ iajs-2288 33 60 the the DET iajs-2288 33 61 convers conver NOUN iajs-2288 33 62 is be AUX iajs-2288 33 63 not not PART iajs-2288 33 64 true true ADJ iajs-2288 33 65 in in ADP iajs-2288 33 66 general general ADJ iajs-2288 33 67 as as ADP iajs-2288 33 68 the the DET iajs-2288 33 69 following follow VERB iajs-2288 33 70 example example NOUN iajs-2288 33 71 shows show VERB iajs-2288 33 72 that that PRON iajs-2288 33 73 . . PUNCT iajs-2288 34 1 consider consider VERB iajs-2288 34 2 the the DET iajs-2288 34 3 𝑍-module 𝑍-module PROPN iajs-2288 34 4 𝑍12 𝑍12 PROPN iajs-2288 34 5 and and CCONJ iajs-2288 34 6 𝐿 𝐿 PROPN iajs-2288 34 7 = = PROPN iajs-2288 34 8 〈 〈 PROPN iajs-2288 34 9 0Μ… 0Μ… PROPN iajs-2288 34 10 βŒͺ βŒͺ NOUN iajs-2288 34 11 be be VERB iajs-2288 34 12 a a DET iajs-2288 34 13 submodule submodule NOUN iajs-2288 34 14 of of ADP iajs-2288 34 15 𝑍12 𝑍12 PROPN iajs-2288 34 16 . . PUNCT iajs-2288 35 1 π‘ π‘œπ‘(𝑍12 π‘ π‘œπ‘(𝑍12 X iajs-2288 35 2 ) ) PUNCT iajs-2288 35 3 = = PRON iajs-2288 35 4 { { PUNCT iajs-2288 36 1 0Μ… 0Μ… PROPN iajs-2288 36 2 , , PUNCT iajs-2288 36 3 2Μ… 2Μ… NUM iajs-2288 36 4 , , PUNCT iajs-2288 36 5 4Μ… 4Μ… PROPN iajs-2288 36 6 , , PUNCT iajs-2288 36 7 6Μ… 6Μ… PROPN iajs-2288 36 8 , , PUNCT iajs-2288 36 9 8Μ… 8Μ… NUM iajs-2288 36 10 , , PUNCT iajs-2288 36 11 10Μ…Μ…Μ…Μ… 10Μ…Μ…Μ…Μ… NUM iajs-2288 36 12 } } PUNCT iajs-2288 36 13 . . PUNCT iajs-2288 37 1 𝐿 𝐿 PROPN iajs-2288 37 2 is be AUX iajs-2288 37 3 not not PART iajs-2288 37 4 semi semi ADJ iajs-2288 37 5 - - ADJ iajs-2288 37 6 prime prime ADJ iajs-2288 37 7 in in ADP iajs-2288 37 8 𝑍12 𝑍12 PROPN iajs-2288 37 9 because because SCONJ iajs-2288 37 10 22 22 NUM iajs-2288 37 11 . . PUNCT iajs-2288 38 1 3Μ… 3Μ… NUM iajs-2288 38 2 ∈ ∈ PROPN iajs-2288 38 3 𝐿 𝐿 PROPN iajs-2288 38 4 , , PUNCT iajs-2288 38 5 but but CCONJ iajs-2288 38 6 2 2 NUM iajs-2288 38 7 . . X iajs-2288 38 8 3Μ… 3Μ… NUM iajs-2288 38 9 = = SYM iajs-2288 38 10 6 6 NUM iajs-2288 38 11 βˆ‰ βˆ‰ ADV iajs-2288 38 12 𝐿. 𝐿. VERB iajs-2288 38 13 but but CCONJ iajs-2288 38 14 𝐿 𝐿 PROPN iajs-2288 38 15 is be AUX iajs-2288 38 16 an an DET iajs-2288 38 17 app app ADJ iajs-2288 38 18 - - PUNCT iajs-2288 38 19 semi semi NOUN iajs-2288 38 20 - - ADJ iajs-2288 38 21 prime prime ADJ iajs-2288 38 22 in in ADP iajs-2288 38 23 𝑍12 𝑍12 PROPN iajs-2288 38 24 since since SCONJ iajs-2288 38 25 whenever whenever SCONJ iajs-2288 38 26 π‘Ž2𝑑̅ π‘Ž2𝑑̅ PROPN iajs-2288 38 27 ∈ ∈ PROPN iajs-2288 38 28 𝐿 𝐿 PROPN iajs-2288 38 29 , , PUNCT iajs-2288 38 30 for for ADP iajs-2288 38 31 π‘Ž π‘Ž PROPN iajs-2288 38 32 ∈ ∈ PROPN iajs-2288 38 33 𝑅 𝑅 PROPN iajs-2288 38 34 , , PUNCT iajs-2288 38 35 𝑑̅ 𝑑̅ PROPN iajs-2288 38 36 ∈ ∈ PROPN iajs-2288 38 37 𝑍12 𝑍12 PROPN iajs-2288 38 38 , , PUNCT iajs-2288 38 39 implies imply VERB iajs-2288 38 40 that that SCONJ iajs-2288 38 41 π‘Žπ‘‘Μ… π‘Žπ‘‘Μ… VERB iajs-2288 38 42 ∈ ∈ PROPN iajs-2288 38 43 𝐿 𝐿 PROPN iajs-2288 38 44 + + CCONJ iajs-2288 38 45 π‘ π‘œπ‘(𝑍12 π‘ π‘œπ‘(𝑍12 PROPN iajs-2288 38 46 ) ) PUNCT iajs-2288 38 47 = = PRON iajs-2288 38 48 { { PUNCT iajs-2288 38 49 0Μ… 0Μ… PROPN iajs-2288 38 50 , , PUNCT iajs-2288 38 51 2Μ… 2Μ… NUM iajs-2288 38 52 , , PUNCT iajs-2288 38 53 4Μ… 4Μ… PROPN iajs-2288 38 54 , , PUNCT iajs-2288 38 55 6Μ… 6Μ… PROPN iajs-2288 38 56 , , PUNCT iajs-2288 38 57 8Μ… 8Μ… NUM iajs-2288 38 58 , , PUNCT iajs-2288 38 59 10Μ…Μ…Μ…Μ… 10Μ…Μ…Μ…Μ… NUM iajs-2288 38 60 } } PUNCT iajs-2288 38 61 . . PUNCT iajs-2288 39 1 2 2 X iajs-2288 39 2 ) ) PUNCT iajs-2288 39 3 it it PRON iajs-2288 39 4 is be AUX iajs-2288 39 5 clear clear ADJ iajs-2288 39 6 that that SCONJ iajs-2288 39 7 every every DET iajs-2288 39 8 prime prime ADJ iajs-2288 39 9 submodule submodule NOUN iajs-2288 39 10 of of ADP iajs-2288 39 11 an an DET iajs-2288 39 12 𝑅-module 𝑅-module PROPN iajs-2288 39 13 𝑇 𝑇 PROPN iajs-2288 39 14 is be AUX iajs-2288 39 15 an an DET iajs-2288 39 16 app app ADJ iajs-2288 39 17 - - PUNCT iajs-2288 39 18 semi semi ADJ iajs-2288 39 19 - - ADJ iajs-2288 39 20 prime prime ADJ iajs-2288 39 21 submodule submodule NOUN iajs-2288 39 22 while while SCONJ iajs-2288 39 23 the the DET iajs-2288 39 24 convers conver NOUN iajs-2288 39 25 is be AUX iajs-2288 39 26 not not PART iajs-2288 39 27 true true ADJ iajs-2288 39 28 in in ADP iajs-2288 39 29 general general ADJ iajs-2288 39 30 as as ADP iajs-2288 39 31 the the DET iajs-2288 39 32 following follow VERB iajs-2288 39 33 example example NOUN iajs-2288 39 34 shows show VERB iajs-2288 39 35 that that PRON iajs-2288 39 36 . . PUNCT iajs-2288 40 1 consider consider VERB iajs-2288 40 2 the the DET iajs-2288 40 3 𝑍-module 𝑍-module ADJ iajs-2288 40 4 𝑍4 𝑍4 NOUN iajs-2288 40 5 and and CCONJ iajs-2288 40 6 𝐿 𝐿 PROPN iajs-2288 40 7 = = PROPN iajs-2288 40 8 〈 〈 PROPN iajs-2288 40 9 0Μ… 0Μ… PROPN iajs-2288 40 10 βŒͺ βŒͺ NOUN iajs-2288 40 11 be be VERB iajs-2288 40 12 a a DET iajs-2288 40 13 submodule submodule NOUN iajs-2288 40 14 of of ADP iajs-2288 40 15 𝑍4 𝑍4 NOUN iajs-2288 40 16 . . PUNCT iajs-2288 41 1 𝐿 𝐿 NOUN iajs-2288 41 2 is be AUX iajs-2288 41 3 not not PART iajs-2288 41 4 prime prime ADJ iajs-2288 41 5 but but CCONJ iajs-2288 41 6 𝐿 𝐿 PROPN iajs-2288 41 7 is be AUX iajs-2288 41 8 an an DET iajs-2288 41 9 appsemi appsemi NOUN iajs-2288 41 10 - - NOUN iajs-2288 41 11 prime prime NOUN iajs-2288 41 12 in in ADP iajs-2288 41 13 𝑍4 𝑍4 NOUN iajs-2288 41 14 because because SCONJ iajs-2288 41 15 22 22 NUM iajs-2288 41 16 . . PUNCT iajs-2288 42 1 1Μ… 1Μ… NUM iajs-2288 42 2 ∈ ∈ PROPN iajs-2288 42 3 𝐿 𝐿 PROPN iajs-2288 42 4 but but CCONJ iajs-2288 42 5 2.1 2.1 NUM iajs-2288 42 6 βˆ‰ βˆ‰ PROPN iajs-2288 42 7 𝐿 𝐿 PROPN iajs-2288 42 8 , , PUNCT iajs-2288 42 9 while while SCONJ iajs-2288 42 10 22 22 NUM iajs-2288 42 11 . . PUNCT iajs-2288 43 1 1Μ… 1Μ… NUM iajs-2288 43 2 ∈ ∈ PROPN iajs-2288 43 3 𝐿 𝐿 PROPN iajs-2288 43 4 , , PUNCT iajs-2288 43 5 implies imply VERB iajs-2288 43 6 that that SCONJ iajs-2288 43 7 2 2 X iajs-2288 43 8 . . X iajs-2288 43 9 1Μ… 1Μ… NUM iajs-2288 43 10 = = SYM iajs-2288 43 11 2 2 NUM iajs-2288 43 12 ∈ ∈ PROPN iajs-2288 43 13 𝐿 𝐿 PROPN iajs-2288 43 14 + + NOUN iajs-2288 43 15 π‘ π‘œπ‘(𝑍4 π‘ π‘œπ‘(𝑍4 NUM iajs-2288 43 16 ) ) PUNCT iajs-2288 43 17 = = PUNCT iajs-2288 43 18 〈 〈 PROPN iajs-2288 43 19 0Μ… 0Μ… NUM iajs-2288 43 20 βŒͺ βŒͺ NOUN iajs-2288 43 21 + + CCONJ iajs-2288 43 22 { { PUNCT iajs-2288 43 23 0Μ… 0Μ… NOUN iajs-2288 43 24 , , PUNCT iajs-2288 43 25 2Μ… 2Μ… NOUN iajs-2288 43 26 } } PUNCT iajs-2288 43 27 = = PUNCT iajs-2288 43 28 { { PUNCT iajs-2288 43 29 0Μ… 0Μ… NOUN iajs-2288 43 30 , , PUNCT iajs-2288 43 31 2Μ… 2Μ… PROPN iajs-2288 43 32 } } PUNCT iajs-2288 43 33 . . PUNCT iajs-2288 44 1 that that PRON iajs-2288 44 2 is be AUX iajs-2288 44 3 for for ADP iajs-2288 44 4 all all PRON iajs-2288 44 5 π‘Ž π‘Ž PRON iajs-2288 44 6 ∈ ∈ PROPN iajs-2288 44 7 𝑍 𝑍 NOUN iajs-2288 44 8 , , PUNCT iajs-2288 44 9 𝑑̅ 𝑑̅ PROPN iajs-2288 44 10 ∈ ∈ PROPN iajs-2288 44 11 𝑍4 𝑍4 NOUN iajs-2288 44 12 with with ADP iajs-2288 44 13 π‘Ž2𝑑̅ π‘Ž2𝑑̅ PROPN iajs-2288 44 14 ∈ ∈ PROPN iajs-2288 44 15 𝐿 𝐿 PROPN iajs-2288 44 16 , , PUNCT iajs-2288 44 17 implies imply VERB iajs-2288 44 18 that that SCONJ iajs-2288 44 19 π‘Žπ‘‘Μ… π‘Žπ‘‘Μ… PROPN iajs-2288 44 20 ∈ ∈ PROPN iajs-2288 44 21 𝐿 𝐿 PROPN iajs-2288 44 22 + + NOUN iajs-2288 44 23 π‘ π‘œπ‘(𝑍4 π‘ π‘œπ‘(𝑍4 NUM iajs-2288 44 24 ) ) PUNCT iajs-2288 44 25 . . PUNCT iajs-2288 45 1 3 3 X iajs-2288 45 2 ) ) PUNCT iajs-2288 45 3 it it PRON iajs-2288 45 4 is be AUX iajs-2288 45 5 clear clear ADJ iajs-2288 45 6 that that SCONJ iajs-2288 45 7 every every DET iajs-2288 45 8 quasi quasi ADJ iajs-2288 45 9 - - ADJ iajs-2288 45 10 prime prime ADJ iajs-2288 45 11 submodule submodule NOUN iajs-2288 45 12 of of ADP iajs-2288 45 13 an an DET iajs-2288 45 14 𝑅-module 𝑅-module PROPN iajs-2288 45 15 𝑇 𝑇 PROPN iajs-2288 45 16 is be AUX iajs-2288 45 17 an an DET iajs-2288 45 18 app app ADJ iajs-2288 45 19 - - PUNCT iajs-2288 45 20 semi semi ADJ iajs-2288 45 21 - - ADJ iajs-2288 45 22 prime prime ADJ iajs-2288 45 23 submodule submodule NOUN iajs-2288 45 24 while while SCONJ iajs-2288 45 25 the the DET iajs-2288 45 26 convers conver NOUN iajs-2288 45 27 is be AUX iajs-2288 45 28 not not PART iajs-2288 45 29 true true ADJ iajs-2288 45 30 in in ADP iajs-2288 45 31 general general ADJ iajs-2288 45 32 as as ADP iajs-2288 45 33 an an DET iajs-2288 45 34 example example NOUN iajs-2288 45 35 shows show VERB iajs-2288 45 36 that that PRON iajs-2288 45 37 . . PUNCT iajs-2288 46 1 consider consider VERB iajs-2288 46 2 the the DET iajs-2288 46 3 𝑍-module 𝑍-module PROPN iajs-2288 46 4 𝑍24 𝑍24 PROPN iajs-2288 46 5 and and CCONJ iajs-2288 46 6 the the DET iajs-2288 46 7 submodule submodule NOUN iajs-2288 46 8 𝐿 𝐿 PROPN iajs-2288 46 9 = = PROPN iajs-2288 46 10 〈 〈 PROPN iajs-2288 46 11 6Μ… 6Μ… NOUN iajs-2288 46 12 βŒͺ βŒͺ NOUN iajs-2288 46 13 = = SYM iajs-2288 46 14 { { PUNCT iajs-2288 46 15 0Μ… 0Μ… PROPN iajs-2288 46 16 , , PUNCT iajs-2288 46 17 6Μ… 6Μ… PROPN iajs-2288 46 18 , , PUNCT iajs-2288 46 19 12Μ…Μ…Μ…Μ… 12Μ…Μ…Μ…Μ… PROPN iajs-2288 46 20 , , PUNCT iajs-2288 46 21 18Μ…Μ…Μ…Μ… 18Μ…Μ…Μ…Μ… NUM iajs-2288 46 22 } } PUNCT iajs-2288 46 23 . . PUNCT iajs-2288 47 1 𝐿 𝐿 PROPN iajs-2288 47 2 is be AUX iajs-2288 47 3 not not PART iajs-2288 47 4 quasi quasi ADJ iajs-2288 47 5 - - NOUN iajs-2288 47 6 prime prime ADJ iajs-2288 47 7 in in ADP iajs-2288 47 8 𝑍24 𝑍24 PROPN iajs-2288 47 9 because because SCONJ iajs-2288 47 10 2.3 2.3 NUM iajs-2288 47 11 . . PUNCT iajs-2288 48 1 1Μ… 1Μ… NUM iajs-2288 48 2 ∈ ∈ PROPN iajs-2288 48 3 𝐿 𝐿 PROPN iajs-2288 48 4 , , PUNCT iajs-2288 48 5 but but CCONJ iajs-2288 48 6 2 2 X iajs-2288 48 7 . . X iajs-2288 48 8 1Μ… 1Μ… NUM iajs-2288 48 9 βˆ‰ βˆ‰ PROPN iajs-2288 48 10 𝐿 𝐿 PROPN iajs-2288 48 11 and and CCONJ iajs-2288 48 12 3 3 NUM iajs-2288 48 13 . . PUNCT iajs-2288 49 1 1Μ… 1Μ… NUM iajs-2288 49 2 βˆ‰ βˆ‰ PROPN iajs-2288 49 3 𝐿. 𝐿. NOUN iajs-2288 49 4 but but CCONJ iajs-2288 49 5 𝐿 𝐿 PROPN iajs-2288 49 6 is be AUX iajs-2288 49 7 an an DET iajs-2288 49 8 app app ADJ iajs-2288 49 9 - - PUNCT iajs-2288 49 10 semi semi NOUN iajs-2288 49 11 - - ADJ iajs-2288 49 12 prime prime ADJ iajs-2288 49 13 in in ADP iajs-2288 49 14 𝑍24 𝑍24 PROPN iajs-2288 49 15 since since SCONJ iajs-2288 49 16 whenever whenever SCONJ iajs-2288 49 17 π‘Ž2𝑑̅ π‘Ž2𝑑̅ PROPN iajs-2288 49 18 ∈ ∈ PROPN iajs-2288 49 19 𝐿 𝐿 PROPN iajs-2288 49 20 for for ADP iajs-2288 49 21 π‘Ž π‘Ž PROPN iajs-2288 49 22 ∈ ∈ PROPN iajs-2288 49 23 𝑅 𝑅 PROPN iajs-2288 49 24 , , PUNCT iajs-2288 49 25 𝑑̅ 𝑑̅ PROPN iajs-2288 49 26 ∈ ∈ PROPN iajs-2288 49 27 𝑍24 𝑍24 PROPN iajs-2288 49 28 , , PUNCT iajs-2288 49 29 implies imply VERB iajs-2288 49 30 that that SCONJ iajs-2288 49 31 π‘Žπ‘‘Μ… π‘Žπ‘‘Μ… VERB iajs-2288 49 32 ∈ ∈ PROPN iajs-2288 49 33 𝐿 𝐿 PROPN iajs-2288 49 34 + + CCONJ iajs-2288 49 35 π‘ π‘œπ‘(𝑍24 π‘ π‘œπ‘(𝑍24 PROPN iajs-2288 49 36 ) ) PUNCT iajs-2288 49 37 = = SYM iajs-2288 49 38 { { PUNCT iajs-2288 49 39 0Μ… 0Μ… PROPN iajs-2288 49 40 , , PUNCT iajs-2288 49 41 6Μ… 6Μ… PROPN iajs-2288 49 42 , , PUNCT iajs-2288 49 43 12Μ…Μ…Μ…Μ… 12Μ…Μ…Μ…Μ… PROPN iajs-2288 49 44 , , PUNCT iajs-2288 49 45 18Μ…Μ…Μ…Μ… 18Μ…Μ…Μ…Μ… PROPN iajs-2288 49 46 } } PUNCT iajs-2288 49 47 + + CCONJ iajs-2288 49 48 { { PUNCT iajs-2288 49 49 0Μ… 0Μ… NOUN iajs-2288 49 50 , , PUNCT iajs-2288 49 51 2Μ… 2Μ… NUM iajs-2288 49 52 , , PUNCT iajs-2288 49 53 4Μ… 4Μ… PROPN iajs-2288 49 54 , , PUNCT iajs-2288 49 55 6Μ… 6Μ… PROPN iajs-2288 49 56 , , PUNCT iajs-2288 49 57 8Μ… 8Μ… NUM iajs-2288 49 58 , , PUNCT iajs-2288 49 59 10Μ…Μ…Μ…Μ… 10Μ…Μ…Μ…Μ… NUM iajs-2288 49 60 , , PUNCT iajs-2288 49 61 12Μ…Μ…Μ…Μ… 12Μ…Μ…Μ…Μ… PROPN iajs-2288 49 62 , , PUNCT iajs-2288 49 63 14Μ…Μ…Μ…Μ… 14Μ…Μ…Μ…Μ… PROPN iajs-2288 49 64 , , PUNCT iajs-2288 49 65 16Μ…Μ…Μ…Μ… 16Μ…Μ…Μ…Μ… NUM iajs-2288 49 66 , , PUNCT iajs-2288 49 67 18Μ…Μ…Μ…Μ… 18Μ…Μ…Μ…Μ… NUM iajs-2288 49 68 , , PUNCT iajs-2288 49 69 20Μ…Μ…Μ…Μ… 20Μ…Μ…Μ…Μ… PROPN iajs-2288 49 70 , , PUNCT iajs-2288 49 71 22Μ…Μ…Μ…Μ… 22Μ…Μ…Μ…Μ… PROPN iajs-2288 49 72 } } PUNCT iajs-2288 49 73 = = SYM iajs-2288 49 74 { { PUNCT iajs-2288 49 75 0Μ… 0Μ… PROPN iajs-2288 49 76 , , PUNCT iajs-2288 49 77 2Μ… 2Μ… NUM iajs-2288 49 78 , , PUNCT iajs-2288 49 79 4Μ… 4Μ… PROPN iajs-2288 49 80 , , PUNCT iajs-2288 49 81 6Μ… 6Μ… PROPN iajs-2288 49 82 , , PUNCT iajs-2288 49 83 8Μ… 8Μ… NUM iajs-2288 49 84 , , PUNCT iajs-2288 49 85 10Μ…Μ…Μ…Μ… 10Μ…Μ…Μ…Μ… NUM iajs-2288 49 86 , , PUNCT iajs-2288 49 87 12Μ…Μ…Μ…Μ… 12Μ…Μ…Μ…Μ… PROPN iajs-2288 49 88 , , PUNCT iajs-2288 49 89 14Μ…Μ…Μ…Μ… 14Μ…Μ…Μ…Μ… PROPN iajs-2288 49 90 , , PUNCT iajs-2288 49 91 16Μ…Μ…Μ…Μ… 16Μ…Μ…Μ…Μ… NUM iajs-2288 49 92 , , PUNCT iajs-2288 49 93 18Μ…Μ…Μ…Μ… 18Μ…Μ…Μ…Μ… NUM iajs-2288 49 94 , , PUNCT iajs-2288 49 95 20Μ…Μ…Μ…Μ… 20Μ…Μ…Μ…Μ… PROPN iajs-2288 49 96 , , PUNCT iajs-2288 49 97 22Μ…Μ…Μ…Μ… 22Μ…Μ…Μ…Μ… PROPN iajs-2288 49 98 } } PUNCT iajs-2288 49 99 . . PUNCT iajs-2288 50 1 4 4 X iajs-2288 50 2 ) ) PUNCT iajs-2288 50 3 it it PRON iajs-2288 50 4 is be AUX iajs-2288 50 5 clear clear ADJ iajs-2288 50 6 that that SCONJ iajs-2288 50 7 every every DET iajs-2288 50 8 approximaitly approximaitly ADV iajs-2288 50 9 prime prime ADJ iajs-2288 50 10 submodule submodule NOUN iajs-2288 50 11 of of ADP iajs-2288 50 12 an an DET iajs-2288 50 13 𝑅-module 𝑅-module PROPN iajs-2288 50 14 𝑇 𝑇 PROPN iajs-2288 50 15 is be AUX iajs-2288 50 16 an an DET iajs-2288 50 17 app app ADJ iajs-2288 50 18 - - PUNCT iajs-2288 50 19 semiprime semiprime NOUN iajs-2288 50 20 submodule submodule NOUN iajs-2288 50 21 , , PUNCT iajs-2288 50 22 but but CCONJ iajs-2288 50 23 the the DET iajs-2288 50 24 convers conver NOUN iajs-2288 50 25 is be AUX iajs-2288 50 26 not not PART iajs-2288 50 27 true true ADJ iajs-2288 50 28 . . PUNCT iajs-2288 51 1 the the DET iajs-2288 51 2 following follow VERB iajs-2288 51 3 example example NOUN iajs-2288 51 4 shows show VERB iajs-2288 51 5 that that PRON iajs-2288 51 6 . . PUNCT iajs-2288 52 1 the the DET iajs-2288 52 2 submodule submodule PROPN iajs-2288 52 3 𝐿 𝐿 PROPN iajs-2288 52 4 = = PROPN iajs-2288 52 5 6𝑍 6𝑍 NOUN iajs-2288 52 6 of of ADP iajs-2288 52 7 a a DET iajs-2288 52 8 𝑍-module 𝑍-module PROPN iajs-2288 52 9 𝑍 𝑍 PROPN iajs-2288 52 10 is be AUX iajs-2288 52 11 an an DET iajs-2288 52 12 app app ADJ iajs-2288 52 13 - - PUNCT iajs-2288 52 14 semi semi ADJ iajs-2288 52 15 - - ADJ iajs-2288 52 16 prime prime ADJ iajs-2288 52 17 submodule submodule NOUN iajs-2288 52 18 of of ADP iajs-2288 52 19 𝑍 𝑍 PROPN iajs-2288 52 20 ( ( PUNCT iajs-2288 52 21 because because SCONJ iajs-2288 52 22 𝐿 𝐿 PROPN iajs-2288 52 23 is be AUX iajs-2288 52 24 a a DET iajs-2288 52 25 semi semi ADJ iajs-2288 52 26 - - ADJ iajs-2288 52 27 prime prime ADJ iajs-2288 52 28 submodule submodule NOUN iajs-2288 52 29 of of ADP iajs-2288 52 30 𝑍 𝑍 PROPN iajs-2288 52 31 ) ) PUNCT iajs-2288 52 32 , , PUNCT iajs-2288 52 33 but but CCONJ iajs-2288 52 34 𝐿 𝐿 PROPN iajs-2288 52 35 is be AUX iajs-2288 52 36 not not PART iajs-2288 52 37 an an DET iajs-2288 52 38 approximaitly approximaitly ADV iajs-2288 52 39 prime prime ADJ iajs-2288 52 40 submodule submodule NOUN iajs-2288 52 41 of of ADP iajs-2288 52 42 𝑍 𝑍 PROPN iajs-2288 52 43 because because SCONJ iajs-2288 52 44 , , PUNCT iajs-2288 52 45 if if SCONJ iajs-2288 52 46 2,3 2,3 NUM iajs-2288 52 47 ∈ ∈ NOUN iajs-2288 52 48 𝑍 𝑍 VERB iajs-2288 52 49 such such ADJ iajs-2288 52 50 that that SCONJ iajs-2288 52 51 2.3 2.3 NUM iajs-2288 52 52 ∈ ∈ NOUN iajs-2288 52 53 6𝑍 6𝑍 NOUN iajs-2288 52 54 , , PUNCT iajs-2288 52 55 but but CCONJ iajs-2288 52 56 3 3 NUM iajs-2288 52 57 βˆ‰ βˆ‰ NOUN iajs-2288 52 58 6𝑍 6𝑍 NOUN iajs-2288 52 59 + + CCONJ iajs-2288 52 60 π‘ π‘œπ‘(𝑍 π‘ π‘œπ‘(𝑍 NUM iajs-2288 52 61 ) ) PUNCT iajs-2288 53 1 = = NOUN iajs-2288 53 2 6𝑍 6𝑍 NOUN iajs-2288 53 3 and and CCONJ iajs-2288 53 4 2 2 NUM iajs-2288 53 5 βˆ‰ βˆ‰ NOUN iajs-2288 53 6 [ [ X iajs-2288 53 7 6𝑍 6𝑍 NOUN iajs-2288 53 8 + + NOUN iajs-2288 53 9 π‘ π‘œπ‘(𝑍 π‘ π‘œπ‘(𝑍 NUM iajs-2288 53 10 ): ): PUNCT iajs-2288 53 11 𝑍 𝑍 PROPN iajs-2288 53 12 ] ] PUNCT iajs-2288 53 13 = = SYM iajs-2288 53 14 6𝑍 6𝑍 NOUN iajs-2288 53 15 , , PUNCT iajs-2288 53 16 because because SCONJ iajs-2288 53 17 π‘ π‘œπ‘(𝑍 π‘ π‘œπ‘(𝑍 NUM iajs-2288 53 18 ) ) PUNCT iajs-2288 53 19 = = SYM iajs-2288 53 20 ( ( PUNCT iajs-2288 53 21 0 0 NUM iajs-2288 53 22 ) ) PUNCT iajs-2288 53 23 . . PUNCT iajs-2288 54 1 the the DET iajs-2288 54 2 following follow VERB iajs-2288 54 3 results result NOUN iajs-2288 54 4 are be AUX iajs-2288 54 5 characterizations characterization NOUN iajs-2288 54 6 of of ADP iajs-2288 54 7 app app NOUN iajs-2288 54 8 - - PUNCT iajs-2288 54 9 semi semi ADJ iajs-2288 54 10 - - ADJ iajs-2288 54 11 prime prime ADJ iajs-2288 54 12 submodules submodule NOUN iajs-2288 54 13 . . PUNCT iajs-2288 55 1 119 119 NUM iajs-2288 55 2 ibn ibn PROPN iajs-2288 55 3 al al PROPN iajs-2288 55 4 - - PUNCT iajs-2288 55 5 haitham haitham PROPN iajs-2288 55 6 jour jour X iajs-2288 55 7 . . PROPN iajs-2288 55 8 for for ADP iajs-2288 55 9 pure pure ADJ iajs-2288 55 10 & & CCONJ iajs-2288 55 11 appl appl PROPN iajs-2288 55 12 . . PUNCT iajs-2288 56 1 sci sci PROPN iajs-2288 56 2 . . PROPN iajs-2288 56 3 32 32 NUM iajs-2288 56 4 ( ( PUNCT iajs-2288 56 5 3 3 NUM iajs-2288 56 6 ) ) SYM iajs-2288 56 7 2019 2019 NUM iajs-2288 56 8 proposition proposition NOUN iajs-2288 56 9 ( ( PUNCT iajs-2288 56 10 3 3 X iajs-2288 56 11 ) ) PUNCT iajs-2288 56 12 let let VERB iajs-2288 56 13 𝐿 𝐿 PROPN iajs-2288 56 14 be be AUX iajs-2288 56 15 a a DET iajs-2288 56 16 proper proper ADJ iajs-2288 56 17 submodule submodule NOUN iajs-2288 56 18 of of ADP iajs-2288 56 19 an an DET iajs-2288 56 20 𝑅-module 𝑅-module PROPN iajs-2288 56 21 𝑇. 𝑇. PROPN iajs-2288 56 22 then then ADV iajs-2288 56 23 𝐿 𝐿 PROPN iajs-2288 56 24 is be AUX iajs-2288 56 25 an an DET iajs-2288 56 26 app app ADJ iajs-2288 56 27 - - PUNCT iajs-2288 56 28 semi semi ADJ iajs-2288 56 29 - - ADJ iajs-2288 56 30 prime prime ADJ iajs-2288 56 31 submodule submodule NOUN iajs-2288 56 32 of of ADP iajs-2288 56 33 𝑇 𝑇 PROPN iajs-2288 57 1 if if SCONJ iajs-2288 58 1 and and CCONJ iajs-2288 58 2 only only ADV iajs-2288 58 3 if if SCONJ iajs-2288 58 4 𝐽𝑛𝐸 𝐽𝑛𝐸 NOUN iajs-2288 58 5 βŠ† βŠ† NUM iajs-2288 58 6 𝐿 𝐿 PROPN iajs-2288 58 7 where where SCONJ iajs-2288 58 8 𝐽 𝐽 PROPN iajs-2288 58 9 is be AUX iajs-2288 58 10 an an DET iajs-2288 58 11 ideal ideal NOUN iajs-2288 58 12 of of ADP iajs-2288 58 13 𝑅 𝑅 PROPN iajs-2288 58 14 , , PUNCT iajs-2288 58 15 𝐸 𝐸 PROPN iajs-2288 58 16 is be AUX iajs-2288 58 17 a a DET iajs-2288 58 18 submodule submodule NOUN iajs-2288 58 19 of of ADP iajs-2288 58 20 𝑇 𝑇 PROPN iajs-2288 58 21 and and CCONJ iajs-2288 58 22 𝑛 𝑛 DET iajs-2288 58 23 ∈ ∈ PROPN iajs-2288 58 24 𝑍+ 𝑍+ NOUN iajs-2288 58 25 , , PUNCT iajs-2288 58 26 implies imply VERB iajs-2288 58 27 that that SCONJ iajs-2288 58 28 𝐽𝐸 𝐽𝐸 VERB iajs-2288 58 29 βŠ† βŠ† NUM iajs-2288 58 30 𝐿 𝐿 PROPN iajs-2288 58 31 + + NOUN iajs-2288 58 32 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 58 33 ) ) PUNCT iajs-2288 58 34 . . PUNCT iajs-2288 59 1 proof proof NOUN iajs-2288 59 2 ( ( PUNCT iajs-2288 59 3 β‡’ β‡’ PROPN iajs-2288 59 4 ) ) PUNCT iajs-2288 59 5 suppose suppose VERB iajs-2288 59 6 that that SCONJ iajs-2288 59 7 𝐽𝑛𝐸 𝐽𝑛𝐸 NOUN iajs-2288 59 8 βŠ† βŠ† NUM iajs-2288 59 9 𝐿 𝐿 PROPN iajs-2288 59 10 , , PUNCT iajs-2288 59 11 where where SCONJ iajs-2288 59 12 𝐽 𝐽 PROPN iajs-2288 59 13 is be AUX iajs-2288 59 14 an an DET iajs-2288 59 15 ideal ideal NOUN iajs-2288 59 16 of of ADP iajs-2288 59 17 𝑅 𝑅 PROPN iajs-2288 59 18 , , PUNCT iajs-2288 59 19 𝐸 𝐸 PROPN iajs-2288 59 20 is be AUX iajs-2288 59 21 a a DET iajs-2288 59 22 submodule submodule NOUN iajs-2288 59 23 of of ADP iajs-2288 59 24 𝑇 𝑇 PROPN iajs-2288 59 25 and and CCONJ iajs-2288 59 26 𝑛 𝑛 DET iajs-2288 59 27 ∈ ∈ PROPN iajs-2288 59 28 𝑍+ 𝑍+ NOUN iajs-2288 59 29 . . PUNCT iajs-2288 60 1 now now ADV iajs-2288 60 2 , , PUNCT iajs-2288 60 3 let let VERB iajs-2288 60 4 𝑑 𝑑 PRON iajs-2288 60 5 ∈ ∈ PROPN iajs-2288 60 6 𝐽𝐸 𝐽𝐸 NOUN iajs-2288 60 7 , , PUNCT iajs-2288 60 8 then then ADV iajs-2288 60 9 𝑑 𝑑 X iajs-2288 60 10 = = VERB iajs-2288 60 11 π‘Ž1𝑑1 π‘Ž1𝑑1 PUNCT iajs-2288 60 12 + + ADJ iajs-2288 60 13 π‘Ž2𝑑2 π‘Ž2𝑑2 X iajs-2288 60 14 + + ADJ iajs-2288 60 15 β‹―+ β‹―+ NOUN iajs-2288 60 16 π‘Žπ‘›π‘‘π‘› π‘Žπ‘›π‘‘π‘› VERB iajs-2288 60 17 , , PUNCT iajs-2288 60 18 where where SCONJ iajs-2288 60 19 π‘Žπ‘– π‘Žπ‘– ADP iajs-2288 60 20 ∈ ∈ PROPN iajs-2288 60 21 𝐽 𝐽 PROPN iajs-2288 60 22 , , PUNCT iajs-2288 60 23 𝑑𝑖 𝑑𝑖 PROPN iajs-2288 60 24 ∈ ∈ PROPN iajs-2288 60 25 𝐸 𝐸 PROPN iajs-2288 60 26 , , PUNCT iajs-2288 60 27 𝑖 𝑖 NOUN iajs-2288 60 28 = = SYM iajs-2288 60 29 1,2 1,2 NUM iajs-2288 60 30 , , PUNCT iajs-2288 60 31 … … PUNCT iajs-2288 60 32 , , PUNCT iajs-2288 60 33 𝑛 𝑛 PROPN iajs-2288 60 34 , , PUNCT iajs-2288 60 35 that that PRON iajs-2288 60 36 is be AUX iajs-2288 60 37 π‘Žπ‘–π‘‘π‘– π‘Žπ‘–π‘‘π‘– NOUN iajs-2288 60 38 ∈ ∈ PROPN iajs-2288 60 39 𝐽𝐸 𝐽𝐸 NOUN iajs-2288 60 40 for for ADP iajs-2288 60 41 each each PRON iajs-2288 60 42 𝑖 𝑖 NOUN iajs-2288 60 43 = = SYM iajs-2288 60 44 1,2 1,2 NUM iajs-2288 60 45 , , PUNCT iajs-2288 60 46 … … PUNCT iajs-2288 60 47 , , PUNCT iajs-2288 60 48 𝑛 𝑛 X iajs-2288 60 49 , , PUNCT iajs-2288 60 50 it it PRON iajs-2288 60 51 follows follow VERB iajs-2288 60 52 that that SCONJ iajs-2288 60 53 π‘Žπ‘›π‘–π‘‘π‘– π‘Žπ‘›π‘–π‘‘π‘– NOUN iajs-2288 60 54 ∈ ∈ PROPN iajs-2288 60 55 𝐽𝑛𝐸 𝐽𝑛𝐸 NOUN iajs-2288 60 56 βŠ† βŠ† NUM iajs-2288 60 57 𝐿 𝐿 PROPN iajs-2288 60 58 , , PUNCT iajs-2288 60 59 that that PRON iajs-2288 60 60 is is ADV iajs-2288 60 61 π‘Žπ‘›π‘–π‘‘π‘– π‘Žπ‘›π‘–π‘‘π‘– NOUN iajs-2288 60 62 ∈ ∈ NOUN iajs-2288 60 63 𝐿. 𝐿. NOUN iajs-2288 60 64 but but CCONJ iajs-2288 60 65 𝐿 𝐿 PROPN iajs-2288 60 66 is be AUX iajs-2288 60 67 an an DET iajs-2288 60 68 app app ADJ iajs-2288 60 69 - - PUNCT iajs-2288 60 70 semi semi ADJ iajs-2288 60 71 - - ADJ iajs-2288 60 72 prime prime ADJ iajs-2288 60 73 submodule submodule NOUN iajs-2288 60 74 of of ADP iajs-2288 60 75 𝑇 𝑇 PROPN iajs-2288 60 76 , , PUNCT iajs-2288 60 77 implies imply VERB iajs-2288 60 78 that that SCONJ iajs-2288 60 79 π‘Žπ‘–π‘‘π‘– π‘Žπ‘–π‘‘π‘– PROPN iajs-2288 60 80 ∈ ∈ PROPN iajs-2288 60 81 𝐿 𝐿 PROPN iajs-2288 60 82 + + NOUN iajs-2288 60 83 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 60 84 ) ) PUNCT iajs-2288 60 85 for for ADP iajs-2288 60 86 each each PRON iajs-2288 60 87 𝑖 𝑖 NOUN iajs-2288 60 88 = = SYM iajs-2288 60 89 1,2 1,2 NUM iajs-2288 60 90 , , PUNCT iajs-2288 60 91 … … PUNCT iajs-2288 60 92 , , PUNCT iajs-2288 60 93 𝑛 𝑛 NOUN iajs-2288 60 94 , , PUNCT iajs-2288 60 95 hence hence ADV iajs-2288 60 96 𝑑 𝑑 PROPN iajs-2288 60 97 ∈ ∈ PROPN iajs-2288 60 98 𝐿 𝐿 PROPN iajs-2288 60 99 + + NOUN iajs-2288 60 100 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 60 101 ) ) PUNCT iajs-2288 60 102 , , PUNCT iajs-2288 60 103 it it PRON iajs-2288 60 104 follows follow VERB iajs-2288 60 105 that that SCONJ iajs-2288 60 106 𝐽𝐸 𝐽𝐸 VERB iajs-2288 60 107 βŠ† βŠ† NUM iajs-2288 60 108 𝐿 𝐿 PROPN iajs-2288 60 109 + + NOUN iajs-2288 60 110 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 60 111 ) ) PUNCT iajs-2288 60 112 . . PUNCT iajs-2288 61 1 ( ( PUNCT iajs-2288 61 2 ⇐ ⇐ PROPN iajs-2288 61 3 ) ) PUNCT iajs-2288 61 4 let let VERB iajs-2288 61 5 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN iajs-2288 61 6 ∈ ∈ PROPN iajs-2288 61 7 𝐿 𝐿 PROPN iajs-2288 61 8 , , PUNCT iajs-2288 61 9 where where SCONJ iajs-2288 61 10 π‘Ž π‘Ž PROPN iajs-2288 61 11 ∈ ∈ PROPN iajs-2288 61 12 𝑅 𝑅 PROPN iajs-2288 61 13 , , PUNCT iajs-2288 61 14 𝑑 𝑑 PROPN iajs-2288 61 15 ∈ ∈ PROPN iajs-2288 61 16 𝑇 𝑇 PROPN iajs-2288 61 17 and and CCONJ iajs-2288 61 18 𝑛 𝑛 DET iajs-2288 61 19 ∈ ∈ PROPN iajs-2288 61 20 𝑍+ 𝑍+ NOUN iajs-2288 61 21 , , PUNCT iajs-2288 61 22 implies imply VERB iajs-2288 61 23 that that SCONJ iajs-2288 61 24 〈 〈 PROPN iajs-2288 61 25 π‘Žπ‘›βŒͺ𝑅𝑑 π‘Žπ‘›βŒͺ𝑅𝑑 PUNCT iajs-2288 61 26 βŠ† βŠ† NUM iajs-2288 61 27 𝐿 𝐿 PROPN iajs-2288 61 28 , , PUNCT iajs-2288 61 29 that that PRON iajs-2288 61 30 is be AUX iajs-2288 61 31 〈 〈 PROPN iajs-2288 61 32 π‘ŽβŒͺ𝑛𝑅𝑑 π‘ŽβŒͺ𝑛𝑅𝑑 PROPN iajs-2288 61 33 βŠ† βŠ† NUM iajs-2288 61 34 𝐿. 𝐿. VERB iajs-2288 61 35 thus thus ADV iajs-2288 61 36 by by ADP iajs-2288 61 37 hypothesis hypothesis NOUN iajs-2288 61 38 we we PRON iajs-2288 61 39 haveβŒ©π‘ŽβŒͺ𝑅𝑑 haveβŒ©π‘ŽβŒͺ𝑅𝑑 VERB iajs-2288 61 40 βŠ† βŠ† NUM iajs-2288 61 41 𝐿 𝐿 PROPN iajs-2288 61 42 + + NOUN iajs-2288 61 43 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 61 44 ) ) PUNCT iajs-2288 61 45 , , PUNCT iajs-2288 61 46 implies imply VERB iajs-2288 61 47 that that SCONJ iajs-2288 61 48 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 61 49 ∈ ∈ PROPN iajs-2288 62 1 〈 〈 PRON iajs-2288 62 2 π‘ŽβŒͺ𝑅𝑑 π‘ŽβŒͺ𝑅𝑑 PROPN iajs-2288 62 3 βŠ† βŠ† NUM iajs-2288 62 4 𝐿 𝐿 PROPN iajs-2288 62 5 + + NOUN iajs-2288 62 6 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 62 7 ) ) PUNCT iajs-2288 62 8 , , PUNCT iajs-2288 62 9 thus thus ADV iajs-2288 62 10 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 62 11 ∈ ∈ PROPN iajs-2288 62 12 𝐿 𝐿 PROPN iajs-2288 62 13 + + NOUN iajs-2288 62 14 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 62 15 ) ) PUNCT iajs-2288 62 16 . . PUNCT iajs-2288 63 1 therefore therefore ADV iajs-2288 63 2 𝐿 𝐿 PROPN iajs-2288 63 3 is be AUX iajs-2288 63 4 an an DET iajs-2288 63 5 app app ADJ iajs-2288 63 6 - - PUNCT iajs-2288 63 7 semi semi ADJ iajs-2288 63 8 - - ADJ iajs-2288 63 9 prime prime ADJ iajs-2288 63 10 submodule submodule NOUN iajs-2288 63 11 of of ADP iajs-2288 63 12 𝑇. 𝑇. PROPN iajs-2288 63 13 corollary corollary NOUN iajs-2288 63 14 ( ( PUNCT iajs-2288 63 15 4 4 NUM iajs-2288 63 16 ) ) PUNCT iajs-2288 63 17 let let VERB iajs-2288 63 18 𝐿 𝐿 PROPN iajs-2288 63 19 be be AUX iajs-2288 63 20 a a DET iajs-2288 63 21 proper proper ADJ iajs-2288 63 22 submodule submodule NOUN iajs-2288 63 23 of of ADP iajs-2288 63 24 an an DET iajs-2288 63 25 𝑅-module 𝑅-module PROPN iajs-2288 63 26 𝑇. 𝑇. PROPN iajs-2288 63 27 then then ADV iajs-2288 63 28 𝐿 𝐿 PROPN iajs-2288 63 29 is be AUX iajs-2288 63 30 an an DET iajs-2288 63 31 app app ADJ iajs-2288 63 32 - - PUNCT iajs-2288 63 33 semi semi ADJ iajs-2288 63 34 - - ADJ iajs-2288 63 35 prime prime ADJ iajs-2288 63 36 submodule submodule NOUN iajs-2288 63 37 of of ADP iajs-2288 63 38 𝑇 𝑇 PROPN iajs-2288 63 39 if if SCONJ iajs-2288 63 40 and and CCONJ iajs-2288 63 41 only only ADV iajs-2288 63 42 if if SCONJ iajs-2288 63 43 𝐽𝑛𝑇 𝐽𝑛𝑇 PROPN iajs-2288 63 44 βŠ† βŠ† NUM iajs-2288 63 45 𝐿 𝐿 PROPN iajs-2288 63 46 where where SCONJ iajs-2288 63 47 𝐽 𝐽 PROPN iajs-2288 63 48 is be AUX iajs-2288 63 49 an an DET iajs-2288 63 50 ideal ideal NOUN iajs-2288 63 51 of of ADP iajs-2288 63 52 𝑅 𝑅 PROPN iajs-2288 63 53 , , PUNCT iajs-2288 63 54 𝑛 𝑛 PRON iajs-2288 63 55 ∈ ∈ PROPN iajs-2288 63 56 𝑍+ 𝑍+ NOUN iajs-2288 63 57 , , PUNCT iajs-2288 63 58 implies imply VERB iajs-2288 63 59 that that SCONJ iajs-2288 63 60 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 63 61 βŠ† βŠ† NUM iajs-2288 63 62 𝐿 𝐿 PROPN iajs-2288 63 63 + + NOUN iajs-2288 63 64 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 63 65 ) ) PUNCT iajs-2288 63 66 . . PUNCT iajs-2288 64 1 proof proof NOUN iajs-2288 64 2 it it PRON iajs-2288 64 3 follows follow VERB iajs-2288 64 4 by by ADP iajs-2288 64 5 proposition proposition NOUN iajs-2288 64 6 ( ( PUNCT iajs-2288 64 7 2.3 2.3 NUM iajs-2288 64 8 ) ) PUNCT iajs-2288 64 9 . . PUNCT iajs-2288 65 1 corollary corollary ADJ iajs-2288 65 2 ( ( PUNCT iajs-2288 65 3 5 5 NUM iajs-2288 65 4 ) ) PUNCT iajs-2288 65 5 let let VERB iajs-2288 65 6 𝐿 𝐿 PROPN iajs-2288 65 7 be be AUX iajs-2288 65 8 a a DET iajs-2288 65 9 proper proper ADJ iajs-2288 65 10 submodule submodule NOUN iajs-2288 65 11 of of ADP iajs-2288 65 12 an an DET iajs-2288 65 13 𝑅-module 𝑅-module PROPN iajs-2288 65 14 𝑇. 𝑇. PROPN iajs-2288 65 15 then then ADV iajs-2288 65 16 𝐿 𝐿 PROPN iajs-2288 65 17 is be AUX iajs-2288 65 18 an an DET iajs-2288 65 19 app app ADJ iajs-2288 65 20 - - PUNCT iajs-2288 65 21 semi semi ADJ iajs-2288 65 22 - - ADJ iajs-2288 65 23 prime prime ADJ iajs-2288 65 24 submodule submodule NOUN iajs-2288 65 25 of of ADP iajs-2288 65 26 𝑇 𝑇 PROPN iajs-2288 66 1 if if SCONJ iajs-2288 66 2 and and CCONJ iajs-2288 66 3 only only ADV iajs-2288 66 4 if if SCONJ iajs-2288 66 5 𝐽2𝐸 𝐽2𝐸 ADP iajs-2288 66 6 βŠ† βŠ† NUM iajs-2288 66 7 𝐿 𝐿 PROPN iajs-2288 66 8 where where SCONJ iajs-2288 66 9 𝐽 𝐽 PROPN iajs-2288 66 10 is be AUX iajs-2288 66 11 an an DET iajs-2288 66 12 ideal ideal NOUN iajs-2288 66 13 of of ADP iajs-2288 66 14 𝑅 𝑅 PROPN iajs-2288 66 15 and and CCONJ iajs-2288 66 16 𝐸 𝐸 PROPN iajs-2288 66 17 is be AUX iajs-2288 66 18 a a DET iajs-2288 66 19 submodule submodule NOUN iajs-2288 66 20 of of ADP iajs-2288 66 21 𝑇 𝑇 PROPN iajs-2288 66 22 , , PUNCT iajs-2288 66 23 implies imply VERB iajs-2288 66 24 that that SCONJ iajs-2288 66 25 𝐽𝐸 𝐽𝐸 VERB iajs-2288 66 26 βŠ† βŠ† NUM iajs-2288 66 27 𝐿 𝐿 PROPN iajs-2288 66 28 + + NOUN iajs-2288 66 29 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 66 30 ) ) PUNCT iajs-2288 66 31 . . PUNCT iajs-2288 67 1 proposition proposition NOUN iajs-2288 67 2 ( ( PUNCT iajs-2288 67 3 6 6 NUM iajs-2288 67 4 ) ) PUNCT iajs-2288 67 5 let let VERB iajs-2288 67 6 𝐿 𝐿 PROPN iajs-2288 67 7 be be AUX iajs-2288 67 8 a a DET iajs-2288 67 9 proper proper ADJ iajs-2288 67 10 submodule submodule NOUN iajs-2288 67 11 of of ADP iajs-2288 67 12 an an DET iajs-2288 67 13 𝑅-module 𝑅-module PROPN iajs-2288 67 14 𝑇. 𝑇. PROPN iajs-2288 67 15 then then ADV iajs-2288 67 16 𝐿 𝐿 PROPN iajs-2288 67 17 + + CCONJ iajs-2288 67 18 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 67 19 ) ) PUNCT iajs-2288 67 20 is be AUX iajs-2288 67 21 an an DET iajs-2288 67 22 app app ADJ iajs-2288 67 23 - - PUNCT iajs-2288 67 24 semi semi ADJ iajs-2288 67 25 - - ADJ iajs-2288 67 26 prime prime ADJ iajs-2288 67 27 submodule submodule NOUN iajs-2288 67 28 of of ADP iajs-2288 67 29 𝑇 𝑇 PROPN iajs-2288 68 1 if if SCONJ iajs-2288 68 2 and and CCONJ iajs-2288 68 3 only only ADV iajs-2288 68 4 if if SCONJ iajs-2288 68 5 [ [ X iajs-2288 68 6 𝐿 𝐿 PROPN iajs-2288 68 7 + + NOUN iajs-2288 68 8 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 68 9 ): ): PUNCT iajs-2288 68 10 𝑇 𝑇 PROPN iajs-2288 68 11 ] ] PUNCT iajs-2288 68 12 is be AUX iajs-2288 68 13 a a DET iajs-2288 68 14 semi semi ADJ iajs-2288 68 15 - - ADJ iajs-2288 68 16 prime prime ADJ iajs-2288 68 17 ideal ideal NOUN iajs-2288 68 18 of of ADP iajs-2288 68 19 𝑅 𝑅 PROPN iajs-2288 68 20 ( ( PUNCT iajs-2288 68 21 hence hence ADV iajs-2288 68 22 an an DET iajs-2288 68 23 app app ADJ iajs-2288 68 24 - - PUNCT iajs-2288 68 25 semiprime semiprime NOUN iajs-2288 68 26 ) ) PUNCT iajs-2288 68 27 . . PUNCT iajs-2288 69 1 proof proof NOUN iajs-2288 69 2 ( ( PUNCT iajs-2288 69 3 β‡’ β‡’ PROPN iajs-2288 69 4 ) ) PUNCT iajs-2288 69 5 let let VERB iajs-2288 69 6 π‘Ÿ π‘Ÿ PRON iajs-2288 69 7 ∈ ∈ NOUN iajs-2288 69 8 √[𝐿 √[𝐿 PROPN iajs-2288 69 9 + + NUM iajs-2288 69 10 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 69 11 ): ): PUNCT iajs-2288 69 12 𝑇 𝑇 PROPN iajs-2288 69 13 ] ] PUNCT iajs-2288 69 14 , , PUNCT iajs-2288 69 15 implies imply VERB iajs-2288 69 16 that that SCONJ iajs-2288 69 17 π‘Ÿπ‘› π‘Ÿπ‘› NUM iajs-2288 69 18 ∈ ∈ PROPN iajs-2288 69 19 [ [ X iajs-2288 69 20 𝐿 𝐿 PROPN iajs-2288 69 21 + + NOUN iajs-2288 69 22 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 69 23 ): ): PUNCT iajs-2288 69 24 𝑇 𝑇 PROPN iajs-2288 69 25 ] ] PUNCT iajs-2288 69 26 for for ADP iajs-2288 69 27 some some DET iajs-2288 69 28 𝑛 𝑛 PRON iajs-2288 69 29 ∈ ∈ PROPN iajs-2288 69 30 𝑍+ 𝑍+ NOUN iajs-2288 69 31 , , PUNCT iajs-2288 69 32 it it PRON iajs-2288 69 33 follows follow VERB iajs-2288 69 34 that that PRON iajs-2288 69 35 π‘Ÿπ‘›π‘‡ π‘Ÿπ‘›π‘‡ VERB iajs-2288 69 36 βŠ† βŠ† NUM iajs-2288 69 37 𝐿 𝐿 PROPN iajs-2288 69 38 + + NOUN iajs-2288 69 39 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 69 40 ) ) PUNCT iajs-2288 69 41 , , PUNCT iajs-2288 69 42 then then ADV iajs-2288 69 43 π‘Ÿπ‘›π‘‘ π‘Ÿπ‘›π‘‘ PROPN iajs-2288 69 44 ∈ ∈ PROPN iajs-2288 69 45 𝐿 𝐿 PROPN iajs-2288 69 46 + + NOUN iajs-2288 69 47 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 69 48 ) ) PUNCT iajs-2288 69 49 for for ADP iajs-2288 69 50 all all DET iajs-2288 69 51 𝑑 𝑑 PROPN iajs-2288 69 52 ∈ ∈ PROPN iajs-2288 69 53 𝑇. 𝑇. PROPN iajs-2288 69 54 but but CCONJ iajs-2288 69 55 𝐿 𝐿 PROPN iajs-2288 69 56 + + NOUN iajs-2288 69 57 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 69 58 ) ) PUNCT iajs-2288 69 59 is be AUX iajs-2288 69 60 an an DET iajs-2288 69 61 app app ADJ iajs-2288 69 62 - - PUNCT iajs-2288 69 63 semi semi NOUN iajs-2288 69 64 - - ADJ iajs-2288 69 65 prime prime ADJ iajs-2288 69 66 , , PUNCT iajs-2288 69 67 implies imply VERB iajs-2288 69 68 that that SCONJ iajs-2288 69 69 π‘Ÿπ‘‘ π‘Ÿπ‘‘ PROPN iajs-2288 69 70 ∈ ∈ PROPN iajs-2288 69 71 𝐿 𝐿 PROPN iajs-2288 69 72 + + NOUN iajs-2288 69 73 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 69 74 ) ) PUNCT iajs-2288 69 75 + + NUM iajs-2288 69 76 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 69 77 ) ) PUNCT iajs-2288 69 78 = = SYM iajs-2288 69 79 𝐿 𝐿 PROPN iajs-2288 69 80 + + NOUN iajs-2288 69 81 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 69 82 ) ) PUNCT iajs-2288 69 83 for for ADP iajs-2288 69 84 all all DET iajs-2288 69 85 𝑑 𝑑 PROPN iajs-2288 69 86 ∈ ∈ PROPN iajs-2288 69 87 𝑇. 𝑇. PROPN iajs-2288 69 88 that that PRON iajs-2288 69 89 is be AUX iajs-2288 69 90 π‘Ÿπ‘‡ π‘Ÿπ‘‡ NOUN iajs-2288 69 91 βŠ† βŠ† NUM iajs-2288 69 92 𝐿 𝐿 PROPN iajs-2288 69 93 + + NOUN iajs-2288 69 94 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 69 95 ) ) PUNCT iajs-2288 69 96 , , PUNCT iajs-2288 69 97 implies imply VERB iajs-2288 69 98 that that SCONJ iajs-2288 69 99 π‘Ÿ π‘Ÿ X iajs-2288 69 100 ∈ ∈ PROPN iajs-2288 69 101 [ [ X iajs-2288 69 102 𝐿 𝐿 PROPN iajs-2288 69 103 + + NOUN iajs-2288 69 104 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 69 105 ): ): PUNCT iajs-2288 69 106 𝑇 𝑇 PROPN iajs-2288 69 107 ] ] PUNCT iajs-2288 69 108 . . PUNCT iajs-2288 70 1 thus thus ADV iajs-2288 70 2 √[𝐿 √[𝐿 NUM iajs-2288 70 3 + + NUM iajs-2288 70 4 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 70 5 ): ): PUNCT iajs-2288 70 6 𝑇 𝑇 PROPN iajs-2288 70 7 ] ] PUNCT iajs-2288 70 8 βŠ† βŠ† NUM iajs-2288 70 9 [ [ X iajs-2288 70 10 𝐿 𝐿 PROPN iajs-2288 70 11 + + NOUN iajs-2288 70 12 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 70 13 ): ): PUNCT iajs-2288 70 14 𝑇 𝑇 PROPN iajs-2288 70 15 ] ] PUNCT iajs-2288 70 16 , , PUNCT iajs-2288 70 17 but but CCONJ iajs-2288 70 18 [ [ X iajs-2288 70 19 𝐿 𝐿 PROPN iajs-2288 70 20 + + NOUN iajs-2288 70 21 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 70 22 ): ): PUNCT iajs-2288 70 23 𝑇 𝑇 PROPN iajs-2288 70 24 ] ] PUNCT iajs-2288 70 25 βŠ† βŠ† NUM iajs-2288 70 26 √[𝐿 √[𝐿 NOUN iajs-2288 70 27 + + NUM iajs-2288 70 28 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 70 29 ): ): PUNCT iajs-2288 70 30 𝑇 𝑇 PROPN iajs-2288 70 31 ] ] PUNCT iajs-2288 70 32 , , PUNCT iajs-2288 70 33 it it PRON iajs-2288 70 34 follows follow VERB iajs-2288 70 35 that that SCONJ iajs-2288 70 36 [ [ X iajs-2288 70 37 𝐿 𝐿 PROPN iajs-2288 70 38 + + NOUN iajs-2288 70 39 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 70 40 ): ): PUNCT iajs-2288 70 41 𝑇 𝑇 PROPN iajs-2288 70 42 ] ] PUNCT iajs-2288 70 43 = = X iajs-2288 70 44 √[𝐿 √[𝐿 PUNCT iajs-2288 70 45 + + NUM iajs-2288 70 46 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 70 47 ): ): PUNCT iajs-2288 70 48 𝑇 𝑇 PROPN iajs-2288 70 49 ] ] PUNCT iajs-2288 70 50 , , PUNCT iajs-2288 70 51 hence hence ADV iajs-2288 70 52 [ [ X iajs-2288 70 53 𝐿 𝐿 PROPN iajs-2288 70 54 + + NOUN iajs-2288 70 55 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 70 56 ): ): PUNCT iajs-2288 70 57 𝑇 𝑇 PROPN iajs-2288 70 58 ] ] PUNCT iajs-2288 70 59 is be AUX iajs-2288 70 60 a a DET iajs-2288 70 61 semi semi ADJ iajs-2288 70 62 - - ADJ iajs-2288 70 63 prime prime ADJ iajs-2288 70 64 ideal ideal NOUN iajs-2288 70 65 of of ADP iajs-2288 70 66 𝑅 𝑅 PROPN iajs-2288 70 67 ( ( PUNCT iajs-2288 70 68 hence hence ADV iajs-2288 71 1 [ [ X iajs-2288 71 2 𝐿 𝐿 PROPN iajs-2288 71 3 + + NOUN iajs-2288 71 4 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 71 5 ): ): PUNCT iajs-2288 71 6 𝑇 𝑇 PROPN iajs-2288 71 7 ] ] PUNCT iajs-2288 71 8 is be AUX iajs-2288 71 9 an an DET iajs-2288 71 10 app app ADJ iajs-2288 71 11 - - PUNCT iajs-2288 71 12 semi semi ADJ iajs-2288 71 13 - - ADJ iajs-2288 71 14 prime prime ADJ iajs-2288 71 15 ) ) PUNCT iajs-2288 71 16 . . PUNCT iajs-2288 72 1 ( ( PUNCT iajs-2288 72 2 ⇐ ⇐ PROPN iajs-2288 72 3 ) ) PUNCT iajs-2288 72 4 let let VERB iajs-2288 72 5 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN iajs-2288 72 6 ∈ ∈ PROPN iajs-2288 72 7 𝐿 𝐿 PROPN iajs-2288 72 8 + + NOUN iajs-2288 72 9 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 72 10 ) ) PUNCT iajs-2288 72 11 , , PUNCT iajs-2288 72 12 where where SCONJ iajs-2288 72 13 π‘Ž π‘Ž PROPN iajs-2288 72 14 ∈ ∈ PROPN iajs-2288 72 15 𝑅 𝑅 PROPN iajs-2288 72 16 , , PUNCT iajs-2288 72 17 𝑑 𝑑 PROPN iajs-2288 72 18 ∈ ∈ PROPN iajs-2288 72 19 𝑇 𝑇 PROPN iajs-2288 72 20 and and CCONJ iajs-2288 72 21 𝑛 𝑛 DET iajs-2288 72 22 ∈ ∈ PROPN iajs-2288 72 23 𝑍+ 𝑍+ NOUN iajs-2288 72 24 , , PUNCT iajs-2288 72 25 implies imply VERB iajs-2288 72 26 that that SCONJ iajs-2288 72 27 π‘Žπ‘›π‘‡ π‘Žπ‘›π‘‡ PROPN iajs-2288 72 28 βŠ† βŠ† NUM iajs-2288 72 29 𝐿 𝐿 PROPN iajs-2288 72 30 + + NOUN iajs-2288 72 31 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 72 32 ) ) PUNCT iajs-2288 72 33 , , PUNCT iajs-2288 72 34 that that PRON iajs-2288 72 35 is be AUX iajs-2288 72 36 π‘Žπ‘› π‘Žπ‘› PRON iajs-2288 72 37 ∈ ∈ PROPN iajs-2288 73 1 [ [ X iajs-2288 73 2 𝐿 𝐿 PROPN iajs-2288 73 3 + + NOUN iajs-2288 73 4 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 73 5 ): ): PUNCT iajs-2288 73 6 𝑇 𝑇 PROPN iajs-2288 73 7 ] ] PUNCT iajs-2288 73 8 . . PUNCT iajs-2288 74 1 since since SCONJ iajs-2288 74 2 [ [ X iajs-2288 74 3 𝐿 𝐿 PROPN iajs-2288 74 4 + + NOUN iajs-2288 74 5 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 74 6 ): ): PUNCT iajs-2288 74 7 𝑇 𝑇 PROPN iajs-2288 74 8 ] ] PUNCT iajs-2288 74 9 is be AUX iajs-2288 74 10 a a DET iajs-2288 74 11 semi semi ADJ iajs-2288 74 12 - - ADJ iajs-2288 74 13 prime prime ADJ iajs-2288 74 14 ideal ideal NOUN iajs-2288 74 15 of of ADP iajs-2288 74 16 𝑅 𝑅 PROPN iajs-2288 74 17 then then ADV iajs-2288 74 18 π‘Ž π‘Ž PROPN iajs-2288 74 19 ∈ ∈ NOUN iajs-2288 74 20 [ [ X iajs-2288 74 21 𝐿 𝐿 PROPN iajs-2288 74 22 + + NOUN iajs-2288 74 23 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 74 24 ): ): PUNCT iajs-2288 74 25 𝑇 𝑇 PROPN iajs-2288 74 26 ] ] PUNCT iajs-2288 74 27 , , PUNCT iajs-2288 74 28 it it PRON iajs-2288 74 29 follows follow VERB iajs-2288 74 30 that that SCONJ iajs-2288 74 31 π‘Žπ‘‡ π‘Žπ‘‡ VERB iajs-2288 74 32 βŠ† βŠ† NUM iajs-2288 74 33 𝐿 𝐿 PROPN iajs-2288 74 34 + + NOUN iajs-2288 74 35 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 74 36 ) ) PUNCT iajs-2288 74 37 , , PUNCT iajs-2288 74 38 implies imply VERB iajs-2288 74 39 that that SCONJ iajs-2288 74 40 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 74 41 ∈ ∈ PROPN iajs-2288 74 42 𝐿 𝐿 PROPN iajs-2288 74 43 + + NOUN iajs-2288 74 44 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 74 45 ) ) PUNCT iajs-2288 74 46 for for ADP iajs-2288 74 47 all all DET iajs-2288 74 48 𝑑 𝑑 PROPN iajs-2288 74 49 ∈ ∈ PROPN iajs-2288 74 50 𝑇. 𝑇. PROPN iajs-2288 74 51 therefore therefore ADV iajs-2288 74 52 𝐿 𝐿 PROPN iajs-2288 74 53 + + NOUN iajs-2288 74 54 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 74 55 ) ) PUNCT iajs-2288 74 56 is be AUX iajs-2288 74 57 an an DET iajs-2288 74 58 app app ADJ iajs-2288 74 59 - - PUNCT iajs-2288 74 60 semi semi ADJ iajs-2288 74 61 - - ADJ iajs-2288 74 62 prime prime ADJ iajs-2288 74 63 submodule submodule NOUN iajs-2288 74 64 of of ADP iajs-2288 74 65 𝑇. 𝑇. PROPN iajs-2288 74 66 proposition proposition NOUN iajs-2288 74 67 ( ( PUNCT iajs-2288 74 68 7 7 X iajs-2288 74 69 ) ) PUNCT iajs-2288 74 70 let let VERB iajs-2288 74 71 𝐿 𝐿 PROPN iajs-2288 74 72 be be AUX iajs-2288 74 73 a a DET iajs-2288 74 74 proper proper ADJ iajs-2288 74 75 submodule submodule NOUN iajs-2288 74 76 of of ADP iajs-2288 74 77 an an DET iajs-2288 74 78 𝑅-module 𝑅-module PROPN iajs-2288 74 79 𝑇 𝑇 PROPN iajs-2288 74 80 with with ADP iajs-2288 74 81 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 74 82 ) ) PUNCT iajs-2288 74 83 βŠ† βŠ† NUM iajs-2288 74 84 𝐿. 𝐿. VERB iajs-2288 74 85 then then ADV iajs-2288 74 86 𝐿 𝐿 PROPN iajs-2288 74 87 is be AUX iajs-2288 74 88 an an DET iajs-2288 74 89 app app ADJ iajs-2288 74 90 - - PUNCT iajs-2288 74 91 semiprime semiprime NOUN iajs-2288 74 92 submodule submodule NOUN iajs-2288 74 93 of of ADP iajs-2288 74 94 𝑇 𝑇 PROPN iajs-2288 74 95 if if SCONJ iajs-2288 74 96 and and CCONJ iajs-2288 74 97 only only ADV iajs-2288 74 98 if if SCONJ iajs-2288 74 99 [ [ X iajs-2288 74 100 𝐿:𝑇 𝐿:𝑇 X iajs-2288 74 101 π‘Ž π‘Ž PART iajs-2288 74 102 𝑛 𝑛 NOUN iajs-2288 74 103 ] ] PUNCT iajs-2288 74 104 = = SYM iajs-2288 75 1 [ [ X iajs-2288 75 2 𝐿:𝑇 𝐿:𝑇 X iajs-2288 75 3 π‘Ž π‘Ž X iajs-2288 75 4 ] ] X iajs-2288 75 5 for for ADP iajs-2288 75 6 π‘Ž π‘Ž PROPN iajs-2288 75 7 ∈ ∈ PROPN iajs-2288 75 8 𝑅 𝑅 PROPN iajs-2288 75 9 and and CCONJ iajs-2288 75 10 some some PRON iajs-2288 75 11 𝑛 𝑛 DET iajs-2288 75 12 ∈ ∈ PROPN iajs-2288 75 13 𝑍+ 𝑍+ NOUN iajs-2288 75 14 . . PUNCT iajs-2288 76 1 120 120 NUM iajs-2288 76 2 ibn ibn PROPN iajs-2288 76 3 al al PROPN iajs-2288 76 4 - - PUNCT iajs-2288 76 5 haitham haitham PROPN iajs-2288 76 6 jour jour X iajs-2288 76 7 . . PROPN iajs-2288 76 8 for for ADP iajs-2288 76 9 pure pure ADJ iajs-2288 76 10 & & CCONJ iajs-2288 76 11 appl appl PROPN iajs-2288 76 12 . . PUNCT iajs-2288 77 1 sci sci PROPN iajs-2288 77 2 . . PROPN iajs-2288 77 3 32 32 NUM iajs-2288 77 4 ( ( PUNCT iajs-2288 77 5 3 3 NUM iajs-2288 77 6 ) ) PUNCT iajs-2288 77 7 2019 2019 NUM iajs-2288 77 8 proof proof NOUN iajs-2288 77 9 ( ( PUNCT iajs-2288 77 10 β‡’ β‡’ PROPN iajs-2288 77 11 ) ) PUNCT iajs-2288 77 12 let let VERB iajs-2288 77 13 𝑑 𝑑 PRON iajs-2288 77 14 ∈ ∈ PROPN iajs-2288 77 15 [ [ X iajs-2288 77 16 𝐿:𝑇 𝐿:𝑇 PROPN iajs-2288 77 17 π‘Ž π‘Ž DET iajs-2288 77 18 𝑛 𝑛 NOUN iajs-2288 77 19 ] ] PUNCT iajs-2288 77 20 , , PUNCT iajs-2288 77 21 implies imply VERB iajs-2288 77 22 that that SCONJ iajs-2288 77 23 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 77 24 ∈ ∈ PROPN iajs-2288 77 25 𝐿. 𝐿. PROPN iajs-2288 77 26 but but CCONJ iajs-2288 77 27 𝐿 𝐿 PROPN iajs-2288 77 28 is be AUX iajs-2288 77 29 an an DET iajs-2288 77 30 app app ADJ iajs-2288 77 31 - - PUNCT iajs-2288 77 32 semi semi ADJ iajs-2288 77 33 - - ADJ iajs-2288 77 34 prime prime ADJ iajs-2288 77 35 submodule submodule NOUN iajs-2288 77 36 of of ADP iajs-2288 77 37 𝑇 𝑇 PROPN iajs-2288 77 38 , , PUNCT iajs-2288 77 39 then then ADV iajs-2288 77 40 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 77 41 ∈ ∈ PROPN iajs-2288 77 42 𝐿 𝐿 PROPN iajs-2288 77 43 + + NOUN iajs-2288 77 44 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 77 45 ) ) PUNCT iajs-2288 77 46 . . PUNCT iajs-2288 78 1 but but CCONJ iajs-2288 78 2 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 78 3 ) ) PUNCT iajs-2288 78 4 βŠ† βŠ† X iajs-2288 78 5 𝐿 𝐿 PROPN iajs-2288 78 6 , , PUNCT iajs-2288 78 7 implies imply VERB iajs-2288 78 8 that that SCONJ iajs-2288 78 9 𝐿 𝐿 PROPN iajs-2288 78 10 + + NOUN iajs-2288 78 11 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 78 12 ) ) PUNCT iajs-2288 78 13 = = SYM iajs-2288 78 14 𝐿 𝐿 PROPN iajs-2288 78 15 , , PUNCT iajs-2288 78 16 thus thus ADV iajs-2288 78 17 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 78 18 ∈ ∈ PROPN iajs-2288 78 19 𝐿 𝐿 PROPN iajs-2288 78 20 , , PUNCT iajs-2288 78 21 it it PRON iajs-2288 78 22 follows follow VERB iajs-2288 78 23 that that SCONJ iajs-2288 78 24 𝑑 𝑑 PROPN iajs-2288 78 25 ∈ ∈ PROPN iajs-2288 79 1 [ [ X iajs-2288 79 2 𝐿:𝑇 𝐿:𝑇 PROPN iajs-2288 79 3 π‘Ž π‘Ž X iajs-2288 79 4 ] ] X iajs-2288 79 5 , , PUNCT iajs-2288 79 6 hence hence ADV iajs-2288 79 7 [ [ X iajs-2288 79 8 𝐿:𝑇 𝐿:𝑇 PROPN iajs-2288 79 9 π‘Ž π‘Ž PART iajs-2288 79 10 𝑛 𝑛 NOUN iajs-2288 79 11 ] ] PUNCT iajs-2288 79 12 βŠ† βŠ† NUM iajs-2288 79 13 [ [ X iajs-2288 79 14 𝐿:𝑇 𝐿:𝑇 PROPN iajs-2288 79 15 π‘Ž π‘Ž X iajs-2288 79 16 ] ] X iajs-2288 79 17 . . PUNCT iajs-2288 80 1 but but CCONJ iajs-2288 80 2 [ [ X iajs-2288 80 3 𝐿:𝑇 𝐿:𝑇 X iajs-2288 80 4 π‘Ž π‘Ž X iajs-2288 80 5 ] ] PUNCT iajs-2288 80 6 βŠ† βŠ† NUM iajs-2288 80 7 [ [ X iajs-2288 80 8 𝐿:𝑇 𝐿:𝑇 PROPN iajs-2288 80 9 π‘Ž π‘Ž DET iajs-2288 80 10 𝑛 𝑛 NOUN iajs-2288 80 11 ] ] PUNCT iajs-2288 80 12 , , PUNCT iajs-2288 80 13 so so SCONJ iajs-2288 80 14 we we PRON iajs-2288 80 15 get get VERB iajs-2288 80 16 [ [ X iajs-2288 80 17 𝐿:𝑇 𝐿:𝑇 PROPN iajs-2288 80 18 π‘Ž π‘Ž PART iajs-2288 80 19 𝑛 𝑛 NOUN iajs-2288 80 20 ] ] PUNCT iajs-2288 80 21 = = SYM iajs-2288 81 1 [ [ X iajs-2288 81 2 𝐿:𝑇 𝐿:𝑇 PROPN iajs-2288 81 3 π‘Ž π‘Ž X iajs-2288 81 4 ] ] X iajs-2288 81 5 . . PUNCT iajs-2288 82 1 ( ( PUNCT iajs-2288 82 2 ⇐ ⇐ PROPN iajs-2288 82 3 ) ) PUNCT iajs-2288 82 4 suppose suppose VERB iajs-2288 82 5 that that SCONJ iajs-2288 82 6 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 82 7 ∈ ∈ PROPN iajs-2288 82 8 𝐿 𝐿 PROPN iajs-2288 82 9 , , PUNCT iajs-2288 82 10 where where SCONJ iajs-2288 82 11 π‘Ž π‘Ž PROPN iajs-2288 82 12 ∈ ∈ PROPN iajs-2288 82 13 𝑅 𝑅 PROPN iajs-2288 82 14 , , PUNCT iajs-2288 82 15 𝑑 𝑑 PROPN iajs-2288 82 16 ∈ ∈ PROPN iajs-2288 82 17 𝑇 𝑇 PROPN iajs-2288 82 18 and and CCONJ iajs-2288 82 19 𝑛 𝑛 PRON iajs-2288 82 20 ∈ ∈ PROPN iajs-2288 82 21 𝑍+ 𝑍+ NOUN iajs-2288 82 22 , , PUNCT iajs-2288 82 23 it it PRON iajs-2288 82 24 follows follow VERB iajs-2288 82 25 that that SCONJ iajs-2288 82 26 𝑑 𝑑 PROPN iajs-2288 82 27 ∈ ∈ PROPN iajs-2288 82 28 [ [ X iajs-2288 82 29 𝐿:𝑇 𝐿:𝑇 PROPN iajs-2288 82 30 π‘Ž π‘Ž PROPN iajs-2288 82 31 𝑛 𝑛 NOUN iajs-2288 82 32 ] ] PUNCT iajs-2288 82 33 = = SYM iajs-2288 83 1 [ [ X iajs-2288 83 2 𝐿:𝑇 𝐿:𝑇 PROPN iajs-2288 83 3 π‘Ž π‘Ž X iajs-2288 83 4 ] ] X iajs-2288 83 5 , , PUNCT iajs-2288 83 6 implies imply VERB iajs-2288 83 7 that that SCONJ iajs-2288 83 8 𝑑 𝑑 PROPN iajs-2288 83 9 ∈ ∈ PROPN iajs-2288 83 10 [ [ X iajs-2288 83 11 𝐿:𝑇 𝐿:𝑇 PROPN iajs-2288 83 12 π‘Ž π‘Ž X iajs-2288 83 13 ] ] X iajs-2288 83 14 , , PUNCT iajs-2288 83 15 so so CCONJ iajs-2288 83 16 π‘Žπ‘‘ π‘Žπ‘‘ SCONJ iajs-2288 83 17 ∈ ∈ PROPN iajs-2288 83 18 𝐿 𝐿 PROPN iajs-2288 83 19 βŠ† βŠ† NUM iajs-2288 83 20 𝐿 𝐿 PROPN iajs-2288 83 21 + + NOUN iajs-2288 83 22 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 83 23 ) ) PUNCT iajs-2288 83 24 . . PUNCT iajs-2288 84 1 that that PRON iajs-2288 84 2 is be AUX iajs-2288 84 3 π‘Žπ‘‘ π‘Žπ‘‘ PRON iajs-2288 84 4 ∈ ∈ PROPN iajs-2288 84 5 𝐿 𝐿 PROPN iajs-2288 84 6 + + NOUN iajs-2288 84 7 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 84 8 ) ) PUNCT iajs-2288 84 9 , , PUNCT iajs-2288 84 10 hence hence ADV iajs-2288 84 11 𝐿 𝐿 PROPN iajs-2288 84 12 is be AUX iajs-2288 84 13 an an DET iajs-2288 84 14 app app ADJ iajs-2288 84 15 - - PUNCT iajs-2288 84 16 semi semi ADJ iajs-2288 84 17 - - ADJ iajs-2288 84 18 prime prime ADJ iajs-2288 84 19 submodule submodule NOUN iajs-2288 84 20 of of ADP iajs-2288 84 21 𝑇. 𝑇. PROPN iajs-2288 84 22 we we PRON iajs-2288 84 23 recall recall VERB iajs-2288 84 24 the the DET iajs-2288 84 25 following following NOUN iajs-2288 84 26 lemmas lemma NOUN iajs-2288 84 27 before before SCONJ iajs-2288 84 28 we we PRON iajs-2288 84 29 introduce introduce VERB iajs-2288 84 30 the the DET iajs-2288 84 31 next next ADJ iajs-2288 84 32 results result NOUN iajs-2288 84 33 . . PUNCT iajs-2288 85 1 lemma lemma PROPN iajs-2288 85 2 ( ( PUNCT iajs-2288 85 3 8) 8) NUM iajs-2288 85 4 [ [ X iajs-2288 85 5 11 11 NUM iajs-2288 85 6 , , PUNCT iajs-2288 85 7 lemma lemma PROPN iajs-2288 85 8 2.3.15 2.3.15 PROPN iajs-2288 85 9 ] ] PUNCT iajs-2288 85 10 . . PUNCT iajs-2288 86 1 let let VERB iajs-2288 86 2 𝑇 𝑇 PROPN iajs-2288 86 3 be be AUX iajs-2288 86 4 𝑅-module 𝑅-module PROPN iajs-2288 86 5 and and CCONJ iajs-2288 86 6 𝐿 𝐿 PROPN iajs-2288 86 7 , , PUNCT iajs-2288 86 8 𝐾 𝐾 PROPN iajs-2288 86 9 , , PUNCT iajs-2288 86 10 𝐸 𝐸 PROPN iajs-2288 86 11 are be AUX iajs-2288 86 12 submodules submodule NOUN iajs-2288 86 13 of of ADP iajs-2288 86 14 𝑇 𝑇 PROPN iajs-2288 86 15 with with ADP iajs-2288 86 16 𝐾 𝐾 PROPN iajs-2288 86 17 βŠ† βŠ† NUM iajs-2288 86 18 𝐸 𝐸 PROPN iajs-2288 86 19 , , PUNCT iajs-2288 86 20 then then ADV iajs-2288 86 21 ( ( PUNCT iajs-2288 86 22 𝐿 𝐿 PROPN iajs-2288 86 23 + + CCONJ iajs-2288 86 24 𝐾 𝐾 PROPN iajs-2288 86 25 ) ) PUNCT iajs-2288 86 26 ∩ ∩ NOUN iajs-2288 86 27 𝐸 𝐸 NOUN iajs-2288 86 28 = = SYM iajs-2288 86 29 ( ( PUNCT iajs-2288 86 30 𝐿 𝐿 PROPN iajs-2288 86 31 ∩ ∩ X iajs-2288 86 32 𝐸 𝐸 PROPN iajs-2288 86 33 ) ) PUNCT iajs-2288 87 1 + + CCONJ iajs-2288 87 2 ( ( PUNCT iajs-2288 87 3 𝐾 𝐾 PROPN iajs-2288 87 4 ∩ ∩ ADJ iajs-2288 87 5 𝐸 𝐸 PROPN iajs-2288 87 6 ) ) PUNCT iajs-2288 87 7 = = PUNCT iajs-2288 87 8 ( ( PUNCT iajs-2288 87 9 𝐿 𝐿 PROPN iajs-2288 87 10 ∩ ∩ X iajs-2288 87 11 𝐸 𝐸 PROPN iajs-2288 87 12 ) ) PUNCT iajs-2288 87 13 + + CCONJ iajs-2288 87 14 𝐾. 𝐾. PROPN iajs-2288 87 15 lemma lemma PROPN iajs-2288 87 16 ( ( PUNCT iajs-2288 87 17 9 9 NUM iajs-2288 87 18 ) ) PUNCT iajs-2288 87 19 [ [ X iajs-2288 87 20 13 13 NUM iajs-2288 87 21 , , PUNCT iajs-2288 87 22 cor cor NOUN iajs-2288 87 23 . . PROPN iajs-2288 87 24 9.9 9.9 NUM iajs-2288 87 25 ] ] PUNCT iajs-2288 87 26 . . PUNCT iajs-2288 88 1 let let VERB iajs-2288 88 2 𝑇 𝑇 PROPN iajs-2288 88 3 be be AUX iajs-2288 88 4 𝑅-module 𝑅-module PROPN iajs-2288 88 5 and and CCONJ iajs-2288 88 6 𝐿 𝐿 PROPN iajs-2288 88 7 be be AUX iajs-2288 88 8 a a DET iajs-2288 88 9 submodule submodule NOUN iajs-2288 88 10 of of ADP iajs-2288 88 11 𝑇 𝑇 PROPN iajs-2288 88 12 , , PUNCT iajs-2288 88 13 then then ADV iajs-2288 88 14 π‘ π‘œπ‘(𝐿 π‘ π‘œπ‘(𝐿 NUM iajs-2288 88 15 ) ) PUNCT iajs-2288 88 16 = = SYM iajs-2288 88 17 𝐿 𝐿 PROPN iajs-2288 88 18 ∩ ∩ NOUN iajs-2288 88 19 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 88 20 ) ) PUNCT iajs-2288 88 21 . . PUNCT iajs-2288 89 1 proposition proposition NOUN iajs-2288 89 2 ( ( PUNCT iajs-2288 89 3 10 10 NUM iajs-2288 89 4 ) ) PUNCT iajs-2288 89 5 let let VERB iajs-2288 89 6 𝐿 𝐿 PROPN iajs-2288 89 7 and and CCONJ iajs-2288 89 8 𝐸 𝐸 PROPN iajs-2288 89 9 are be AUX iajs-2288 89 10 proper proper ADJ iajs-2288 89 11 submodules submodule NOUN iajs-2288 89 12 of of ADP iajs-2288 89 13 an an DET iajs-2288 89 14 𝑅-module 𝑅-module PROPN iajs-2288 89 15 𝑇 𝑇 PROPN iajs-2288 89 16 withe withe ADJ iajs-2288 89 17 𝐿 𝐿 PROPN iajs-2288 89 18 ⊊ ⊊ VERB iajs-2288 89 19 𝐸 𝐸 PROPN iajs-2288 89 20 and and CCONJ iajs-2288 89 21 𝐿 𝐿 PROPN iajs-2288 89 22 is be AUX iajs-2288 89 23 an an DET iajs-2288 89 24 app app ADJ iajs-2288 89 25 - - PUNCT iajs-2288 89 26 semiprime semiprime NOUN iajs-2288 89 27 submodule submodule NOUN iajs-2288 89 28 of of ADP iajs-2288 89 29 𝑇. 𝑇. PROPN iajs-2288 89 30 then then ADV iajs-2288 89 31 𝐿 𝐿 PROPN iajs-2288 89 32 is be AUX iajs-2288 89 33 an an DET iajs-2288 89 34 app app ADJ iajs-2288 89 35 - - PUNCT iajs-2288 89 36 semi semi ADJ iajs-2288 89 37 - - ADJ iajs-2288 89 38 prime prime ADJ iajs-2288 89 39 submodule submodule NOUN iajs-2288 89 40 of of ADP iajs-2288 89 41 𝐸. 𝐸. PROPN iajs-2288 89 42 proof proof NOUN iajs-2288 89 43 suppose suppose VERB iajs-2288 89 44 that that SCONJ iajs-2288 89 45 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 89 46 ∈ ∈ PROPN iajs-2288 89 47 𝐿 𝐿 PROPN iajs-2288 89 48 , , PUNCT iajs-2288 89 49 where where SCONJ iajs-2288 89 50 π‘Ž π‘Ž PROPN iajs-2288 89 51 ∈ ∈ PROPN iajs-2288 89 52 𝑅 𝑅 PROPN iajs-2288 89 53 , , PUNCT iajs-2288 89 54 𝑑 𝑑 PROPN iajs-2288 89 55 ∈ ∈ PROPN iajs-2288 89 56 𝐸 𝐸 PROPN iajs-2288 89 57 ⊊ ⊊ VERB iajs-2288 89 58 𝑇 𝑇 PROPN iajs-2288 89 59 and and CCONJ iajs-2288 89 60 𝑛 𝑛 PRON iajs-2288 89 61 ∈ ∈ PROPN iajs-2288 89 62 𝑍+ 𝑍+ NOUN iajs-2288 89 63 . . PUNCT iajs-2288 90 1 but but CCONJ iajs-2288 90 2 𝐿 𝐿 PROPN iajs-2288 90 3 is be AUX iajs-2288 90 4 an an DET iajs-2288 90 5 app app ADJ iajs-2288 90 6 - - PUNCT iajs-2288 90 7 semi semi ADJ iajs-2288 90 8 - - ADJ iajs-2288 90 9 prime prime ADJ iajs-2288 90 10 submodule submodule NOUN iajs-2288 90 11 of of ADP iajs-2288 90 12 𝑇 𝑇 PROPN iajs-2288 90 13 , , PUNCT iajs-2288 90 14 implies imply VERB iajs-2288 90 15 that that SCONJ iajs-2288 90 16 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 90 17 ∈ ∈ PROPN iajs-2288 90 18 𝐿 𝐿 PROPN iajs-2288 90 19 + + NOUN iajs-2288 90 20 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 90 21 ) ) PUNCT iajs-2288 90 22 , , PUNCT iajs-2288 90 23 but but CCONJ iajs-2288 90 24 𝑑 𝑑 PROPN iajs-2288 90 25 ∈ ∈ PROPN iajs-2288 90 26 𝐸 𝐸 PROPN iajs-2288 90 27 , , PUNCT iajs-2288 90 28 implies imply VERB iajs-2288 90 29 that that SCONJ iajs-2288 90 30 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 90 31 ∈ ∈ PROPN iajs-2288 90 32 𝐸 𝐸 PROPN iajs-2288 90 33 , , PUNCT iajs-2288 90 34 then then ADV iajs-2288 90 35 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 90 36 ∈ ∈ PROPN iajs-2288 90 37 ( ( PUNCT iajs-2288 90 38 𝐿 𝐿 PROPN iajs-2288 90 39 + + NOUN iajs-2288 90 40 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 90 41 ) ) PUNCT iajs-2288 90 42 ) ) PUNCT iajs-2288 90 43 ∩ ∩ PROPN iajs-2288 90 44 𝐸 𝐸 PROPN iajs-2288 90 45 , , PUNCT iajs-2288 90 46 thus thus ADV iajs-2288 90 47 by by ADP iajs-2288 90 48 lemma(2.8 lemma(2.8 NOUN iajs-2288 90 49 ) ) PUNCT iajs-2288 90 50 we we PRON iajs-2288 90 51 have have VERB iajs-2288 90 52 π‘Žπ‘‘ π‘Žπ‘‘ DET iajs-2288 90 53 ∈ ∈ PROPN iajs-2288 90 54 ( ( PUNCT iajs-2288 90 55 𝐿 𝐿 PROPN iajs-2288 90 56 ∩ ∩ ADJ iajs-2288 90 57 𝐸 𝐸 PROPN iajs-2288 90 58 ) ) PUNCT iajs-2288 91 1 + + CCONJ iajs-2288 91 2 ( ( PUNCT iajs-2288 91 3 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 91 4 ) ) PUNCT iajs-2288 91 5 ∩ ∩ ADJ iajs-2288 91 6 𝐸 𝐸 PROPN iajs-2288 91 7 ) ) PUNCT iajs-2288 91 8 βŠ† βŠ† NUM iajs-2288 91 9 𝐿 𝐿 PROPN iajs-2288 91 10 + + NOUN iajs-2288 91 11 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 91 12 ) ) PUNCT iajs-2288 91 13 ∩ ∩ NOUN iajs-2288 91 14 𝐸. 𝐸. VERB iajs-2288 91 15 so so ADV iajs-2288 91 16 by by ADP iajs-2288 91 17 lemma lemma PROPN iajs-2288 91 18 ( ( PUNCT iajs-2288 91 19 2.9 2.9 NUM iajs-2288 91 20 ) ) PUNCT iajs-2288 91 21 we we PRON iajs-2288 91 22 have have VERB iajs-2288 91 23 π‘Žπ‘‘ π‘Žπ‘‘ DET iajs-2288 91 24 ∈ ∈ PROPN iajs-2288 91 25 𝐿 𝐿 PROPN iajs-2288 91 26 + + PROPN iajs-2288 91 27 π‘ π‘œπ‘(𝐸 π‘ π‘œπ‘(𝐸 PROPN iajs-2288 91 28 ) ) PUNCT iajs-2288 91 29 . . PUNCT iajs-2288 92 1 thus thus ADV iajs-2288 92 2 𝐿 𝐿 PROPN iajs-2288 92 3 is be AUX iajs-2288 92 4 an an DET iajs-2288 92 5 app app ADJ iajs-2288 92 6 - - PUNCT iajs-2288 92 7 semi semi ADJ iajs-2288 92 8 - - ADJ iajs-2288 92 9 prime prime ADJ iajs-2288 92 10 submodule submodule NOUN iajs-2288 92 11 of of ADP iajs-2288 92 12 𝐸. 𝐸. PROPN iajs-2288 92 13 proposition proposition NOUN iajs-2288 92 14 ( ( PUNCT iajs-2288 92 15 11 11 NUM iajs-2288 92 16 ) ) PUNCT iajs-2288 92 17 let let VERB iajs-2288 92 18 𝐿 𝐿 PROPN iajs-2288 92 19 and and CCONJ iajs-2288 92 20 𝐸 𝐸 PROPN iajs-2288 92 21 are be AUX iajs-2288 92 22 proper proper ADJ iajs-2288 92 23 submodules submodule NOUN iajs-2288 92 24 of of ADP iajs-2288 92 25 an an DET iajs-2288 92 26 𝑅-module 𝑅-module PROPN iajs-2288 92 27 𝑇 𝑇 PROPN iajs-2288 92 28 withe withe ADJ iajs-2288 92 29 𝐿 𝐿 PROPN iajs-2288 92 30 ⊈ ⊈ PROPN iajs-2288 92 31 𝐸 𝐸 PROPN iajs-2288 92 32 and and CCONJ iajs-2288 92 33 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 92 34 ) ) PUNCT iajs-2288 92 35 βŠ† βŠ† NUM iajs-2288 92 36 𝐸. 𝐸. PROPN iajs-2288 92 37 if if SCONJ iajs-2288 92 38 𝐿 𝐿 PROPN iajs-2288 92 39 is be AUX iajs-2288 92 40 an an DET iajs-2288 92 41 app app ADJ iajs-2288 92 42 - - PUNCT iajs-2288 92 43 semi semi ADJ iajs-2288 92 44 - - ADJ iajs-2288 92 45 prime prime ADJ iajs-2288 92 46 submodule submodule NOUN iajs-2288 92 47 of of ADP iajs-2288 92 48 𝑇. 𝑇. PROPN iajs-2288 92 49 then then ADV iajs-2288 92 50 𝐿 𝐿 PROPN iajs-2288 92 51 ∩ ∩ NOUN iajs-2288 92 52 𝐸 𝐸 PROPN iajs-2288 92 53 is be AUX iajs-2288 92 54 an an DET iajs-2288 92 55 app app ADJ iajs-2288 92 56 - - PUNCT iajs-2288 92 57 semi semi ADJ iajs-2288 92 58 - - ADJ iajs-2288 92 59 prime prime ADJ iajs-2288 92 60 submodule submodule NOUN iajs-2288 92 61 of of ADP iajs-2288 92 62 𝐸. 𝐸. PROPN iajs-2288 92 63 proof proof NOUN iajs-2288 92 64 since since SCONJ iajs-2288 92 65 𝐿 𝐿 PROPN iajs-2288 92 66 ⊈ ⊈ PROPN iajs-2288 92 67 𝐸 𝐸 PROPN iajs-2288 92 68 then then ADV iajs-2288 92 69 𝐿 𝐿 PROPN iajs-2288 92 70 ∩ ∩ NOUN iajs-2288 92 71 𝐸 𝐸 PROPN iajs-2288 92 72 is be AUX iajs-2288 92 73 a a DET iajs-2288 92 74 proper proper ADJ iajs-2288 92 75 in in ADP iajs-2288 92 76 𝐸. 𝐸. PROPN iajs-2288 92 77 suppose suppose VERB iajs-2288 92 78 that that SCONJ iajs-2288 92 79 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 92 80 ∈ ∈ PROPN iajs-2288 92 81 𝐿 𝐿 PROPN iajs-2288 92 82 ∩ ∩ ADJ iajs-2288 92 83 𝐸 𝐸 PROPN iajs-2288 92 84 , , PUNCT iajs-2288 92 85 where where SCONJ iajs-2288 92 86 π‘Ž π‘Ž PROPN iajs-2288 92 87 ∈ ∈ PROPN iajs-2288 92 88 𝑅 𝑅 PROPN iajs-2288 92 89 , , PUNCT iajs-2288 92 90 𝑑 𝑑 PROPN iajs-2288 92 91 ∈ ∈ PROPN iajs-2288 92 92 𝐸 𝐸 PROPN iajs-2288 92 93 ⊊ ⊊ VERB iajs-2288 92 94 𝑇 𝑇 PROPN iajs-2288 92 95 and and CCONJ iajs-2288 92 96 𝑛 𝑛 DET iajs-2288 92 97 ∈ ∈ PROPN iajs-2288 92 98 𝑍+ 𝑍+ NOUN iajs-2288 92 99 , , PUNCT iajs-2288 92 100 implies imply VERB iajs-2288 92 101 that that SCONJ iajs-2288 92 102 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 92 103 ∈ ∈ PROPN iajs-2288 92 104 𝐿. 𝐿. PROPN iajs-2288 92 105 but but CCONJ iajs-2288 92 106 𝐿 𝐿 PROPN iajs-2288 92 107 is be AUX iajs-2288 92 108 an an DET iajs-2288 92 109 app app ADJ iajs-2288 92 110 - - PUNCT iajs-2288 92 111 semi semi ADJ iajs-2288 92 112 - - ADJ iajs-2288 92 113 prime prime ADJ iajs-2288 92 114 submodule submodule NOUN iajs-2288 92 115 of of ADP iajs-2288 92 116 𝑇 𝑇 PROPN iajs-2288 92 117 , , PUNCT iajs-2288 92 118 then then ADV iajs-2288 92 119 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 92 120 ∈ ∈ PROPN iajs-2288 92 121 𝐿 𝐿 PROPN iajs-2288 92 122 + + NOUN iajs-2288 92 123 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 92 124 ) ) PUNCT iajs-2288 92 125 , , PUNCT iajs-2288 92 126 it it PRON iajs-2288 92 127 follows follow VERB iajs-2288 92 128 that that SCONJ iajs-2288 92 129 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 92 130 ∈ ∈ PROPN iajs-2288 92 131 ( ( PUNCT iajs-2288 92 132 𝐿 𝐿 PROPN iajs-2288 92 133 + + NOUN iajs-2288 92 134 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 92 135 ) ) PUNCT iajs-2288 92 136 ) ) PUNCT iajs-2288 92 137 ∩ ∩ PROPN iajs-2288 92 138 𝐸 𝐸 PROPN iajs-2288 92 139 , , PUNCT iajs-2288 92 140 so so ADV iajs-2288 92 141 by by ADP iajs-2288 92 142 lemma lemma PROPN iajs-2288 92 143 ( ( PUNCT iajs-2288 92 144 2.8 2.8 NUM iajs-2288 92 145 ) ) PUNCT iajs-2288 92 146 we we PRON iajs-2288 92 147 have have VERB iajs-2288 92 148 π‘Žπ‘‘ π‘Žπ‘‘ DET iajs-2288 92 149 ∈ ∈ PROPN iajs-2288 92 150 ( ( PUNCT iajs-2288 92 151 𝐿 𝐿 PROPN iajs-2288 92 152 ∩ ∩ ADJ iajs-2288 92 153 𝐸 𝐸 PROPN iajs-2288 92 154 ) ) PUNCT iajs-2288 93 1 + + CCONJ iajs-2288 93 2 ( ( PUNCT iajs-2288 93 3 𝐸 𝐸 PROPN iajs-2288 93 4 ∩ ∩ NOUN iajs-2288 93 5 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 93 6 ) ) PUNCT iajs-2288 93 7 ) ) PUNCT iajs-2288 93 8 . . PUNCT iajs-2288 94 1 but but CCONJ iajs-2288 94 2 by by ADP iajs-2288 94 3 lemma lemma PROPN iajs-2288 94 4 ( ( PUNCT iajs-2288 94 5 2.9 2.9 NUM iajs-2288 94 6 ) ) PUNCT iajs-2288 94 7 we we PRON iajs-2288 94 8 have have VERB iajs-2288 94 9 𝐸 𝐸 ADJ iajs-2288 94 10 ∩ ∩ NOUN iajs-2288 94 11 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 94 12 ) ) PUNCT iajs-2288 94 13 = = SYM iajs-2288 94 14 π‘ π‘œπ‘(𝐸 π‘ π‘œπ‘(𝐸 NUM iajs-2288 94 15 ) ) PUNCT iajs-2288 94 16 . . PUNCT iajs-2288 95 1 hence hence ADV iajs-2288 95 2 π‘Žπ‘‘ π‘Žπ‘‘ PRON iajs-2288 95 3 ∈ ∈ PROPN iajs-2288 95 4 ( ( PUNCT iajs-2288 95 5 𝐿 𝐿 PROPN iajs-2288 95 6 ∩ ∩ ADJ iajs-2288 95 7 𝐸 𝐸 PROPN iajs-2288 95 8 ) ) PUNCT iajs-2288 95 9 + + NUM iajs-2288 95 10 π‘ π‘œπ‘(𝐸 π‘ π‘œπ‘(𝐸 NUM iajs-2288 95 11 ) ) PUNCT iajs-2288 95 12 . . PUNCT iajs-2288 96 1 thus thus ADV iajs-2288 96 2 𝐿 𝐿 PROPN iajs-2288 96 3 ∩ ∩ NOUN iajs-2288 96 4 𝐸 𝐸 PROPN iajs-2288 96 5 is be AUX iajs-2288 96 6 an an DET iajs-2288 96 7 app app ADJ iajs-2288 96 8 - - PUNCT iajs-2288 96 9 semi semi ADJ iajs-2288 96 10 - - ADJ iajs-2288 96 11 prime prime ADJ iajs-2288 96 12 submodule submodule NOUN iajs-2288 96 13 of of ADP iajs-2288 96 14 𝐸. 𝐸. PROPN iajs-2288 96 15 remark remark NOUN iajs-2288 96 16 ( ( PUNCT iajs-2288 96 17 12 12 NUM iajs-2288 96 18 ) ) PUNCT iajs-2288 96 19 if if SCONJ iajs-2288 96 20 𝐿 𝐿 PROPN iajs-2288 96 21 and and CCONJ iajs-2288 96 22 𝐸 𝐸 PROPN iajs-2288 96 23 are be AUX iajs-2288 96 24 two two NUM iajs-2288 96 25 app app ADJ iajs-2288 96 26 - - PUNCT iajs-2288 96 27 semi semi ADJ iajs-2288 96 28 - - ADJ iajs-2288 96 29 prime prime ADJ iajs-2288 96 30 submodules submodule NOUN iajs-2288 96 31 of of ADP iajs-2288 96 32 an an DET iajs-2288 96 33 𝑅-module 𝑅-module PROPN iajs-2288 96 34 𝑇 𝑇 PROPN iajs-2288 96 35 , , PUNCT iajs-2288 96 36 then then ADV iajs-2288 96 37 𝐿 𝐿 PROPN iajs-2288 96 38 ∩ ∩ NOUN iajs-2288 96 39 𝐸 𝐸 PROPN iajs-2288 96 40 is be AUX iajs-2288 96 41 not not PART iajs-2288 96 42 necessary necessary ADJ iajs-2288 96 43 an an DET iajs-2288 96 44 app app ADJ iajs-2288 96 45 - - PUNCT iajs-2288 96 46 semi semi ADJ iajs-2288 96 47 - - ADJ iajs-2288 96 48 prime prime ADJ iajs-2288 96 49 submodule submodule NOUN iajs-2288 96 50 of of ADP iajs-2288 96 51 𝑇 𝑇 PROPN iajs-2288 96 52 as as ADP iajs-2288 96 53 the the DET iajs-2288 96 54 following follow VERB iajs-2288 96 55 example example NOUN iajs-2288 96 56 shows show VERB iajs-2288 96 57 that that PRON iajs-2288 96 58 . . PUNCT iajs-2288 97 1 let let VERB iajs-2288 97 2 = = SYM iajs-2288 97 3 𝑍⨁𝑍4 𝑍⨁𝑍4 PROPN iajs-2288 97 4 , , PUNCT iajs-2288 97 5 𝑅 𝑅 PROPN iajs-2288 97 6 = = PUNCT iajs-2288 97 7 𝑍 𝑍 PROPN iajs-2288 97 8 , , PUNCT iajs-2288 97 9 𝐿 𝐿 NOUN iajs-2288 97 10 = = SYM iajs-2288 97 11 𝑍(1 𝑍(1 PROPN iajs-2288 97 12 , , PUNCT iajs-2288 97 13 0Μ… 0Μ… PROPN iajs-2288 97 14 ) ) PUNCT iajs-2288 97 15 , , PUNCT iajs-2288 97 16 𝐸 𝐸 PROPN iajs-2288 97 17 = = SYM iajs-2288 97 18 𝑍(1 𝑍(1 PUNCT iajs-2288 97 19 , , PUNCT iajs-2288 97 20 1Μ… 1Μ… NUM iajs-2288 97 21 ) ) PUNCT iajs-2288 97 22 , , PUNCT iajs-2288 97 23 where where SCONJ iajs-2288 97 24 𝐿 𝐿 PROPN iajs-2288 97 25 and and CCONJ iajs-2288 97 26 𝐸 𝐸 PROPN iajs-2288 97 27 are be AUX iajs-2288 97 28 app app ADJ iajs-2288 97 29 - - PUNCT iajs-2288 97 30 semi semi ADJ iajs-2288 97 31 - - ADJ iajs-2288 97 32 prime prime ADJ iajs-2288 97 33 submodules submodule NOUN iajs-2288 97 34 of of ADP iajs-2288 97 35 𝑇. 𝑇. PROPN iajs-2288 97 36 then then ADV iajs-2288 97 37 𝐿 𝐿 PROPN iajs-2288 97 38 ∩ ∩ ADJ iajs-2288 97 39 𝐸 𝐸 NOUN iajs-2288 97 40 = = SYM iajs-2288 97 41 { { PUNCT iajs-2288 97 42 ( ( PUNCT iajs-2288 97 43 0 0 NUM iajs-2288 97 44 , , PUNCT iajs-2288 97 45 0Μ… 0Μ… PROPN iajs-2288 97 46 ) ) PUNCT iajs-2288 97 47 , , PUNCT iajs-2288 97 48 ( ( PUNCT iajs-2288 97 49 4 4 NUM iajs-2288 97 50 , , PUNCT iajs-2288 97 51 0Μ… 0Μ… PROPN iajs-2288 97 52 ) ) PUNCT iajs-2288 97 53 , , PUNCT iajs-2288 97 54 ( ( PUNCT iajs-2288 97 55 8 8 NUM iajs-2288 97 56 , , PUNCT iajs-2288 97 57 0Μ… 0Μ… PROPN iajs-2288 97 58 ) ) PUNCT iajs-2288 97 59 , , PUNCT iajs-2288 97 60 … … PUNCT iajs-2288 97 61 … … PUNCT iajs-2288 97 62 } } PUNCT iajs-2288 97 63 is be AUX iajs-2288 97 64 not not PART iajs-2288 97 65 an an DET iajs-2288 97 66 app app ADJ iajs-2288 97 67 - - PUNCT iajs-2288 97 68 semi semi ADJ iajs-2288 97 69 - - ADJ iajs-2288 97 70 prime prime ADJ iajs-2288 97 71 submodule submodule NOUN iajs-2288 97 72 of of ADP iajs-2288 97 73 𝑇 𝑇 PROPN iajs-2288 97 74 , , PUNCT iajs-2288 97 75 since since SCONJ iajs-2288 97 76 22(1 22(1 PRON iajs-2288 97 77 , , PUNCT iajs-2288 97 78 0Μ… 0Μ… PROPN iajs-2288 97 79 ) ) PUNCT iajs-2288 97 80 ∈ ∈ PROPN iajs-2288 97 81 𝐿 𝐿 PROPN iajs-2288 97 82 ∩ ∩ X iajs-2288 97 83 𝐸 𝐸 PROPN iajs-2288 97 84 , , PUNCT iajs-2288 97 85 but but CCONJ iajs-2288 97 86 2(1 2(1 NUM iajs-2288 97 87 , , PUNCT iajs-2288 97 88 0Μ… 0Μ… NUM iajs-2288 97 89 ) ) PUNCT iajs-2288 97 90 βˆ‰ βˆ‰ PROPN iajs-2288 97 91 𝐿 𝐿 PROPN iajs-2288 97 92 ∩ ∩ ADJ iajs-2288 97 93 𝐸 𝐸 PROPN iajs-2288 97 94 + + CCONJ iajs-2288 97 95 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 97 96 ) ) PUNCT iajs-2288 97 97 where where SCONJ iajs-2288 97 98 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 97 99 ) ) PUNCT iajs-2288 97 100 = = SYM iajs-2288 97 101 ( ( PUNCT iajs-2288 97 102 0 0 NUM iajs-2288 97 103 ) ) PUNCT iajs-2288 97 104 . . PUNCT iajs-2288 98 1 proposition proposition NOUN iajs-2288 98 2 ( ( PUNCT iajs-2288 98 3 13 13 NUM iajs-2288 98 4 ) ) PUNCT iajs-2288 98 5 let let VERB iajs-2288 98 6 𝐿 𝐿 PROPN iajs-2288 98 7 and and CCONJ iajs-2288 98 8 𝐸 𝐸 PROPN iajs-2288 98 9 are be AUX iajs-2288 98 10 two two NUM iajs-2288 98 11 app app ADJ iajs-2288 98 12 - - PUNCT iajs-2288 98 13 semi semi ADJ iajs-2288 98 14 - - ADJ iajs-2288 98 15 prime prime ADJ iajs-2288 98 16 submodules submodule NOUN iajs-2288 98 17 of of ADP iajs-2288 98 18 an an DET iajs-2288 98 19 𝑅-module 𝑅-module PROPN iajs-2288 98 20 𝑇 𝑇 PROPN iajs-2288 98 21 with with ADP iajs-2288 98 22 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 98 23 ) ) PUNCT iajs-2288 98 24 βŠ† βŠ† NUM iajs-2288 98 25 𝐿 𝐿 PROPN iajs-2288 98 26 or or CCONJ iajs-2288 98 27 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 98 28 ) ) PUNCT iajs-2288 98 29 βŠ† βŠ† NUM iajs-2288 98 30 𝐸. 𝐸. PROPN iajs-2288 98 31 then then ADV iajs-2288 98 32 𝐿 𝐿 PROPN iajs-2288 98 33 ∩ ∩ NOUN iajs-2288 98 34 𝐸 𝐸 PROPN iajs-2288 98 35 is be AUX iajs-2288 98 36 an an DET iajs-2288 98 37 app app ADJ iajs-2288 98 38 - - PUNCT iajs-2288 98 39 semi semi ADJ iajs-2288 98 40 - - ADJ iajs-2288 98 41 prime prime ADJ iajs-2288 98 42 submodule submodule NOUN iajs-2288 98 43 of of ADP iajs-2288 98 44 𝑇. 𝑇. PROPN iajs-2288 98 45 121 121 NUM iajs-2288 98 46 ibn ibn PROPN iajs-2288 98 47 al al PROPN iajs-2288 98 48 - - PUNCT iajs-2288 98 49 haitham haitham PROPN iajs-2288 98 50 jour jour X iajs-2288 98 51 . . PROPN iajs-2288 99 1 for for ADP iajs-2288 99 2 pure pure ADJ iajs-2288 99 3 & & CCONJ iajs-2288 99 4 appl appl PROPN iajs-2288 99 5 . . PUNCT iajs-2288 100 1 sci sci PROPN iajs-2288 100 2 . . PROPN iajs-2288 100 3 32 32 NUM iajs-2288 100 4 ( ( PUNCT iajs-2288 100 5 3 3 NUM iajs-2288 100 6 ) ) PUNCT iajs-2288 100 7 2019 2019 NUM iajs-2288 100 8 proof proof NOUN iajs-2288 100 9 suppose suppose VERB iajs-2288 100 10 that that SCONJ iajs-2288 100 11 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 100 12 ∈ ∈ PROPN iajs-2288 100 13 𝐿 𝐿 PROPN iajs-2288 100 14 ∩ ∩ ADJ iajs-2288 100 15 𝐸 𝐸 PROPN iajs-2288 100 16 , , PUNCT iajs-2288 100 17 where where SCONJ iajs-2288 100 18 π‘Ž π‘Ž PROPN iajs-2288 100 19 ∈ ∈ PROPN iajs-2288 100 20 𝑅 𝑅 PROPN iajs-2288 100 21 , , PUNCT iajs-2288 100 22 𝑑 𝑑 PROPN iajs-2288 100 23 ∈ ∈ PROPN iajs-2288 100 24 𝑇 𝑇 PROPN iajs-2288 100 25 and and CCONJ iajs-2288 100 26 𝑛 𝑛 DET iajs-2288 100 27 ∈ ∈ PROPN iajs-2288 100 28 𝑍+ 𝑍+ NOUN iajs-2288 100 29 , , PUNCT iajs-2288 100 30 implies imply VERB iajs-2288 100 31 that that SCONJ iajs-2288 100 32 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 100 33 ∈ ∈ PROPN iajs-2288 100 34 𝐿 𝐿 PROPN iajs-2288 100 35 and and CCONJ iajs-2288 100 36 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 100 37 ∈ ∈ PROPN iajs-2288 100 38 𝐿. 𝐿. PROPN iajs-2288 100 39 but but CCONJ iajs-2288 100 40 𝐿 𝐿 PROPN iajs-2288 100 41 and and CCONJ iajs-2288 100 42 𝐸 𝐸 PROPN iajs-2288 100 43 are be AUX iajs-2288 100 44 app app ADJ iajs-2288 100 45 - - PUNCT iajs-2288 100 46 semi semi ADJ iajs-2288 100 47 - - ADJ iajs-2288 100 48 prime prime ADJ iajs-2288 100 49 submodules submodule NOUN iajs-2288 100 50 of of ADP iajs-2288 100 51 𝑇 𝑇 PROPN iajs-2288 100 52 , , PUNCT iajs-2288 100 53 then then ADV iajs-2288 100 54 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 100 55 ∈ ∈ PROPN iajs-2288 100 56 𝐿 𝐿 PROPN iajs-2288 100 57 + + NOUN iajs-2288 100 58 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 100 59 ) ) PUNCT iajs-2288 100 60 and and CCONJ iajs-2288 100 61 π‘Žπ‘‘ π‘Žπ‘‘ PRON iajs-2288 100 62 ∈ ∈ PROPN iajs-2288 100 63 𝐸 𝐸 PROPN iajs-2288 100 64 + + CCONJ iajs-2288 100 65 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 100 66 ) ) PUNCT iajs-2288 100 67 , , PUNCT iajs-2288 100 68 it it PRON iajs-2288 100 69 follows follow VERB iajs-2288 100 70 that that SCONJ iajs-2288 101 1 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 101 2 ∈ ∈ PROPN iajs-2288 101 3 ( ( PUNCT iajs-2288 101 4 𝐿 𝐿 PROPN iajs-2288 101 5 + + NOUN iajs-2288 101 6 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 101 7 ) ) PUNCT iajs-2288 101 8 ) ) PUNCT iajs-2288 101 9 ∩ ∩ NOUN iajs-2288 101 10 ( ( PUNCT iajs-2288 101 11 𝐸 𝐸 PROPN iajs-2288 101 12 + + NUM iajs-2288 101 13 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 101 14 ) ) PUNCT iajs-2288 101 15 ) ) PUNCT iajs-2288 101 16 . . PUNCT iajs-2288 102 1 if if SCONJ iajs-2288 102 2 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 102 3 ) ) PUNCT iajs-2288 102 4 βŠ† βŠ† NUM iajs-2288 102 5 𝐸 𝐸 PROPN iajs-2288 102 6 , , PUNCT iajs-2288 102 7 then then ADV iajs-2288 102 8 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 102 9 ∈ ∈ PROPN iajs-2288 102 10 ( ( PUNCT iajs-2288 102 11 𝐿 𝐿 PROPN iajs-2288 102 12 + + NOUN iajs-2288 102 13 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 102 14 ) ) PUNCT iajs-2288 102 15 ) ) PUNCT iajs-2288 102 16 ∩ ∩ PROPN iajs-2288 102 17 𝐸 𝐸 PROPN iajs-2288 102 18 and and CCONJ iajs-2288 102 19 by by ADP iajs-2288 102 20 lemma lemma PROPN iajs-2288 102 21 ( ( PUNCT iajs-2288 102 22 2.8 2.8 NUM iajs-2288 102 23 ) ) PUNCT iajs-2288 102 24 we we PRON iajs-2288 102 25 have have VERB iajs-2288 102 26 π‘Žπ‘‘ π‘Žπ‘‘ DET iajs-2288 102 27 ∈ ∈ PROPN iajs-2288 102 28 ( ( PUNCT iajs-2288 102 29 𝐿 𝐿 PROPN iajs-2288 102 30 ∩ ∩ ADJ iajs-2288 102 31 𝐸 𝐸 PROPN iajs-2288 102 32 ) ) PUNCT iajs-2288 102 33 + + NUM iajs-2288 102 34 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 102 35 ) ) PUNCT iajs-2288 102 36 . . PUNCT iajs-2288 103 1 similarly similarly ADV iajs-2288 103 2 if if SCONJ iajs-2288 103 3 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 103 4 ) ) PUNCT iajs-2288 103 5 βŠ† βŠ† X iajs-2288 103 6 𝐿 𝐿 PROPN iajs-2288 103 7 , , PUNCT iajs-2288 103 8 we we PRON iajs-2288 103 9 get get VERB iajs-2288 103 10 π‘Žπ‘‘ π‘Žπ‘‘ DET iajs-2288 103 11 ∈ ∈ PROPN iajs-2288 103 12 ( ( PUNCT iajs-2288 103 13 𝐿 𝐿 PROPN iajs-2288 103 14 ∩ ∩ ADJ iajs-2288 103 15 𝐸 𝐸 PROPN iajs-2288 103 16 ) ) PUNCT iajs-2288 103 17 + + NUM iajs-2288 103 18 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 103 19 ) ) PUNCT iajs-2288 103 20 . . PUNCT iajs-2288 104 1 hence hence ADV iajs-2288 104 2 𝐿 𝐿 PROPN iajs-2288 104 3 ∩ ∩ NOUN iajs-2288 104 4 𝐸 𝐸 PROPN iajs-2288 104 5 is be AUX iajs-2288 104 6 an an DET iajs-2288 104 7 app app ADJ iajs-2288 104 8 - - PUNCT iajs-2288 104 9 semi semi ADJ iajs-2288 104 10 - - ADJ iajs-2288 104 11 prime prime ADJ iajs-2288 104 12 submodule submodule NOUN iajs-2288 104 13 of of ADP iajs-2288 104 14 𝑇. 𝑇. PROPN iajs-2288 104 15 remark remark NOUN iajs-2288 104 16 ( ( PUNCT iajs-2288 104 17 14 14 NUM iajs-2288 104 18 ) ) PUNCT iajs-2288 104 19 if if SCONJ iajs-2288 104 20 𝐿 𝐿 PROPN iajs-2288 104 21 and and CCONJ iajs-2288 104 22 𝐸 𝐸 PROPN iajs-2288 104 23 are be AUX iajs-2288 104 24 submodules submodule NOUN iajs-2288 104 25 of of ADP iajs-2288 104 26 an an DET iajs-2288 104 27 𝑅-module 𝑅-module PROPN iajs-2288 104 28 𝑇 𝑇 PROPN iajs-2288 104 29 with with ADP iajs-2288 104 30 𝐿 𝐿 PROPN iajs-2288 104 31 βŠ† βŠ† NUM iajs-2288 104 32 𝐸 𝐸 PROPN iajs-2288 104 33 , , PUNCT iajs-2288 104 34 and and CCONJ iajs-2288 104 35 𝐸 𝐸 PROPN iajs-2288 104 36 ia ia PROPN iajs-2288 104 37 an an DET iajs-2288 104 38 app app ADJ iajs-2288 104 39 - - PUNCT iajs-2288 104 40 semi semi ADJ iajs-2288 104 41 - - ADJ iajs-2288 104 42 prime prime ADJ iajs-2288 104 43 submodule submodule NOUN iajs-2288 104 44 of of ADP iajs-2288 104 45 𝑇 𝑇 PROPN iajs-2288 104 46 , , PUNCT iajs-2288 104 47 then then ADV iajs-2288 104 48 𝐿 𝐿 PROPN iajs-2288 104 49 is be AUX iajs-2288 104 50 not not PART iajs-2288 104 51 an an DET iajs-2288 104 52 app app ADJ iajs-2288 104 53 - - PUNCT iajs-2288 104 54 semi semi ADJ iajs-2288 104 55 - - ADJ iajs-2288 104 56 prime prime ADJ iajs-2288 104 57 submodule submodule NOUN iajs-2288 104 58 of of ADP iajs-2288 104 59 𝑇 𝑇 PROPN iajs-2288 104 60 , , PUNCT iajs-2288 104 61 the the DET iajs-2288 104 62 following follow VERB iajs-2288 104 63 example example NOUN iajs-2288 104 64 shows show VERB iajs-2288 104 65 that that PRON iajs-2288 104 66 . . PUNCT iajs-2288 105 1 let let VERB iajs-2288 105 2 𝐿 𝐿 PROPN iajs-2288 105 3 = = PROPN iajs-2288 105 4 12𝑍 12𝑍 NUM iajs-2288 105 5 and and CCONJ iajs-2288 105 6 𝐸 𝐸 PROPN iajs-2288 105 7 = = PUNCT iajs-2288 105 8 6𝑍 6𝑍 NOUN iajs-2288 105 9 are be AUX iajs-2288 105 10 submodules submodule NOUN iajs-2288 105 11 of of ADP iajs-2288 105 12 a a DET iajs-2288 105 13 𝑍-module 𝑍-module PROPN iajs-2288 105 14 𝑍 𝑍 NOUN iajs-2288 105 15 , , PUNCT iajs-2288 105 16 𝐿 𝐿 PROPN iajs-2288 105 17 βŠ† βŠ† NUM iajs-2288 105 18 𝐸 𝐸 PROPN iajs-2288 105 19 and and CCONJ iajs-2288 105 20 𝐸 𝐸 PROPN iajs-2288 105 21 is be AUX iajs-2288 105 22 an an DET iajs-2288 105 23 app app ADJ iajs-2288 105 24 - - PUNCT iajs-2288 105 25 semiprime semiprime NOUN iajs-2288 105 26 submodule submodule NOUN iajs-2288 105 27 of of ADP iajs-2288 105 28 z z PROPN iajs-2288 105 29 , , PUNCT iajs-2288 105 30 but but CCONJ iajs-2288 105 31 𝐿 𝐿 PROPN iajs-2288 105 32 = = PROPN iajs-2288 105 33 12𝑍 12𝑍 NUM iajs-2288 105 34 is be AUX iajs-2288 105 35 not not PART iajs-2288 105 36 an an DET iajs-2288 105 37 app app ADJ iajs-2288 105 38 - - PUNCT iajs-2288 105 39 semi semi ADJ iajs-2288 105 40 - - ADJ iajs-2288 105 41 prime prime ADJ iajs-2288 105 42 submodule submodule NOUN iajs-2288 105 43 because because SCONJ iajs-2288 105 44 22 22 NUM iajs-2288 105 45 . . SYM iajs-2288 105 46 3 3 NUM iajs-2288 105 47 ∈ ∈ PROPN iajs-2288 105 48 12𝑍 12𝑍 NUM iajs-2288 105 49 , , PUNCT iajs-2288 105 50 but but CCONJ iajs-2288 105 51 2.3 2.3 NUM iajs-2288 105 52 βˆ‰ βˆ‰ X iajs-2288 105 53 12𝑍 12𝑍 NUM iajs-2288 105 54 + + CCONJ iajs-2288 105 55 π‘ π‘œπ‘(𝑍 π‘ π‘œπ‘(𝑍 NUM iajs-2288 105 56 ) ) PUNCT iajs-2288 105 57 . . PUNCT iajs-2288 106 1 recall recall VERB iajs-2288 106 2 that that SCONJ iajs-2288 106 3 an an DET iajs-2288 106 4 𝑅-module 𝑅-module PROPN iajs-2288 106 5 𝑇 𝑇 PROPN iajs-2288 106 6 is be AUX iajs-2288 106 7 multiplication multiplication NOUN iajs-2288 106 8 if if SCONJ iajs-2288 106 9 every every DET iajs-2288 106 10 submodule submodule NOUN iajs-2288 106 11 𝐿 𝐿 PROPN iajs-2288 106 12 of of ADP iajs-2288 106 13 𝑇 𝑇 PROPN iajs-2288 106 14 is be AUX iajs-2288 106 15 of of ADP iajs-2288 106 16 the the DET iajs-2288 106 17 form form NOUN iajs-2288 106 18 𝐿 𝐿 PROPN iajs-2288 106 19 = = PROPN iajs-2288 106 20 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 106 21 for for ADP iajs-2288 106 22 some some DET iajs-2288 106 23 ideal ideal NOUN iajs-2288 106 24 𝐼 𝐼 ADP iajs-2288 106 25 of of ADP iajs-2288 106 26 𝑅. 𝑅. NOUN iajs-2288 106 27 in in ADP iajs-2288 106 28 particular particular ADJ iajs-2288 106 29 𝐿 𝐿 NOUN iajs-2288 106 30 = = SYM iajs-2288 107 1 [ [ X iajs-2288 107 2 𝐿:𝑅 𝐿:𝑅 X iajs-2288 107 3 𝑇]𝑇 𝑇]𝑇 PROPN iajs-2288 108 1 [ [ X iajs-2288 108 2 14 14 NUM iajs-2288 108 3 ] ] PUNCT iajs-2288 108 4 . . PUNCT iajs-2288 109 1 recall recall VERB iajs-2288 109 2 that that PRON iajs-2288 109 3 for for ADP iajs-2288 109 4 any any DET iajs-2288 109 5 submodules submodule NOUN iajs-2288 109 6 𝐾 𝐾 PROPN iajs-2288 109 7 and and CCONJ iajs-2288 109 8 𝐹 𝐹 PROPN iajs-2288 109 9 of of ADP iajs-2288 109 10 a a DET iajs-2288 109 11 multiplication multiplication NOUN iajs-2288 109 12 𝑅-module 𝑅-module ADP iajs-2288 109 13 𝑇 𝑇 PROPN iajs-2288 109 14 with with ADP iajs-2288 109 15 𝐾 𝐾 PROPN iajs-2288 109 16 = = PUNCT iajs-2288 109 17 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 109 18 and and CCONJ iajs-2288 109 19 𝐹 𝐹 PROPN iajs-2288 109 20 = = SYM iajs-2288 109 21 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 109 22 for for ADP iajs-2288 109 23 some some DET iajs-2288 109 24 ideals ideal NOUN iajs-2288 109 25 𝐼 𝐼 PROPN iajs-2288 109 26 and and CCONJ iajs-2288 109 27 𝐽 𝐽 PROPN iajs-2288 109 28 of of ADP iajs-2288 109 29 𝑅. 𝑅. NOUN iajs-2288 109 30 the the DET iajs-2288 109 31 product product NOUN iajs-2288 109 32 𝐾𝐹 𝐾𝐹 PROPN iajs-2288 109 33 = = SYM iajs-2288 109 34 𝐼𝑇. 𝐼𝑇. X iajs-2288 109 35 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 109 36 = = SYM iajs-2288 109 37 𝐼𝐽𝑇 𝐼𝐽𝑇 PROPN iajs-2288 109 38 that that PRON iajs-2288 109 39 is be AUX iajs-2288 109 40 𝐾𝐹 𝐾𝐹 PROPN iajs-2288 109 41 = = SYM iajs-2288 109 42 𝐼𝐹. 𝐼𝐹. X iajs-2288 109 43 in in ADP iajs-2288 109 44 particular particular ADJ iajs-2288 109 45 𝐾𝑇 𝐾𝑇 PROPN iajs-2288 109 46 = = SYM iajs-2288 109 47 𝐼𝑇𝑇 𝐼𝑇𝑇 PROPN iajs-2288 109 48 = = SYM iajs-2288 109 49 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 109 50 = = SYM iajs-2288 109 51 𝐾. 𝐾. PROPN iajs-2288 109 52 also also ADV iajs-2288 109 53 for for ADP iajs-2288 109 54 any any DET iajs-2288 109 55 𝑑 𝑑 PROPN iajs-2288 109 56 ∈ ∈ PROPN iajs-2288 109 57 𝑇 𝑇 PROPN iajs-2288 109 58 , , PUNCT iajs-2288 109 59 we we PRON iajs-2288 109 60 have have VERB iajs-2288 109 61 𝐾𝑑 𝐾𝑑 X iajs-2288 109 62 = = SYM iajs-2288 109 63 πΎβŒ©π‘‘ πΎβŒ©π‘‘ X iajs-2288 109 64 βŒͺ βŒͺ NOUN iajs-2288 109 65 = = PUNCT iajs-2288 110 1 𝐼𝑑 𝐼𝑑 ADP iajs-2288 110 2 [ [ X iajs-2288 110 3 15 15 NUM iajs-2288 110 4 ] ] PUNCT iajs-2288 110 5 . . PUNCT iajs-2288 111 1 the the DET iajs-2288 111 2 following follow VERB iajs-2288 111 3 propositions proposition NOUN iajs-2288 111 4 are be AUX iajs-2288 111 5 characterizations characterization NOUN iajs-2288 111 6 of of ADP iajs-2288 111 7 app app NOUN iajs-2288 111 8 - - PUNCT iajs-2288 111 9 semi semi ADJ iajs-2288 111 10 - - ADJ iajs-2288 111 11 prime prime ADJ iajs-2288 111 12 submodules submodule NOUN iajs-2288 111 13 in in ADP iajs-2288 111 14 the the DET iajs-2288 111 15 class class NOUN iajs-2288 111 16 of of ADP iajs-2288 111 17 multiplication multiplication NOUN iajs-2288 111 18 modules module NOUN iajs-2288 111 19 . . PUNCT iajs-2288 112 1 proposition proposition NOUN iajs-2288 112 2 ( ( PUNCT iajs-2288 112 3 15 15 NUM iajs-2288 112 4 ) ) PUNCT iajs-2288 112 5 let let VERB iajs-2288 112 6 𝐿 𝐿 PROPN iajs-2288 112 7 be be AUX iajs-2288 112 8 a a DET iajs-2288 112 9 proper proper ADJ iajs-2288 112 10 submodule submodule NOUN iajs-2288 112 11 of of ADP iajs-2288 112 12 a a DET iajs-2288 112 13 multiplication multiplication NOUN iajs-2288 112 14 𝑅-module 𝑅-module PROPN iajs-2288 112 15 𝑇. 𝑇. PROPN iajs-2288 112 16 then then ADV iajs-2288 112 17 𝐿 𝐿 PROPN iajs-2288 112 18 is be AUX iajs-2288 112 19 an an DET iajs-2288 112 20 app app ADJ iajs-2288 112 21 - - PUNCT iajs-2288 112 22 semi semi ADJ iajs-2288 112 23 - - ADJ iajs-2288 112 24 prime prime ADJ iajs-2288 112 25 submodule submodule NOUN iajs-2288 112 26 of of ADP iajs-2288 112 27 𝑇 𝑇 PROPN iajs-2288 113 1 if if SCONJ iajs-2288 113 2 and and CCONJ iajs-2288 113 3 only only ADV iajs-2288 113 4 if if SCONJ iajs-2288 113 5 𝐾𝑛𝐹 𝐾𝑛𝐹 PROPN iajs-2288 113 6 βŠ† βŠ† NUM iajs-2288 113 7 𝐿 𝐿 PROPN iajs-2288 113 8 implies imply VERB iajs-2288 113 9 that that SCONJ iajs-2288 113 10 𝐾𝐹 𝐾𝐹 PROPN iajs-2288 113 11 βŠ† βŠ† NUM iajs-2288 113 12 𝐿 𝐿 PROPN iajs-2288 113 13 + + NOUN iajs-2288 113 14 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 113 15 ) ) PUNCT iajs-2288 113 16 , , PUNCT iajs-2288 113 17 where where SCONJ iajs-2288 113 18 𝐾 𝐾 PROPN iajs-2288 113 19 , , PUNCT iajs-2288 113 20 𝐹 𝐹 PROPN iajs-2288 113 21 are be AUX iajs-2288 113 22 submodules submodule NOUN iajs-2288 113 23 of of ADP iajs-2288 113 24 𝑇 𝑇 PROPN iajs-2288 113 25 , , PUNCT iajs-2288 113 26 𝑛 𝑛 PRON iajs-2288 113 27 ∈ ∈ PROPN iajs-2288 113 28 𝑍+ 𝑍+ NOUN iajs-2288 113 29 . . PUNCT iajs-2288 114 1 proof proof NOUN iajs-2288 114 2 ( ( PUNCT iajs-2288 114 3 β‡’ β‡’ PROPN iajs-2288 114 4 ) ) PUNCT iajs-2288 114 5 suppose suppose VERB iajs-2288 114 6 that that SCONJ iajs-2288 114 7 𝐾𝑛𝐹 𝐾𝑛𝐹 PROPN iajs-2288 114 8 βŠ† βŠ† NUM iajs-2288 114 9 𝐿 𝐿 PROPN iajs-2288 114 10 , , PUNCT iajs-2288 114 11 where where SCONJ iajs-2288 114 12 𝐾 𝐾 PROPN iajs-2288 114 13 , , PUNCT iajs-2288 114 14 𝐹 𝐹 PROPN iajs-2288 114 15 are be AUX iajs-2288 114 16 submodules submodule NOUN iajs-2288 114 17 of of ADP iajs-2288 114 18 𝑇 𝑇 PROPN iajs-2288 114 19 , , PUNCT iajs-2288 114 20 𝑛 𝑛 DET iajs-2288 114 21 ∈ ∈ PROPN iajs-2288 114 22 𝑍+ 𝑍+ NOUN iajs-2288 114 23 . . PUNCT iajs-2288 115 1 since since SCONJ iajs-2288 115 2 𝑇 𝑇 PROPN iajs-2288 115 3 is be AUX iajs-2288 115 4 a a DET iajs-2288 115 5 multiplication multiplication NOUN iajs-2288 115 6 then then ADV iajs-2288 115 7 𝐾 𝐾 PROPN iajs-2288 115 8 = = PUNCT iajs-2288 115 9 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 115 10 and and CCONJ iajs-2288 115 11 𝐹 𝐹 PROPN iajs-2288 115 12 = = SYM iajs-2288 115 13 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 115 14 for for ADP iajs-2288 115 15 some some DET iajs-2288 115 16 ideals ideal NOUN iajs-2288 115 17 𝐼 𝐼 PROPN iajs-2288 115 18 , , PUNCT iajs-2288 115 19 𝐽 𝐽 PROPN iajs-2288 115 20 of of ADP iajs-2288 115 21 𝑅. 𝑅. NOUN iajs-2288 115 22 thus thus ADV iajs-2288 115 23 𝐾𝑛𝐹 𝐾𝑛𝐹 PROPN iajs-2288 115 24 = = SYM iajs-2288 115 25 ( ( PUNCT iajs-2288 115 26 𝐼𝑇)𝑛𝐽𝑇 𝐼𝑇)𝑛𝐽𝑇 PROPN iajs-2288 115 27 = = PUNCT iajs-2288 115 28 𝐼𝑛(𝐽𝑇 𝐼𝑛(𝐽𝑇 PROPN iajs-2288 115 29 ) ) PUNCT iajs-2288 115 30 βŠ† βŠ† NUM iajs-2288 115 31 𝐿. 𝐿. NOUN iajs-2288 115 32 but but CCONJ iajs-2288 115 33 𝐿 𝐿 PROPN iajs-2288 115 34 is be AUX iajs-2288 115 35 an an DET iajs-2288 115 36 app app ADJ iajs-2288 115 37 - - PUNCT iajs-2288 115 38 semi semi NOUN iajs-2288 115 39 - - ADJ iajs-2288 115 40 prime prime ADJ iajs-2288 115 41 , , PUNCT iajs-2288 115 42 then then ADV iajs-2288 115 43 by by ADP iajs-2288 115 44 proposition proposition NOUN iajs-2288 115 45 ( ( PUNCT iajs-2288 115 46 2.3 2.3 NUM iajs-2288 115 47 ) ) PUNCT iajs-2288 115 48 we we PRON iajs-2288 115 49 have have VERB iajs-2288 115 50 𝐼(𝐽𝑇 𝐼(𝐽𝑇 NOUN iajs-2288 115 51 ) ) PUNCT iajs-2288 115 52 βŠ† βŠ† NUM iajs-2288 115 53 𝐿 𝐿 PROPN iajs-2288 115 54 + + NOUN iajs-2288 115 55 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 115 56 ) ) PUNCT iajs-2288 115 57 . . PUNCT iajs-2288 116 1 that that PRON iajs-2288 116 2 is be AUX iajs-2288 116 3 𝐼𝐹 𝐼𝐹 PROPN iajs-2288 116 4 βŠ† βŠ† NUM iajs-2288 116 5 𝐿 𝐿 PROPN iajs-2288 116 6 + + NOUN iajs-2288 116 7 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 116 8 ) ) PUNCT iajs-2288 116 9 , , PUNCT iajs-2288 116 10 so so ADV iajs-2288 116 11 𝐾𝐹 𝐾𝐹 PROPN iajs-2288 116 12 βŠ† βŠ† NUM iajs-2288 116 13 𝐿 𝐿 PROPN iajs-2288 116 14 + + NOUN iajs-2288 116 15 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 116 16 ) ) PUNCT iajs-2288 116 17 . . PUNCT iajs-2288 117 1 ( ( PUNCT iajs-2288 117 2 ⇐ ⇐ PROPN iajs-2288 117 3 ) ) PUNCT iajs-2288 117 4 suppose suppose VERB iajs-2288 117 5 that that SCONJ iajs-2288 117 6 𝐼𝑛𝐹 𝐼𝑛𝐹 PROPN iajs-2288 117 7 βŠ† βŠ† NUM iajs-2288 117 8 𝐿 𝐿 PROPN iajs-2288 117 9 , , PUNCT iajs-2288 117 10 where where SCONJ iajs-2288 117 11 𝐼 𝐼 PROPN iajs-2288 117 12 is be AUX iajs-2288 117 13 an an DET iajs-2288 117 14 ideal ideal NOUN iajs-2288 117 15 of of ADP iajs-2288 117 16 𝑅 𝑅 PROPN iajs-2288 117 17 , , PUNCT iajs-2288 117 18 𝐹 𝐹 PROPN iajs-2288 117 19 is be AUX iajs-2288 117 20 a a DET iajs-2288 117 21 submodule submodule NOUN iajs-2288 117 22 of of ADP iajs-2288 117 23 𝑇 𝑇 PROPN iajs-2288 117 24 and and CCONJ iajs-2288 117 25 𝑛 𝑛 PRON iajs-2288 117 26 ∈ ∈ PROPN iajs-2288 117 27 𝑍+ 𝑍+ NOUN iajs-2288 117 28 . . PUNCT iajs-2288 118 1 since since SCONJ iajs-2288 118 2 𝑇 𝑇 PROPN iajs-2288 118 3 is be AUX iajs-2288 118 4 a a DET iajs-2288 118 5 multiplication multiplication NOUN iajs-2288 118 6 then then ADV iajs-2288 118 7 𝐹 𝐹 PROPN iajs-2288 118 8 = = PUNCT iajs-2288 118 9 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 118 10 for for ADP iajs-2288 118 11 some some DET iajs-2288 118 12 ideal ideal ADJ iajs-2288 118 13 𝐽 𝐽 NOUN iajs-2288 118 14 of of ADP iajs-2288 118 15 𝑅. 𝑅. NOUN iajs-2288 118 16 that that PRON iajs-2288 118 17 is be AUX iajs-2288 118 18 𝐼𝑛𝐽𝑇 𝐼𝑛𝐽𝑇 VERB iajs-2288 118 19 βŠ† βŠ† NUM iajs-2288 118 20 𝐿 𝐿 PROPN iajs-2288 118 21 implies imply VERB iajs-2288 118 22 that that SCONJ iajs-2288 118 23 𝐾𝑛𝐹 𝐾𝑛𝐹 PROPN iajs-2288 118 24 βŠ† βŠ† NUM iajs-2288 118 25 𝐿 𝐿 NOUN iajs-2288 118 26 so so ADV iajs-2288 118 27 by by ADP iajs-2288 118 28 hypothesis hypothesis NOUN iajs-2288 118 29 𝐾𝐹 𝐾𝐹 PROPN iajs-2288 118 30 βŠ† βŠ† NUM iajs-2288 118 31 𝐿 𝐿 PROPN iajs-2288 118 32 + + NOUN iajs-2288 118 33 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 118 34 ) ) PUNCT iajs-2288 118 35 , , PUNCT iajs-2288 118 36 thus thus ADV iajs-2288 118 37 𝐼𝐹 𝐼𝐹 PROPN iajs-2288 118 38 βŠ† βŠ† NUM iajs-2288 118 39 𝐿 𝐿 PROPN iajs-2288 118 40 + + NOUN iajs-2288 118 41 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 118 42 ) ) PUNCT iajs-2288 118 43 . . PUNCT iajs-2288 119 1 hence hence ADV iajs-2288 119 2 by by ADP iajs-2288 119 3 proposition proposition NOUN iajs-2288 119 4 ( ( PUNCT iajs-2288 119 5 2.3 2.3 NUM iajs-2288 119 6 ) ) PUNCT iajs-2288 119 7 𝐿 𝐿 PROPN iajs-2288 119 8 is be AUX iajs-2288 119 9 an an DET iajs-2288 119 10 app app ADJ iajs-2288 119 11 - - PUNCT iajs-2288 119 12 semi semi ADJ iajs-2288 119 13 - - ADJ iajs-2288 119 14 prime prime ADJ iajs-2288 119 15 submodule submodule NOUN iajs-2288 119 16 of of ADP iajs-2288 119 17 𝑇. 𝑇. PROPN iajs-2288 119 18 proposition proposition NOUN iajs-2288 119 19 ( ( PUNCT iajs-2288 119 20 16 16 NUM iajs-2288 119 21 ) ) PUNCT iajs-2288 119 22 let let VERB iajs-2288 119 23 𝐿 𝐿 PROPN iajs-2288 119 24 be be AUX iajs-2288 119 25 a a DET iajs-2288 119 26 proper proper ADJ iajs-2288 119 27 submodule submodule NOUN iajs-2288 119 28 of of ADP iajs-2288 119 29 a a DET iajs-2288 119 30 multiplication multiplication NOUN iajs-2288 119 31 𝑅-module 𝑅-module PROPN iajs-2288 119 32 𝑇. 𝑇. PROPN iajs-2288 119 33 then then ADV iajs-2288 119 34 the the DET iajs-2288 119 35 following follow VERB iajs-2288 119 36 statements statement NOUN iajs-2288 119 37 are be AUX iajs-2288 119 38 equivalent equivalent ADJ iajs-2288 119 39 : : PUNCT iajs-2288 119 40 1 1 X iajs-2288 119 41 ) ) PUNCT iajs-2288 119 42 𝐿 𝐿 PROPN iajs-2288 119 43 is be AUX iajs-2288 119 44 an an DET iajs-2288 119 45 app app ADJ iajs-2288 119 46 - - PUNCT iajs-2288 119 47 semi semi ADJ iajs-2288 119 48 - - ADJ iajs-2288 119 49 prime prime ADJ iajs-2288 119 50 submodule submodule NOUN iajs-2288 119 51 of of ADP iajs-2288 119 52 𝑇. 𝑇. PROPN iajs-2288 119 53 2 2 NUM iajs-2288 119 54 ) ) PUNCT iajs-2288 119 55 𝑑𝑛 𝑑𝑛 VERB iajs-2288 119 56 ∈ ∈ PROPN iajs-2288 119 57 𝐿 𝐿 PROPN iajs-2288 119 58 implies imply VERB iajs-2288 119 59 that that SCONJ iajs-2288 119 60 𝑑 𝑑 PROPN iajs-2288 119 61 ∈ ∈ PROPN iajs-2288 119 62 𝐿 𝐿 PROPN iajs-2288 119 63 + + NOUN iajs-2288 119 64 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 119 65 ) ) PUNCT iajs-2288 119 66 for for ADP iajs-2288 119 67 every every DET iajs-2288 119 68 𝑑 𝑑 PROPN iajs-2288 119 69 ∈ ∈ PROPN iajs-2288 119 70 𝑇. 𝑇. PROPN iajs-2288 119 71 3 3 NUM iajs-2288 119 72 ) ) PUNCT iajs-2288 119 73 √𝐿 √𝐿 NOUN iajs-2288 119 74 βŠ† βŠ† NUM iajs-2288 119 75 𝐿 𝐿 PROPN iajs-2288 119 76 + + NOUN iajs-2288 119 77 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 119 78 ) ) PUNCT iajs-2288 119 79 . . PUNCT iajs-2288 120 1 4 4 X iajs-2288 120 2 ) ) PUNCT iajs-2288 120 3 𝐹1𝐹2 𝐹1𝐹2 NOUN iajs-2288 120 4 … … SYM iajs-2288 120 5 … … PUNCT iajs-2288 120 6 𝐹𝑗 𝐹𝑗 PROPN iajs-2288 120 7 βŠ† βŠ† NUM iajs-2288 120 8 𝐿 𝐿 PROPN iajs-2288 120 9 , , PUNCT iajs-2288 120 10 implies imply VERB iajs-2288 120 11 that that SCONJ iajs-2288 120 12 𝐹1 𝐹1 PROPN iajs-2288 120 13 ∩ ∩ PROPN iajs-2288 120 14 𝐹2 𝐹2 PROPN iajs-2288 120 15 ∩ ∩ NOUN iajs-2288 120 16 … … PUNCT iajs-2288 120 17 … … SYM iajs-2288 120 18 ∩ ∩ X iajs-2288 120 19 𝐹𝑗 𝐹𝑗 PROPN iajs-2288 120 20 βŠ† βŠ† NUM iajs-2288 120 21 𝐿 𝐿 PROPN iajs-2288 120 22 + + NOUN iajs-2288 120 23 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 120 24 ) ) PUNCT iajs-2288 120 25 for for ADP iajs-2288 120 26 every every DET iajs-2288 120 27 submodules submodule NOUN iajs-2288 120 28 𝐹1 𝐹1 NOUN iajs-2288 120 29 , , PUNCT iajs-2288 120 30 𝐹2 𝐹2 NOUN iajs-2288 120 31 , , PUNCT iajs-2288 120 32 … … PUNCT iajs-2288 120 33 … … PUNCT iajs-2288 120 34 , , PUNCT iajs-2288 120 35 𝐹𝑗 𝐹𝑗 PROPN iajs-2288 120 36 of of ADP iajs-2288 120 37 𝑇 𝑇 PROPN iajs-2288 120 38 and and CCONJ iajs-2288 120 39 𝑗 𝑗 PRON iajs-2288 120 40 ∈ ∈ PROPN iajs-2288 120 41 𝑍+ 𝑍+ NOUN iajs-2288 120 42 . . PUNCT iajs-2288 120 43 122 122 NUM iajs-2288 121 1 ibn ibn PROPN iajs-2288 121 2 al al PROPN iajs-2288 121 3 - - PUNCT iajs-2288 121 4 haitham haitham PROPN iajs-2288 121 5 jour jour X iajs-2288 121 6 . . PROPN iajs-2288 121 7 for for ADP iajs-2288 121 8 pure pure ADJ iajs-2288 121 9 & & CCONJ iajs-2288 121 10 appl appl PROPN iajs-2288 121 11 . . PUNCT iajs-2288 122 1 sci sci PROPN iajs-2288 122 2 . . PROPN iajs-2288 122 3 32 32 NUM iajs-2288 122 4 ( ( PUNCT iajs-2288 122 5 3 3 NUM iajs-2288 122 6 ) ) PUNCT iajs-2288 122 7 2019 2019 NUM iajs-2288 122 8 proof proof NOUN iajs-2288 122 9 ( ( PUNCT iajs-2288 122 10 1 1 X iajs-2288 122 11 ) ) PUNCT iajs-2288 122 12 β‡’ β‡’ NOUN iajs-2288 122 13 ( ( PUNCT iajs-2288 122 14 2 2 X iajs-2288 122 15 ) ) PUNCT iajs-2288 122 16 let let VERB iajs-2288 122 17 𝑑𝑛 𝑑𝑛 NUM iajs-2288 122 18 ∈ ∈ PROPN iajs-2288 122 19 𝐿 𝐿 PROPN iajs-2288 123 1 where where SCONJ iajs-2288 123 2 𝑑 𝑑 PROPN iajs-2288 123 3 ∈ ∈ PROPN iajs-2288 123 4 𝑇 𝑇 PROPN iajs-2288 123 5 and and CCONJ iajs-2288 123 6 for for ADP iajs-2288 123 7 some some PRON iajs-2288 123 8 𝑛 𝑛 DET iajs-2288 123 9 ∈ ∈ PROPN iajs-2288 123 10 𝑍+ 𝑍+ NOUN iajs-2288 123 11 , , PUNCT iajs-2288 123 12 then then ADV iajs-2288 123 13 〈 〈 NOUN iajs-2288 123 14 𝑑𝑛 𝑑𝑛 ADJ iajs-2288 123 15 βŒͺ βŒͺ NOUN iajs-2288 123 16 βŠ† βŠ† NUM iajs-2288 123 17 𝐿. 𝐿. VERB iajs-2288 123 18 but but CCONJ iajs-2288 123 19 𝑇 𝑇 PROPN iajs-2288 123 20 is be AUX iajs-2288 123 21 a a DET iajs-2288 123 22 multiplication multiplication NOUN iajs-2288 123 23 𝑅-module 𝑅-module NOUN iajs-2288 123 24 , , PUNCT iajs-2288 123 25 then then ADV iajs-2288 123 26 〈 〈 NOUN iajs-2288 123 27 𝑑 𝑑 X iajs-2288 123 28 βŒͺ βŒͺ ADJ iajs-2288 123 29 = = SYM iajs-2288 123 30 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 123 31 for for ADP iajs-2288 123 32 some some DET iajs-2288 123 33 ideal ideal ADJ iajs-2288 123 34 𝐼 𝐼 PROPN iajs-2288 123 35 of of ADP iajs-2288 123 36 𝑅 𝑅 PROPN iajs-2288 123 37 , , PUNCT iajs-2288 123 38 so so ADV iajs-2288 123 39 〈 〈 NOUN iajs-2288 123 40 𝑑𝑛 𝑑𝑛 ADJ iajs-2288 123 41 βŒͺ βŒͺ NOUN iajs-2288 123 42 = = NOUN iajs-2288 123 43 𝐼𝑛𝑇 𝐼𝑛𝑇 NOUN iajs-2288 123 44 βŠ† βŠ† NUM iajs-2288 123 45 𝐿. 𝐿. VERB iajs-2288 123 46 since since SCONJ iajs-2288 123 47 𝐿 𝐿 PROPN iajs-2288 123 48 is be AUX iajs-2288 123 49 an an DET iajs-2288 123 50 app app ADJ iajs-2288 123 51 - - PUNCT iajs-2288 123 52 semi semi ADJ iajs-2288 123 53 - - ADJ iajs-2288 123 54 prime prime ADJ iajs-2288 123 55 submodule submodule NOUN iajs-2288 123 56 of of ADP iajs-2288 123 57 𝑇 𝑇 PROPN iajs-2288 123 58 , , PUNCT iajs-2288 123 59 then then ADV iajs-2288 123 60 by by ADP iajs-2288 123 61 corollary corollary ADJ iajs-2288 123 62 ( ( PUNCT iajs-2288 123 63 2.4 2.4 NUM iajs-2288 123 64 ) ) PUNCT iajs-2288 123 65 we we PRON iajs-2288 123 66 have have VERB iajs-2288 123 67 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 123 68 βŠ† βŠ† NUM iajs-2288 123 69 𝐿 𝐿 PROPN iajs-2288 123 70 + + NOUN iajs-2288 123 71 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 123 72 ) ) PUNCT iajs-2288 123 73 . . PUNCT iajs-2288 124 1 that that PRON iajs-2288 124 2 is be AUX iajs-2288 124 3 〈 〈 NOUN iajs-2288 124 4 𝑑 𝑑 X iajs-2288 124 5 βŒͺ βŒͺ PRON iajs-2288 124 6 βŠ† βŠ† NUM iajs-2288 124 7 𝐿 𝐿 PROPN iajs-2288 124 8 + + NOUN iajs-2288 124 9 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 124 10 ) ) PUNCT iajs-2288 124 11 , , PUNCT iajs-2288 124 12 implies imply VERB iajs-2288 124 13 that that SCONJ iajs-2288 124 14 𝑑 𝑑 PROPN iajs-2288 124 15 βŠ† βŠ† NUM iajs-2288 124 16 𝐿 𝐿 PROPN iajs-2288 124 17 + + NOUN iajs-2288 124 18 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 124 19 ) ) PUNCT iajs-2288 124 20 . . PUNCT iajs-2288 125 1 ( ( PUNCT iajs-2288 125 2 2 2 X iajs-2288 125 3 ) ) PUNCT iajs-2288 125 4 β‡’ β‡’ NOUN iajs-2288 125 5 ( ( PUNCT iajs-2288 125 6 3 3 X iajs-2288 125 7 ) ) PUNCT iajs-2288 125 8 let let VERB iajs-2288 125 9 𝑑 𝑑 PRON iajs-2288 125 10 ∈ ∈ PROPN iajs-2288 125 11 √𝐿 √𝐿 NOUN iajs-2288 125 12 , , PUNCT iajs-2288 125 13 implies imply VERB iajs-2288 125 14 that that SCONJ iajs-2288 125 15 𝑑𝑛 𝑑𝑛 VERB iajs-2288 125 16 ∈ ∈ PROPN iajs-2288 125 17 𝐿 𝐿 PROPN iajs-2288 125 18 for for ADP iajs-2288 125 19 some some PRON iajs-2288 125 20 𝑛 𝑛 DET iajs-2288 125 21 ∈ ∈ PROPN iajs-2288 125 22 𝑍+ 𝑍+ NOUN iajs-2288 125 23 , , PUNCT iajs-2288 125 24 so so ADV iajs-2288 125 25 by by ADP iajs-2288 125 26 hypothesis hypothesis NOUN iajs-2288 125 27 𝑑 𝑑 PROPN iajs-2288 125 28 ∈ ∈ PROPN iajs-2288 125 29 𝐿 𝐿 PROPN iajs-2288 125 30 + + NOUN iajs-2288 125 31 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 125 32 ) ) PUNCT iajs-2288 125 33 . . PUNCT iajs-2288 126 1 thus thus ADV iajs-2288 126 2 √𝐿 √𝐿 X iajs-2288 126 3 βŠ† βŠ† NUM iajs-2288 126 4 𝐿 𝐿 PROPN iajs-2288 126 5 + + NOUN iajs-2288 126 6 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 126 7 ) ) PUNCT iajs-2288 126 8 . . PUNCT iajs-2288 127 1 ( ( PUNCT iajs-2288 127 2 3 3 X iajs-2288 127 3 ) ) PUNCT iajs-2288 127 4 β‡’ β‡’ NOUN iajs-2288 127 5 ( ( PUNCT iajs-2288 127 6 4 4 X iajs-2288 127 7 ) ) PUNCT iajs-2288 127 8 suppose suppose VERB iajs-2288 127 9 that that SCONJ iajs-2288 127 10 𝐹1𝐹2 𝐹1𝐹2 PROPN iajs-2288 127 11 … … SYM iajs-2288 127 12 … … PUNCT iajs-2288 128 1 𝐹𝑗 𝐹𝑗 PROPN iajs-2288 128 2 βŠ† βŠ† NUM iajs-2288 128 3 𝐿 𝐿 PROPN iajs-2288 128 4 where where SCONJ iajs-2288 128 5 𝐹1 𝐹1 NOUN iajs-2288 128 6 , , PUNCT iajs-2288 128 7 𝐹2 𝐹2 NOUN iajs-2288 128 8 , , PUNCT iajs-2288 128 9 … … PUNCT iajs-2288 128 10 … … PUNCT iajs-2288 128 11 , , PUNCT iajs-2288 128 12 𝐹𝑗 𝐹𝑗 PROPN iajs-2288 128 13 are be AUX iajs-2288 128 14 submodules submodule NOUN iajs-2288 128 15 of of ADP iajs-2288 128 16 𝑇 𝑇 PROPN iajs-2288 128 17 and and CCONJ iajs-2288 128 18 𝑗 𝑗 PRON iajs-2288 128 19 ∈ ∈ NOUN iajs-2288 128 20 𝑍+ 𝑍+ NOUN iajs-2288 128 21 . . PUNCT iajs-2288 129 1 let let VERB iajs-2288 130 1 𝑑 𝑑 PROPN iajs-2288 130 2 ∈ ∈ PROPN iajs-2288 130 3 𝐹1 𝐹1 PROPN iajs-2288 130 4 ∩ ∩ PROPN iajs-2288 130 5 𝐹2 𝐹2 PROPN iajs-2288 130 6 ∩ ∩ NOUN iajs-2288 130 7 … … PUNCT iajs-2288 130 8 … … SYM iajs-2288 130 9 ∩ ∩ X iajs-2288 130 10 𝐹𝑗 𝐹𝑗 PROPN iajs-2288 130 11 then then ADV iajs-2288 130 12 𝑑 𝑑 PROPN iajs-2288 130 13 ∈ ∈ PROPN iajs-2288 130 14 𝐹𝑖 𝐹𝑖 NOUN iajs-2288 130 15 for for ADP iajs-2288 130 16 each each DET iajs-2288 130 17 = = SYM iajs-2288 130 18 1,2 1,2 NUM iajs-2288 130 19 , , PUNCT iajs-2288 130 20 … … PUNCT iajs-2288 130 21 … … PUNCT iajs-2288 130 22 , , PUNCT iajs-2288 130 23 𝑗 𝑗 INTJ iajs-2288 130 24 , , PUNCT iajs-2288 130 25 so so ADV iajs-2288 130 26 𝑑𝑗 𝑑𝑗 PROPN iajs-2288 130 27 ∈ ∈ PROPN iajs-2288 130 28 𝐹1𝐹2 𝐹1𝐹2 PROPN iajs-2288 130 29 … … SYM iajs-2288 130 30 … … PUNCT iajs-2288 130 31 𝐹𝑗 𝐹𝑗 PROPN iajs-2288 130 32 βŠ† βŠ† NUM iajs-2288 130 33 𝐿 𝐿 PROPN iajs-2288 130 34 , , PUNCT iajs-2288 130 35 it it PRON iajs-2288 130 36 follows follow VERB iajs-2288 130 37 that that SCONJ iajs-2288 130 38 𝑑𝑗 𝑑𝑗 PROPN iajs-2288 130 39 ∈ ∈ PROPN iajs-2288 130 40 𝐿 𝐿 PROPN iajs-2288 130 41 , , PUNCT iajs-2288 130 42 so so SCONJ iajs-2288 130 43 𝑑 𝑑 PROPN iajs-2288 130 44 ∈ ∈ PROPN iajs-2288 130 45 √𝐿. √𝐿. NOUN iajs-2288 130 46 but but CCONJ iajs-2288 130 47 by by ADP iajs-2288 130 48 hypothesis hypothesis NOUN iajs-2288 130 49 √𝐿 √𝐿 NOUN iajs-2288 130 50 βŠ† βŠ† NUM iajs-2288 130 51 𝐿 𝐿 PROPN iajs-2288 130 52 + + NOUN iajs-2288 130 53 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 130 54 ) ) PUNCT iajs-2288 130 55 , , PUNCT iajs-2288 130 56 then then ADV iajs-2288 130 57 𝑑 𝑑 PROPN iajs-2288 130 58 ∈ ∈ PROPN iajs-2288 130 59 𝐿 𝐿 PROPN iajs-2288 130 60 + + NOUN iajs-2288 130 61 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 130 62 ) ) PUNCT iajs-2288 130 63 . . PUNCT iajs-2288 131 1 thus thus ADV iajs-2288 131 2 𝐹1 𝐹1 PROPN iajs-2288 131 3 ∩ ∩ PROPN iajs-2288 131 4 𝐹2 𝐹2 PROPN iajs-2288 131 5 ∩ ∩ NOUN iajs-2288 131 6 … … PUNCT iajs-2288 131 7 … … SYM iajs-2288 131 8 ∩ ∩ X iajs-2288 131 9 𝐹𝑗 𝐹𝑗 PROPN iajs-2288 131 10 βŠ† βŠ† NUM iajs-2288 131 11 𝐿 𝐿 PROPN iajs-2288 131 12 + + NOUN iajs-2288 131 13 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 131 14 ) ) PUNCT iajs-2288 131 15 . . PUNCT iajs-2288 132 1 ( ( PUNCT iajs-2288 132 2 3 3 X iajs-2288 132 3 ) ) PUNCT iajs-2288 132 4 β‡’ β‡’ NOUN iajs-2288 132 5 ( ( PUNCT iajs-2288 132 6 4 4 X iajs-2288 132 7 ) ) PUNCT iajs-2288 132 8 let let VERB iajs-2288 132 9 𝐼𝑛𝐸 𝐼𝑛𝐸 PROPN iajs-2288 132 10 βŠ† βŠ† NUM iajs-2288 132 11 𝐿 𝐿 PROPN iajs-2288 132 12 , , PUNCT iajs-2288 132 13 where where SCONJ iajs-2288 132 14 𝐼 𝐼 PROPN iajs-2288 132 15 is be AUX iajs-2288 132 16 an an DET iajs-2288 132 17 ideal ideal NOUN iajs-2288 132 18 of of ADP iajs-2288 132 19 𝑅 𝑅 PROPN iajs-2288 132 20 , , PUNCT iajs-2288 132 21 𝐸 𝐸 PROPN iajs-2288 132 22 is be AUX iajs-2288 132 23 a a DET iajs-2288 132 24 submodule submodule NOUN iajs-2288 132 25 of of ADP iajs-2288 132 26 𝑇 𝑇 PROPN iajs-2288 132 27 , , PUNCT iajs-2288 132 28 and and CCONJ iajs-2288 132 29 𝑛 𝑛 DET iajs-2288 132 30 ∈ ∈ PROPN iajs-2288 132 31 𝑍+ 𝑍+ NOUN iajs-2288 132 32 . . PUNCT iajs-2288 133 1 that that PRON iajs-2288 133 2 is is ADV iajs-2288 133 3 ( ( PUNCT iajs-2288 133 4 𝐼𝑇)(𝐼𝑇 𝐼𝑇)(𝐼𝑇 PROPN iajs-2288 133 5 ) ) PUNCT iajs-2288 133 6 …… …… NOUN iajs-2288 133 7 ( ( PUNCT iajs-2288 133 8 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 133 9 ) ) PUNCT iajs-2288 133 10 βŠ† βŠ† NUM iajs-2288 133 11 𝐿 𝐿 PROPN iajs-2288 133 12 , , PUNCT iajs-2288 133 13 so so ADV iajs-2288 133 14 by by ADP iajs-2288 133 15 hypothesis hypothesis NOUN iajs-2288 133 16 ( ( PUNCT iajs-2288 133 17 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 133 18 ) ) PUNCT iajs-2288 133 19 ∩ ∩ NOUN iajs-2288 133 20 ( ( PUNCT iajs-2288 133 21 𝐼𝑇) 𝐼𝑇) NOUN iajs-2288 133 22 … … SYM iajs-2288 133 23 … … SYM iajs-2288 133 24 ∩ ∩ NOUN iajs-2288 133 25 ( ( PUNCT iajs-2288 133 26 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 133 27 ) ) PUNCT iajs-2288 133 28 βŠ† βŠ† NUM iajs-2288 133 29 𝐿 𝐿 PROPN iajs-2288 133 30 + + NOUN iajs-2288 133 31 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 133 32 ) ) PUNCT iajs-2288 133 33 . . PUNCT iajs-2288 133 34 implies imply VERB iajs-2288 133 35 that that SCONJ iajs-2288 133 36 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 133 37 βŠ† βŠ† NUM iajs-2288 133 38 𝐿 𝐿 PROPN iajs-2288 133 39 + + NOUN iajs-2288 133 40 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 133 41 ) ) PUNCT iajs-2288 133 42 . . PUNCT iajs-2288 134 1 thus thus ADV iajs-2288 134 2 by by ADP iajs-2288 134 3 corollary corollary ADJ iajs-2288 134 4 ( ( PUNCT iajs-2288 134 5 2.4 2.4 NUM iajs-2288 134 6 ) ) PUNCT iajs-2288 134 7 𝐿 𝐿 PROPN iajs-2288 134 8 is be AUX iajs-2288 134 9 an an DET iajs-2288 134 10 app app ADJ iajs-2288 134 11 - - PUNCT iajs-2288 134 12 semi semi ADJ iajs-2288 134 13 - - ADJ iajs-2288 134 14 prime prime ADJ iajs-2288 134 15 submodule submodule NOUN iajs-2288 134 16 of of ADP iajs-2288 134 17 𝑇. 𝑇. PROPN iajs-2288 134 18 remark remark NOUN iajs-2288 134 19 ( ( PUNCT iajs-2288 134 20 17 17 NUM iajs-2288 134 21 ) ) PUNCT iajs-2288 134 22 if if SCONJ iajs-2288 134 23 𝐿 𝐿 PROPN iajs-2288 134 24 is be AUX iajs-2288 134 25 an an DET iajs-2288 134 26 app app ADJ iajs-2288 134 27 - - PUNCT iajs-2288 134 28 semi semi ADJ iajs-2288 134 29 - - ADJ iajs-2288 134 30 prime prime ADJ iajs-2288 134 31 submodule submodule NOUN iajs-2288 134 32 of of ADP iajs-2288 134 33 an an DET iajs-2288 134 34 𝑅-module 𝑅-module PROPN iajs-2288 134 35 𝑇 𝑇 PROPN iajs-2288 134 36 , , PUNCT iajs-2288 134 37 then then ADV iajs-2288 134 38 [ [ X iajs-2288 134 39 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 134 40 𝑇 𝑇 PROPN iajs-2288 134 41 ] ] PUNCT iajs-2288 134 42 is be AUX iajs-2288 134 43 not not PART iajs-2288 134 44 necessary necessary ADJ iajs-2288 134 45 app app ADJ iajs-2288 134 46 - - PUNCT iajs-2288 134 47 semi semi ADJ iajs-2288 134 48 - - ADJ iajs-2288 134 49 prime prime ADJ iajs-2288 134 50 ideal ideal NOUN iajs-2288 134 51 of of ADP iajs-2288 134 52 𝑅. 𝑅. NOUN iajs-2288 134 53 the the DET iajs-2288 134 54 following follow VERB iajs-2288 134 55 example example NOUN iajs-2288 134 56 shows show VERB iajs-2288 134 57 that that PRON iajs-2288 134 58 . . PUNCT iajs-2288 135 1 consider consider VERB iajs-2288 135 2 the the DET iajs-2288 135 3 𝑍-module 𝑍-module PROPN iajs-2288 135 4 𝑍8 𝑍8 NOUN iajs-2288 135 5 and and CCONJ iajs-2288 135 6 a a DET iajs-2288 135 7 submodule submodule NOUN iajs-2288 135 8 𝐿 𝐿 PROPN iajs-2288 135 9 = = PROPN iajs-2288 135 10 〈 〈 PROPN iajs-2288 135 11 0Μ… 0Μ… PROPN iajs-2288 135 12 βŒͺ βŒͺ NOUN iajs-2288 135 13 . . PUNCT iajs-2288 136 1 𝐿 𝐿 PROPN iajs-2288 136 2 is be AUX iajs-2288 136 3 an an DET iajs-2288 136 4 app app ADJ iajs-2288 136 5 - - PUNCT iajs-2288 136 6 semi semi ADJ iajs-2288 136 7 - - ADJ iajs-2288 136 8 prime prime ADJ iajs-2288 136 9 submodule submodule NOUN iajs-2288 136 10 since since SCONJ iajs-2288 136 11 π‘ π‘œπ‘(𝑍8 π‘ π‘œπ‘(𝑍8 NOUN iajs-2288 136 12 ) ) PUNCT iajs-2288 136 13 = = PUNCT iajs-2288 136 14 〈 〈 PROPN iajs-2288 136 15 2Μ… 2Μ… NOUN iajs-2288 136 16 βŒͺ βŒͺ NOUN iajs-2288 136 17 = = PUNCT iajs-2288 136 18 { { PUNCT iajs-2288 136 19 0Μ… 0Μ… PROPN iajs-2288 136 20 , , PUNCT iajs-2288 136 21 2Μ… 2Μ… NUM iajs-2288 136 22 , , PUNCT iajs-2288 136 23 4Μ… 4Μ… PROPN iajs-2288 136 24 , , PUNCT iajs-2288 136 25 6Μ… 6Μ… PROPN iajs-2288 136 26 } } PUNCT iajs-2288 136 27 and and CCONJ iajs-2288 136 28 22 22 NUM iajs-2288 136 29 . . PUNCT iajs-2288 137 1 2Μ… 2Μ… NUM iajs-2288 137 2 ∈ ∈ PROPN iajs-2288 137 3 𝐿 𝐿 PROPN iajs-2288 137 4 , , PUNCT iajs-2288 137 5 implies imply VERB iajs-2288 137 6 that that SCONJ iajs-2288 137 7 2 2 X iajs-2288 137 8 . . X iajs-2288 137 9 2Μ… 2Μ… NUM iajs-2288 137 10 = = SYM iajs-2288 137 11 4 4 NUM iajs-2288 137 12 ∈ ∈ PROPN iajs-2288 137 13 𝐿 𝐿 PROPN iajs-2288 137 14 + + CCONJ iajs-2288 137 15 π‘ π‘œπ‘(𝑍8 π‘ π‘œπ‘(𝑍8 NOUN iajs-2288 137 16 ) ) PUNCT iajs-2288 137 17 = = PRON iajs-2288 137 18 { { PUNCT iajs-2288 137 19 0Μ… 0Μ… PROPN iajs-2288 137 20 , , PUNCT iajs-2288 137 21 2Μ… 2Μ… NUM iajs-2288 137 22 , , PUNCT iajs-2288 137 23 4Μ… 4Μ… PROPN iajs-2288 137 24 , , PUNCT iajs-2288 137 25 6Μ… 6Μ… PROPN iajs-2288 137 26 } } PUNCT iajs-2288 137 27 , , PUNCT iajs-2288 137 28 where where SCONJ iajs-2288 137 29 2 2 NUM iajs-2288 137 30 ∈ ∈ NOUN iajs-2288 137 31 𝑍 𝑍 NOUN iajs-2288 137 32 , , PUNCT iajs-2288 137 33 2Μ… 2Μ… PROPN iajs-2288 137 34 ∈ ∈ PROPN iajs-2288 137 35 𝑍8 𝑍8 NOUN iajs-2288 137 36 . . PUNCT iajs-2288 138 1 so so ADV iajs-2288 138 2 for for ADP iajs-2288 138 3 all all DET iajs-2288 138 4 π‘Ž π‘Ž PRON iajs-2288 138 5 ∈ ∈ PROPN iajs-2288 138 6 𝑅 𝑅 PROPN iajs-2288 138 7 , , PUNCT iajs-2288 138 8 𝑑̅ 𝑑̅ PROPN iajs-2288 138 9 ∈ ∈ PROPN iajs-2288 138 10 𝑍8 𝑍8 NOUN iajs-2288 138 11 , , PUNCT iajs-2288 138 12 such such ADJ iajs-2288 138 13 that that SCONJ iajs-2288 138 14 π‘Žπ‘›π‘‘Μ… π‘Žπ‘›π‘‘Μ… PROPN iajs-2288 138 15 ∈ ∈ PROPN iajs-2288 138 16 𝐿 𝐿 PROPN iajs-2288 138 17 for for ADP iajs-2288 138 18 some some PRON iajs-2288 138 19 𝑛 𝑛 PRON iajs-2288 138 20 ∈ ∈ PROPN iajs-2288 138 21 𝑍+ 𝑍+ NOUN iajs-2288 138 22 , , PUNCT iajs-2288 138 23 implies imply VERB iajs-2288 138 24 that that SCONJ iajs-2288 138 25 π‘Žπ‘‘Μ… π‘Žπ‘‘Μ… VERB iajs-2288 138 26 ∈ ∈ PROPN iajs-2288 138 27 𝐿 𝐿 PROPN iajs-2288 138 28 + + CCONJ iajs-2288 138 29 π‘ π‘œπ‘(𝑍8 π‘ π‘œπ‘(𝑍8 NOUN iajs-2288 138 30 ) ) PUNCT iajs-2288 138 31 . . PUNCT iajs-2288 139 1 but but CCONJ iajs-2288 139 2 [ [ X iajs-2288 139 3 𝐿:𝑍 𝐿:𝑍 PUNCT iajs-2288 139 4 𝑇 𝑇 PROPN iajs-2288 139 5 ] ] PUNCT iajs-2288 139 6 = = PUNCT iajs-2288 140 1 [ [ X iajs-2288 140 2 〈 〈 NOUN iajs-2288 140 3 0Μ…βŒͺ:𝑍 0Μ…βŒͺ:𝑍 NOUN iajs-2288 140 4 𝑍8 𝑍8 NOUN iajs-2288 140 5 ] ] X iajs-2288 140 6 = = SYM iajs-2288 140 7 8𝑍 8𝑍 NOUN iajs-2288 140 8 is be AUX iajs-2288 140 9 not not PART iajs-2288 140 10 app app ADJ iajs-2288 140 11 - - PUNCT iajs-2288 140 12 semi semi ADJ iajs-2288 140 13 - - ADJ iajs-2288 140 14 prime prime ADJ iajs-2288 140 15 ideal ideal NOUN iajs-2288 140 16 in in ADP iajs-2288 140 17 𝑍 𝑍 NOUN iajs-2288 140 18 , , PUNCT iajs-2288 140 19 since since SCONJ iajs-2288 140 20 22 22 NUM iajs-2288 140 21 . . SYM iajs-2288 140 22 2 2 NUM iajs-2288 140 23 ∈ ∈ PROPN iajs-2288 140 24 8𝑍 8𝑍 NOUN iajs-2288 140 25 but but CCONJ iajs-2288 140 26 2.2 2.2 NUM iajs-2288 140 27 βˆ‰ βˆ‰ PROPN iajs-2288 140 28 8𝑍 8𝑍 PROPN iajs-2288 140 29 + + PROPN iajs-2288 140 30 π‘ π‘œπ‘(𝑍 π‘ π‘œπ‘(𝑍 PROPN iajs-2288 140 31 ) ) PUNCT iajs-2288 140 32 = = SYM iajs-2288 140 33 8𝑍 8𝑍 NOUN iajs-2288 140 34 + + CCONJ iajs-2288 140 35 ( ( PUNCT iajs-2288 140 36 0 0 NUM iajs-2288 140 37 ) ) PUNCT iajs-2288 140 38 = = NOUN iajs-2288 140 39 8𝑍. 8𝑍. NUM iajs-2288 140 40 proposition proposition NOUN iajs-2288 140 41 ( ( PUNCT iajs-2288 140 42 18 18 NUM iajs-2288 140 43 ) ) PUNCT iajs-2288 140 44 let let VERB iajs-2288 140 45 𝐿 𝐿 PROPN iajs-2288 140 46 be be AUX iajs-2288 140 47 an an DET iajs-2288 140 48 app app ADJ iajs-2288 140 49 - - PUNCT iajs-2288 140 50 semi semi ADJ iajs-2288 140 51 - - ADJ iajs-2288 140 52 prime prime ADJ iajs-2288 140 53 submodule submodule NOUN iajs-2288 140 54 of of ADP iajs-2288 140 55 an an DET iajs-2288 140 56 𝑅-module 𝑅-module PROPN iajs-2288 140 57 𝑇 𝑇 PROPN iajs-2288 140 58 , , PUNCT iajs-2288 140 59 with with ADP iajs-2288 140 60 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 140 61 ) ) PUNCT iajs-2288 140 62 βŠ† βŠ† NUM iajs-2288 140 63 𝐿. 𝐿. VERB iajs-2288 140 64 then then ADV iajs-2288 140 65 [ [ X iajs-2288 140 66 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 140 67 𝑇 𝑇 PROPN iajs-2288 140 68 ] ] PUNCT iajs-2288 140 69 is be AUX iajs-2288 140 70 an an DET iajs-2288 140 71 app app ADJ iajs-2288 140 72 - - PUNCT iajs-2288 140 73 semi semi ADJ iajs-2288 140 74 - - ADJ iajs-2288 140 75 prime prime ADJ iajs-2288 140 76 ideal ideal NOUN iajs-2288 140 77 of of ADP iajs-2288 140 78 𝑅. 𝑅. ADJ iajs-2288 140 79 proof proof NOUN iajs-2288 140 80 suppose suppose VERB iajs-2288 140 81 that that SCONJ iajs-2288 140 82 π‘Žπ‘›π‘  π‘Žπ‘›π‘  ADJ iajs-2288 140 83 ∈ ∈ PROPN iajs-2288 140 84 [ [ X iajs-2288 140 85 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 140 86 𝑇 𝑇 PROPN iajs-2288 140 87 ] ] PUNCT iajs-2288 140 88 , , PUNCT iajs-2288 140 89 where where SCONJ iajs-2288 140 90 π‘Ž π‘Ž X iajs-2288 140 91 , , PUNCT iajs-2288 140 92 𝑠 𝑠 PROPN iajs-2288 140 93 ∈ ∈ PROPN iajs-2288 140 94 𝑅 𝑅 PROPN iajs-2288 140 95 , , PUNCT iajs-2288 140 96 𝑛 𝑛 PRON iajs-2288 140 97 ∈ ∈ NOUN iajs-2288 140 98 𝑍+ 𝑍+ NOUN iajs-2288 140 99 , , PUNCT iajs-2288 140 100 then then ADV iajs-2288 140 101 π‘Žπ‘›π‘  π‘Žπ‘›π‘  VERB iajs-2288 140 102 𝑇 𝑇 PROPN iajs-2288 140 103 βŠ† βŠ† ADV iajs-2288 140 104 𝐿. 𝐿. PROPN iajs-2288 140 105 that that PRON iajs-2288 140 106 is be AUX iajs-2288 140 107 π‘Žπ‘›(𝑠 π‘Žπ‘›(𝑠 PUNCT iajs-2288 140 108 𝑇 𝑇 PROPN iajs-2288 140 109 ) ) PUNCT iajs-2288 140 110 βŠ† βŠ† NUM iajs-2288 140 111 𝐿 𝐿 PROPN iajs-2288 140 112 , , PUNCT iajs-2288 140 113 implies imply VERB iajs-2288 140 114 that that SCONJ iajs-2288 140 115 π‘Žπ‘›(𝑠 π‘Žπ‘›(𝑠 PUNCT iajs-2288 140 116 𝑑 𝑑 NOUN iajs-2288 140 117 ) ) PUNCT iajs-2288 140 118 ∈ ∈ PROPN iajs-2288 140 119 𝐿 𝐿 PROPN iajs-2288 140 120 for for ADP iajs-2288 140 121 all all DET iajs-2288 140 122 𝑑 𝑑 ADP iajs-2288 140 123 ∈ ∈ PROPN iajs-2288 140 124 𝑇. 𝑇. PROPN iajs-2288 140 125 but but CCONJ iajs-2288 140 126 𝐿 𝐿 PROPN iajs-2288 140 127 is be AUX iajs-2288 140 128 an an DET iajs-2288 140 129 app app ADJ iajs-2288 140 130 - - PUNCT iajs-2288 140 131 semi semi ADJ iajs-2288 140 132 - - ADJ iajs-2288 140 133 prime prime ADJ iajs-2288 140 134 submodule submodule NOUN iajs-2288 140 135 of of ADP iajs-2288 140 136 𝑇 𝑇 PROPN iajs-2288 140 137 , , PUNCT iajs-2288 140 138 implies imply VERB iajs-2288 140 139 that that SCONJ iajs-2288 140 140 π‘Ž(𝑠 π‘Ž(𝑠 PROPN iajs-2288 140 141 𝑑 𝑑 PROPN iajs-2288 140 142 ) ) PUNCT iajs-2288 140 143 ∈ ∈ PROPN iajs-2288 140 144 𝐿 𝐿 PROPN iajs-2288 140 145 + + NOUN iajs-2288 140 146 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 140 147 ) ) PUNCT iajs-2288 140 148 , , PUNCT iajs-2288 140 149 that that PRON iajs-2288 140 150 is be AUX iajs-2288 140 151 π‘Žπ‘  π‘Žπ‘  ADP iajs-2288 140 152 𝑇 𝑇 PROPN iajs-2288 140 153 βŠ† βŠ† NUM iajs-2288 140 154 𝐿 𝐿 PROPN iajs-2288 140 155 + + NOUN iajs-2288 140 156 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 140 157 ) ) PUNCT iajs-2288 140 158 . . PUNCT iajs-2288 141 1 but but CCONJ iajs-2288 141 2 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 141 3 ) ) PUNCT iajs-2288 141 4 βŠ† βŠ† X iajs-2288 141 5 𝐿 𝐿 PROPN iajs-2288 141 6 , , PUNCT iajs-2288 141 7 implies imply VERB iajs-2288 141 8 that that SCONJ iajs-2288 141 9 𝐿 𝐿 PROPN iajs-2288 141 10 + + NOUN iajs-2288 141 11 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 141 12 ) ) PUNCT iajs-2288 141 13 = = SYM iajs-2288 141 14 𝐿 𝐿 PROPN iajs-2288 141 15 , , PUNCT iajs-2288 141 16 thus thus ADV iajs-2288 141 17 π‘Žπ‘  π‘Žπ‘  ADP iajs-2288 141 18 𝑇 𝑇 PROPN iajs-2288 141 19 βŠ† βŠ† NUM iajs-2288 141 20 𝐿 𝐿 PROPN iajs-2288 141 21 , , PUNCT iajs-2288 141 22 it it PRON iajs-2288 141 23 follows follow VERB iajs-2288 141 24 that that SCONJ iajs-2288 141 25 π‘Žπ‘  π‘Žπ‘  PROPN iajs-2288 141 26 ∈ ∈ PROPN iajs-2288 142 1 [ [ X iajs-2288 142 2 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 142 3 𝑇 𝑇 PROPN iajs-2288 142 4 ] ] PUNCT iajs-2288 142 5 βŠ† βŠ† NUM iajs-2288 142 6 [ [ X iajs-2288 142 7 𝐿:𝑅 𝐿:𝑅 X iajs-2288 142 8 𝑇 𝑇 PROPN iajs-2288 142 9 ] ] X iajs-2288 142 10 + + NUM iajs-2288 142 11 π‘ π‘œπ‘(𝑅 π‘ π‘œπ‘(𝑅 NUM iajs-2288 142 12 ) ) PUNCT iajs-2288 142 13 . . PUNCT iajs-2288 143 1 therefore therefore ADV iajs-2288 143 2 [ [ X iajs-2288 143 3 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 143 4 𝑇 𝑇 PROPN iajs-2288 143 5 ] ] PUNCT iajs-2288 143 6 is be AUX iajs-2288 143 7 an an DET iajs-2288 143 8 app app ADJ iajs-2288 143 9 - - PUNCT iajs-2288 143 10 semi semi ADJ iajs-2288 143 11 - - ADJ iajs-2288 143 12 prime prime ADJ iajs-2288 143 13 ideal ideal NOUN iajs-2288 143 14 of of ADP iajs-2288 143 15 𝑅. 𝑅. NOUN iajs-2288 143 16 we we PRON iajs-2288 143 17 need need VERB iajs-2288 143 18 to to PART iajs-2288 143 19 introduce introduce VERB iajs-2288 143 20 the the DET iajs-2288 143 21 following follow VERB iajs-2288 143 22 lemma lemma PROPN iajs-2288 143 23 which which PRON iajs-2288 143 24 appear appear VERB iajs-2288 143 25 in in ADP iajs-2288 143 26 [ [ X iajs-2288 143 27 14 14 NUM iajs-2288 143 28 ] ] PUNCT iajs-2288 143 29 . . PUNCT iajs-2288 144 1 lemma lemma PROPN iajs-2288 144 2 ( ( PUNCT iajs-2288 144 3 19)[14 19)[14 PROPN iajs-2288 144 4 , , PUNCT iajs-2288 144 5 coro coro NOUN iajs-2288 144 6 . . PUNCT iajs-2288 145 1 2.14 2.14 NUM iajs-2288 145 2 ] ] PUNCT iajs-2288 145 3 . . PUNCT iajs-2288 146 1 let let VERB iajs-2288 146 2 𝑇 𝑇 PROPN iajs-2288 146 3 be be AUX iajs-2288 146 4 a a DET iajs-2288 146 5 faithful faithful ADJ iajs-2288 146 6 multiplication multiplication NOUN iajs-2288 146 7 𝑅-module 𝑅-module PROPN iajs-2288 146 8 then then ADV iajs-2288 146 9 π‘ π‘œπ‘(𝑅)𝑇 π‘ π‘œπ‘(𝑅)𝑇 PROPN iajs-2288 146 10 = = SYM iajs-2288 146 11 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 146 12 ) ) PUNCT iajs-2288 146 13 . . PUNCT iajs-2288 147 1 proposition proposition NOUN iajs-2288 147 2 ( ( PUNCT iajs-2288 147 3 20 20 NUM iajs-2288 147 4 ) ) PUNCT iajs-2288 147 5 let let VERB iajs-2288 147 6 𝑇 𝑇 PROPN iajs-2288 147 7 be be AUX iajs-2288 147 8 a a DET iajs-2288 147 9 faithful faithful ADJ iajs-2288 147 10 multiplication multiplication NOUN iajs-2288 147 11 𝑅-module 𝑅-module PROPN iajs-2288 147 12 and and CCONJ iajs-2288 147 13 𝐿 𝐿 PROPN iajs-2288 147 14 be be AUX iajs-2288 147 15 a a DET iajs-2288 147 16 proper proper ADJ iajs-2288 147 17 submodule submodule NOUN iajs-2288 147 18 of of ADP iajs-2288 147 19 𝑇. 𝑇. PROPN iajs-2288 147 20 then then ADV iajs-2288 147 21 𝐿 𝐿 PROPN iajs-2288 147 22 is be AUX iajs-2288 147 23 an an DET iajs-2288 147 24 app app ADJ iajs-2288 147 25 - - PUNCT iajs-2288 147 26 semi semi ADJ iajs-2288 147 27 - - ADJ iajs-2288 147 28 prime prime ADJ iajs-2288 147 29 submodule submodule NOUN iajs-2288 147 30 of of ADP iajs-2288 147 31 𝑇 𝑇 PROPN iajs-2288 148 1 if if SCONJ iajs-2288 148 2 and and CCONJ iajs-2288 148 3 only only ADV iajs-2288 148 4 if if SCONJ iajs-2288 148 5 [ [ X iajs-2288 148 6 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 148 7 𝑇 𝑇 PROPN iajs-2288 148 8 ] ] PUNCT iajs-2288 148 9 is be AUX iajs-2288 148 10 an an DET iajs-2288 148 11 app app ADJ iajs-2288 148 12 - - PUNCT iajs-2288 148 13 semi semi ADJ iajs-2288 148 14 - - ADJ iajs-2288 148 15 prime prime ADJ iajs-2288 148 16 ideal ideal NOUN iajs-2288 148 17 of of ADP iajs-2288 148 18 𝑅. 𝑅. SYM iajs-2288 148 19 123 123 NUM iajs-2288 148 20 ibn ibn PROPN iajs-2288 148 21 al al PROPN iajs-2288 148 22 - - PUNCT iajs-2288 148 23 haitham haitham PROPN iajs-2288 148 24 jour jour X iajs-2288 148 25 . . PROPN iajs-2288 148 26 for for ADP iajs-2288 148 27 pure pure ADJ iajs-2288 148 28 & & CCONJ iajs-2288 148 29 appl appl PROPN iajs-2288 148 30 . . PUNCT iajs-2288 149 1 sci sci PROPN iajs-2288 149 2 . . PROPN iajs-2288 149 3 32 32 NUM iajs-2288 149 4 ( ( PUNCT iajs-2288 149 5 3 3 NUM iajs-2288 149 6 ) ) PUNCT iajs-2288 149 7 2019 2019 NUM iajs-2288 149 8 proof proof NOUN iajs-2288 149 9 ( ( PUNCT iajs-2288 149 10 β‡’ β‡’ PROPN iajs-2288 149 11 ) ) PUNCT iajs-2288 149 12 suppose suppose VERB iajs-2288 149 13 that that SCONJ iajs-2288 149 14 𝐿 𝐿 PROPN iajs-2288 149 15 is be AUX iajs-2288 149 16 an an DET iajs-2288 149 17 app app ADJ iajs-2288 149 18 - - PUNCT iajs-2288 149 19 semi semi ADJ iajs-2288 149 20 - - ADJ iajs-2288 149 21 prime prime ADJ iajs-2288 149 22 submodule submodule NOUN iajs-2288 149 23 of of ADP iajs-2288 149 24 𝑇 𝑇 PROPN iajs-2288 149 25 , , PUNCT iajs-2288 149 26 to to PART iajs-2288 149 27 prove prove VERB iajs-2288 149 28 that that SCONJ iajs-2288 149 29 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2288 149 30 𝑇 𝑇 PROPN iajs-2288 149 31 ] ] PUNCT iajs-2288 149 32 βŠ† βŠ† NUM iajs-2288 149 33 [ [ X iajs-2288 149 34 𝐿:𝑅 𝐿:𝑅 X iajs-2288 149 35 𝑇 𝑇 PROPN iajs-2288 149 36 ] ] X iajs-2288 149 37 + + ADJ iajs-2288 149 38 π‘ π‘œπ‘(𝑅 π‘ π‘œπ‘(𝑅 NUM iajs-2288 149 39 ) ) PUNCT iajs-2288 149 40 by by ADP iajs-2288 149 41 proposition proposition NOUN iajs-2288 149 42 ( ( PUNCT iajs-2288 149 43 2.16 2.16 NUM iajs-2288 149 44 ) ) PUNCT iajs-2288 149 45 , , PUNCT iajs-2288 149 46 we we PRON iajs-2288 149 47 get get VERB iajs-2288 149 48 [ [ X iajs-2288 149 49 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 149 50 𝑇 𝑇 PROPN iajs-2288 149 51 ] ] PUNCT iajs-2288 149 52 is be AUX iajs-2288 149 53 an an DET iajs-2288 149 54 app app ADJ iajs-2288 149 55 - - PUNCT iajs-2288 149 56 semi semi ADJ iajs-2288 149 57 - - ADJ iajs-2288 149 58 prime prime ADJ iajs-2288 149 59 ideal ideal NOUN iajs-2288 149 60 of of ADP iajs-2288 149 61 𝑅. 𝑅. NOUN iajs-2288 149 62 let let VERB iajs-2288 149 63 π‘Ž π‘Ž PROPN iajs-2288 149 64 ∈ ∈ PROPN iajs-2288 149 65 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2288 149 66 𝑇 𝑇 PROPN iajs-2288 149 67 ] ] PUNCT iajs-2288 149 68 , , PUNCT iajs-2288 149 69 implies imply VERB iajs-2288 149 70 that that SCONJ iajs-2288 149 71 π‘Žπ‘› π‘Žπ‘› PRON iajs-2288 149 72 ∈ ∈ PROPN iajs-2288 149 73 [ [ X iajs-2288 149 74 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 149 75 𝑇 𝑇 PROPN iajs-2288 149 76 ] ] PUNCT iajs-2288 149 77 for for ADP iajs-2288 149 78 some some DET iajs-2288 149 79 𝑛 𝑛 PRON iajs-2288 149 80 ∈ ∈ PROPN iajs-2288 149 81 𝑍+ 𝑍+ NOUN iajs-2288 149 82 , , PUNCT iajs-2288 149 83 it it PRON iajs-2288 149 84 follows follow VERB iajs-2288 149 85 that that SCONJ iajs-2288 149 86 π‘Žπ‘›π‘‡ π‘Žπ‘›π‘‡ PROPN iajs-2288 149 87 βŠ† βŠ† NUM iajs-2288 149 88 𝐿 𝐿 PROPN iajs-2288 149 89 , , PUNCT iajs-2288 149 90 that that PRON iajs-2288 149 91 is is ADV iajs-2288 149 92 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN iajs-2288 149 93 ∈ ∈ PROPN iajs-2288 149 94 𝐿 𝐿 PROPN iajs-2288 149 95 for for ADP iajs-2288 149 96 all all DET iajs-2288 149 97 𝑑 𝑑 ADP iajs-2288 149 98 ∈ ∈ PROPN iajs-2288 149 99 𝑇. 𝑇. PROPN iajs-2288 149 100 but but CCONJ iajs-2288 149 101 𝐿 𝐿 PROPN iajs-2288 149 102 is be AUX iajs-2288 149 103 an an DET iajs-2288 149 104 app app ADJ iajs-2288 149 105 - - PUNCT iajs-2288 149 106 semi semi ADJ iajs-2288 149 107 - - ADJ iajs-2288 149 108 prime prime ADJ iajs-2288 149 109 submodule submodule NOUN iajs-2288 149 110 of of ADP iajs-2288 149 111 𝑇 𝑇 PROPN iajs-2288 149 112 , , PUNCT iajs-2288 149 113 then then ADV iajs-2288 149 114 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 149 115 ∈ ∈ PROPN iajs-2288 149 116 𝐿 𝐿 PROPN iajs-2288 149 117 + + NOUN iajs-2288 149 118 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 149 119 ) ) PUNCT iajs-2288 149 120 for for ADP iajs-2288 149 121 all all DET iajs-2288 149 122 𝑑 𝑑 PROPN iajs-2288 149 123 ∈ ∈ PROPN iajs-2288 149 124 𝑇. 𝑇. PROPN iajs-2288 149 125 that that PRON iajs-2288 149 126 is be AUX iajs-2288 149 127 π‘Žπ‘‡ π‘Žπ‘‡ NOUN iajs-2288 149 128 βŠ† βŠ† NUM iajs-2288 149 129 𝐿 𝐿 PROPN iajs-2288 149 130 + + NOUN iajs-2288 149 131 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 149 132 ) ) PUNCT iajs-2288 149 133 . . PUNCT iajs-2288 150 1 since since SCONJ iajs-2288 150 2 𝑇 𝑇 PROPN iajs-2288 150 3 is be AUX iajs-2288 150 4 a a DET iajs-2288 150 5 multiplication𝑅-module multiplication𝑅-module NOUN iajs-2288 150 6 , , PUNCT iajs-2288 150 7 so so ADV iajs-2288 150 8 𝐿 𝐿 PROPN iajs-2288 150 9 = = SYM iajs-2288 150 10 [ [ X iajs-2288 150 11 𝐿:𝑅 𝐿:𝑅 X iajs-2288 150 12 𝑇]𝑇 𝑇]𝑇 PROPN iajs-2288 150 13 , , PUNCT iajs-2288 150 14 and and CCONJ iajs-2288 150 15 since since SCONJ iajs-2288 150 16 𝑇 𝑇 PROPN iajs-2288 150 17 is be AUX iajs-2288 150 18 faithful faithful ADJ iajs-2288 150 19 multiplication multiplication NOUN iajs-2288 150 20 , , PUNCT iajs-2288 150 21 so so ADV iajs-2288 150 22 by by ADP iajs-2288 150 23 lemma lemma PROPN iajs-2288 150 24 ( ( PUNCT iajs-2288 150 25 2.19 2.19 NUM iajs-2288 150 26 ) ) PUNCT iajs-2288 150 27 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 150 28 ) ) PUNCT iajs-2288 150 29 = = SYM iajs-2288 151 1 π‘ π‘œπ‘(𝑅)𝑇. π‘ π‘œπ‘(𝑅)𝑇. PROPN iajs-2288 151 2 thus thus ADV iajs-2288 151 3 π‘Žπ‘‡ π‘Žπ‘‡ VERB iajs-2288 151 4 βŠ† βŠ† PRON iajs-2288 151 5 [ [ X iajs-2288 151 6 𝐿:𝑅 𝐿:𝑅 X iajs-2288 151 7 𝑇]𝑇 𝑇]𝑇 PROPN iajs-2288 151 8 + + CCONJ iajs-2288 151 9 π‘ π‘œπ‘(𝑅)𝑇 π‘ π‘œπ‘(𝑅)𝑇 NOUN iajs-2288 151 10 , , PUNCT iajs-2288 151 11 it it PRON iajs-2288 151 12 follows follow VERB iajs-2288 151 13 that that SCONJ iajs-2288 151 14 π‘Ž π‘Ž X iajs-2288 151 15 ∈ ∈ PROPN iajs-2288 151 16 [ [ X iajs-2288 151 17 𝐿:𝑅 𝐿:𝑅 X iajs-2288 151 18 𝑇 𝑇 PROPN iajs-2288 151 19 ] ] X iajs-2288 151 20 + + NUM iajs-2288 151 21 π‘ π‘œπ‘(𝑅 π‘ π‘œπ‘(𝑅 NUM iajs-2288 151 22 ) ) PUNCT iajs-2288 151 23 . . PUNCT iajs-2288 152 1 hence hence ADV iajs-2288 152 2 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2288 152 3 𝑇 𝑇 PROPN iajs-2288 152 4 ] ] PUNCT iajs-2288 152 5 βŠ† βŠ† NUM iajs-2288 152 6 [ [ X iajs-2288 152 7 𝐿:𝑅 𝐿:𝑅 X iajs-2288 152 8 𝑇 𝑇 PROPN iajs-2288 152 9 ] ] X iajs-2288 152 10 + + NUM iajs-2288 152 11 π‘ π‘œπ‘(𝑅 π‘ π‘œπ‘(𝑅 NUM iajs-2288 152 12 ) ) PUNCT iajs-2288 152 13 , , PUNCT iajs-2288 152 14 therefore therefore ADV iajs-2288 152 15 [ [ X iajs-2288 152 16 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 152 17 𝑇 𝑇 PROPN iajs-2288 152 18 ] ] PUNCT iajs-2288 152 19 is be AUX iajs-2288 152 20 an an DET iajs-2288 152 21 app app ADJ iajs-2288 152 22 - - PUNCT iajs-2288 152 23 semi semi ADJ iajs-2288 152 24 - - ADJ iajs-2288 152 25 prime prime ADJ iajs-2288 152 26 ideal ideal NOUN iajs-2288 152 27 of of ADP iajs-2288 152 28 𝑅. 𝑅. NOUN iajs-2288 152 29 ( ( PUNCT iajs-2288 152 30 ⇐ ⇐ PROPN iajs-2288 152 31 ) ) PUNCT iajs-2288 152 32 suppose suppose VERB iajs-2288 152 33 that that SCONJ iajs-2288 152 34 [ [ X iajs-2288 152 35 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 152 36 𝑇 𝑇 PROPN iajs-2288 152 37 ] ] PUNCT iajs-2288 152 38 is be AUX iajs-2288 152 39 an an DET iajs-2288 152 40 app app ADJ iajs-2288 152 41 - - PUNCT iajs-2288 152 42 semi semi ADJ iajs-2288 152 43 - - ADJ iajs-2288 152 44 prime prime ADJ iajs-2288 152 45 ideal ideal NOUN iajs-2288 152 46 of of ADP iajs-2288 152 47 𝑅 𝑅 PROPN iajs-2288 152 48 , , PUNCT iajs-2288 152 49 and and CCONJ iajs-2288 152 50 let let VERB iajs-2288 152 51 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN iajs-2288 152 52 ∈ ∈ PROPN iajs-2288 152 53 𝐿 𝐿 PROPN iajs-2288 152 54 , , PUNCT iajs-2288 152 55 where where SCONJ iajs-2288 152 56 π‘Ž π‘Ž PROPN iajs-2288 152 57 ∈ ∈ PROPN iajs-2288 152 58 𝑅 𝑅 PROPN iajs-2288 152 59 , , PUNCT iajs-2288 152 60 𝑑 𝑑 PROPN iajs-2288 152 61 ∈ ∈ PROPN iajs-2288 152 62 𝑇 𝑇 PROPN iajs-2288 152 63 and and CCONJ iajs-2288 152 64 𝑛 𝑛 PRON iajs-2288 152 65 ∈ ∈ PROPN iajs-2288 152 66 𝑍+ 𝑍+ NOUN iajs-2288 152 67 , , PUNCT iajs-2288 152 68 it it PRON iajs-2288 152 69 follows follow VERB iajs-2288 152 70 that that SCONJ iajs-2288 152 71 π‘Žπ‘›π‘‡ π‘Žπ‘›π‘‡ PROPN iajs-2288 152 72 βŠ† βŠ† NUM iajs-2288 152 73 𝐿 𝐿 PROPN iajs-2288 152 74 , , PUNCT iajs-2288 152 75 implies imply VERB iajs-2288 152 76 that that SCONJ iajs-2288 152 77 π‘Žπ‘› π‘Žπ‘› PRON iajs-2288 152 78 ∈ ∈ PROPN iajs-2288 153 1 [ [ X iajs-2288 153 2 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 153 3 𝑇 𝑇 PROPN iajs-2288 153 4 ] ] X iajs-2288 153 5 so so SCONJ iajs-2288 153 6 π‘Ž π‘Ž PROPN iajs-2288 153 7 ∈ ∈ PROPN iajs-2288 153 8 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2288 153 9 𝑇 𝑇 PROPN iajs-2288 153 10 ] ] PUNCT iajs-2288 153 11 . . PUNCT iajs-2288 154 1 since since SCONJ iajs-2288 154 2 [ [ X iajs-2288 154 3 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 154 4 𝑇 𝑇 PROPN iajs-2288 154 5 ] ] PUNCT iajs-2288 154 6 is be AUX iajs-2288 154 7 an an DET iajs-2288 154 8 app app ADJ iajs-2288 154 9 - - PUNCT iajs-2288 154 10 semi semi ADJ iajs-2288 154 11 - - ADJ iajs-2288 154 12 prime prime ADJ iajs-2288 154 13 ideal ideal NOUN iajs-2288 154 14 of of ADP iajs-2288 154 15 𝑅 𝑅 PROPN iajs-2288 154 16 then then ADV iajs-2288 154 17 by by ADP iajs-2288 154 18 proposition proposition NOUN iajs-2288 154 19 ( ( PUNCT iajs-2288 154 20 2.16 2.16 NUM iajs-2288 154 21 ) ) PUNCT iajs-2288 154 22 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2288 154 23 𝑇 𝑇 PROPN iajs-2288 154 24 ] ] PUNCT iajs-2288 154 25 βŠ† βŠ† NUM iajs-2288 154 26 [ [ X iajs-2288 154 27 𝐿:𝑅 𝐿:𝑅 X iajs-2288 154 28 𝑇 𝑇 PROPN iajs-2288 154 29 ] ] X iajs-2288 154 30 + + NUM iajs-2288 154 31 π‘ π‘œπ‘(𝑅 π‘ π‘œπ‘(𝑅 NUM iajs-2288 154 32 ) ) PUNCT iajs-2288 154 33 , , PUNCT iajs-2288 154 34 it it PRON iajs-2288 154 35 follows follow VERB iajs-2288 154 36 that that SCONJ iajs-2288 154 37 π‘Ž π‘Ž X iajs-2288 154 38 ∈ ∈ PROPN iajs-2288 154 39 [ [ X iajs-2288 154 40 𝐿:𝑅 𝐿:𝑅 X iajs-2288 154 41 𝑇 𝑇 PROPN iajs-2288 154 42 ] ] X iajs-2288 154 43 + + NUM iajs-2288 154 44 π‘ π‘œπ‘(𝑅 π‘ π‘œπ‘(𝑅 NUM iajs-2288 154 45 ) ) PUNCT iajs-2288 154 46 , , PUNCT iajs-2288 154 47 so so ADV iajs-2288 154 48 π‘Žπ‘‡ π‘Žπ‘‡ VERB iajs-2288 154 49 βŠ† βŠ† PRON iajs-2288 154 50 [ [ X iajs-2288 154 51 𝐿:𝑅 𝐿:𝑅 X iajs-2288 154 52 𝑇]𝑇 𝑇]𝑇 PROPN iajs-2288 154 53 + + CCONJ iajs-2288 155 1 π‘ π‘œπ‘(𝑅)𝑇. π‘ π‘œπ‘(𝑅)𝑇. PROPN iajs-2288 155 2 since since SCONJ iajs-2288 155 3 𝑇 𝑇 PROPN iajs-2288 155 4 is be AUX iajs-2288 155 5 faithful faithful ADJ iajs-2288 155 6 multiplication multiplication NOUN iajs-2288 155 7 , , PUNCT iajs-2288 155 8 then then ADV iajs-2288 155 9 by by ADP iajs-2288 155 10 lemma lemma PROPN iajs-2288 155 11 ( ( PUNCT iajs-2288 155 12 2.19 2.19 NUM iajs-2288 155 13 ) ) PUNCT iajs-2288 155 14 we we PRON iajs-2288 155 15 get get VERB iajs-2288 155 16 π‘Žπ‘‡ π‘Žπ‘‡ NOUN iajs-2288 155 17 βŠ† βŠ† NUM iajs-2288 155 18 𝐿 𝐿 PROPN iajs-2288 155 19 + + NOUN iajs-2288 155 20 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 155 21 ) ) PUNCT iajs-2288 155 22 , , PUNCT iajs-2288 155 23 hence hence ADV iajs-2288 155 24 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 155 25 ∈ ∈ PROPN iajs-2288 155 26 𝐿 𝐿 PROPN iajs-2288 155 27 + + NOUN iajs-2288 155 28 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 155 29 ) ) PUNCT iajs-2288 155 30 for for ADP iajs-2288 155 31 all all DET iajs-2288 155 32 𝑑 𝑑 PROPN iajs-2288 155 33 ∈ ∈ PROPN iajs-2288 155 34 𝑇. 𝑇. PROPN iajs-2288 155 35 thus thus ADV iajs-2288 155 36 𝐿 𝐿 PROPN iajs-2288 155 37 is be AUX iajs-2288 155 38 an an DET iajs-2288 155 39 app app ADJ iajs-2288 155 40 - - PUNCT iajs-2288 155 41 semi semi ADJ iajs-2288 155 42 - - ADJ iajs-2288 155 43 prime prime ADJ iajs-2288 155 44 submodule submodule NOUN iajs-2288 155 45 of of ADP iajs-2288 155 46 𝑇. 𝑇. PROPN iajs-2288 155 47 recall recall NOUN iajs-2288 155 48 that that SCONJ iajs-2288 155 49 an an DET iajs-2288 155 50 𝑅-module 𝑅-module PROPN iajs-2288 155 51 𝑇 𝑇 PROPN iajs-2288 155 52 is be AUX iajs-2288 155 53 a a DET iajs-2288 155 54 non non ADJ iajs-2288 155 55 - - ADJ iajs-2288 155 56 singular singular ADJ iajs-2288 155 57 provided provide VERB iajs-2288 155 58 that that SCONJ iajs-2288 155 59 𝑍(𝑇 𝑍(𝑇 VERB iajs-2288 155 60 ) ) PUNCT iajs-2288 155 61 = = SYM iajs-2288 155 62 𝑇 𝑇 PROPN iajs-2288 155 63 , , PUNCT iajs-2288 155 64 where where SCONJ iajs-2288 155 65 𝑍(𝑇 𝑍(𝑇 VERB iajs-2288 155 66 ) ) PUNCT iajs-2288 155 67 = = PRON iajs-2288 155 68 { { PUNCT iajs-2288 155 69 π‘₯ π‘₯ PUNCT iajs-2288 155 70 ∈ ∈ PROPN iajs-2288 155 71 𝑇 𝑇 PROPN iajs-2288 155 72 ∢ ∢ NOUN iajs-2288 155 73 π‘₯𝐼 π‘₯𝐼 PROPN iajs-2288 156 1 = = PUNCT iajs-2288 156 2 0 0 NUM iajs-2288 156 3 for for ADP iajs-2288 156 4 some some DET iajs-2288 156 5 essential essential ADJ iajs-2288 156 6 ideal ideal NOUN iajs-2288 156 7 𝐼 𝐼 PROPN iajs-2288 156 8 of of ADP iajs-2288 156 9 𝑅 𝑅 PROPN iajs-2288 156 10 } } PUNCT iajs-2288 156 11 [ [ X iajs-2288 156 12 12 12 NUM iajs-2288 156 13 ] ] PUNCT iajs-2288 156 14 . . PUNCT iajs-2288 157 1 we we PRON iajs-2288 157 2 need need VERB iajs-2288 157 3 the the DET iajs-2288 157 4 following follow VERB iajs-2288 157 5 lemma lemma PROPN iajs-2288 157 6 which which PRON iajs-2288 157 7 appears appear VERB iajs-2288 157 8 in in ADP iajs-2288 157 9 [ [ X iajs-2288 157 10 12 12 NUM iajs-2288 157 11 ] ] PUNCT iajs-2288 157 12 . . PUNCT iajs-2288 158 1 before before SCONJ iajs-2288 158 2 we we PRON iajs-2288 158 3 introduced introduce VERB iajs-2288 158 4 the the DET iajs-2288 158 5 next next ADJ iajs-2288 158 6 result result NOUN iajs-2288 158 7 . . PUNCT iajs-2288 159 1 lemma lemma PROPN iajs-2288 159 2 ( ( PUNCT iajs-2288 159 3 21 21 NUM iajs-2288 159 4 ) ) PUNCT iajs-2288 160 1 [ [ X iajs-2288 160 2 12 12 NUM iajs-2288 160 3 , , PUNCT iajs-2288 160 4 coro coro NOUN iajs-2288 160 5 . . PUNCT iajs-2288 161 1 1.26 1.26 NUM iajs-2288 161 2 ] ] PUNCT iajs-2288 161 3 . . PUNCT iajs-2288 162 1 if if SCONJ iajs-2288 162 2 𝑇 𝑇 PROPN iajs-2288 162 3 is be AUX iajs-2288 162 4 a a DET iajs-2288 162 5 non non ADJ iajs-2288 162 6 - - ADJ iajs-2288 162 7 singular singular ADJ iajs-2288 162 8 𝑅-module 𝑅-module NOUN iajs-2288 162 9 , , PUNCT iajs-2288 162 10 then then ADV iajs-2288 162 11 π‘ π‘œπ‘(𝑅)𝑇 π‘ π‘œπ‘(𝑅)𝑇 PROPN iajs-2288 162 12 = = SYM iajs-2288 162 13 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 162 14 ) ) PUNCT iajs-2288 162 15 . . PUNCT iajs-2288 163 1 proposition proposition NOUN iajs-2288 163 2 ( ( PUNCT iajs-2288 163 3 22 22 NUM iajs-2288 163 4 ) ) PUNCT iajs-2288 163 5 let let VERB iajs-2288 163 6 𝐿 𝐿 PROPN iajs-2288 163 7 be be AUX iajs-2288 163 8 a a DET iajs-2288 163 9 proper proper ADJ iajs-2288 163 10 submodule submodule NOUN iajs-2288 163 11 of of ADP iajs-2288 163 12 non non ADJ iajs-2288 163 13 - - ADJ iajs-2288 163 14 singular singular ADJ iajs-2288 163 15 multiplication multiplication NOUN iajs-2288 163 16 𝑅-module 𝑅-module PROPN iajs-2288 163 17 𝑇. 𝑇. PROPN iajs-2288 163 18 then then ADV iajs-2288 163 19 𝐿 𝐿 PROPN iajs-2288 163 20 is be AUX iajs-2288 163 21 an an DET iajs-2288 163 22 app app ADJ iajs-2288 163 23 - - PUNCT iajs-2288 163 24 semi semi ADJ iajs-2288 163 25 - - ADJ iajs-2288 163 26 prime prime ADJ iajs-2288 163 27 submodule submodule NOUN iajs-2288 163 28 of of ADP iajs-2288 163 29 𝑇 𝑇 PROPN iajs-2288 164 1 if if SCONJ iajs-2288 164 2 and and CCONJ iajs-2288 164 3 only only ADV iajs-2288 164 4 if if SCONJ iajs-2288 164 5 [ [ X iajs-2288 164 6 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 164 7 𝑇 𝑇 PROPN iajs-2288 164 8 ] ] PUNCT iajs-2288 164 9 is be AUX iajs-2288 164 10 an an DET iajs-2288 164 11 app app ADJ iajs-2288 164 12 - - PUNCT iajs-2288 164 13 semi semi ADJ iajs-2288 164 14 - - ADJ iajs-2288 164 15 prime prime ADJ iajs-2288 164 16 ideal ideal NOUN iajs-2288 164 17 of of ADP iajs-2288 164 18 𝑅. 𝑅. ADJ iajs-2288 164 19 proof proof NOUN iajs-2288 164 20 follows follow VERB iajs-2288 164 21 by by ADP iajs-2288 164 22 similar similar ADJ iajs-2288 164 23 steps step NOUN iajs-2288 164 24 of of ADP iajs-2288 164 25 proposition proposition NOUN iajs-2288 164 26 ( ( PUNCT iajs-2288 164 27 2.20 2.20 NUM iajs-2288 164 28 ) ) PUNCT iajs-2288 164 29 and and CCONJ iajs-2288 164 30 used use VERB iajs-2288 164 31 lemma lemma PROPN iajs-2288 164 32 ( ( PUNCT iajs-2288 164 33 2.21 2.21 NUM iajs-2288 164 34 ) ) PUNCT iajs-2288 164 35 , , PUNCT iajs-2288 164 36 and and CCONJ iajs-2288 164 37 proposition proposition NOUN iajs-2288 164 38 ( ( PUNCT iajs-2288 164 39 2.16 2.16 NUM iajs-2288 164 40 ) ) PUNCT iajs-2288 164 41 . . PUNCT iajs-2288 165 1 we we PRON iajs-2288 165 2 need need VERB iajs-2288 165 3 the the DET iajs-2288 165 4 following follow VERB iajs-2288 165 5 lemma lemma PROPN iajs-2288 165 6 which which PRON iajs-2288 165 7 appear appear VERB iajs-2288 165 8 in in ADP iajs-2288 165 9 [ [ X iajs-2288 165 10 16 16 NUM iajs-2288 165 11 ] ] PUNCT iajs-2288 165 12 . . PUNCT iajs-2288 166 1 before before SCONJ iajs-2288 166 2 we we PRON iajs-2288 166 3 introduce introduce VERB iajs-2288 166 4 the the DET iajs-2288 166 5 next next ADJ iajs-2288 166 6 result result NOUN iajs-2288 166 7 . . PUNCT iajs-2288 167 1 lemma lemma PROPN iajs-2288 167 2 ( ( PUNCT iajs-2288 167 3 23)[16 23)[16 NUM iajs-2288 167 4 , , PUNCT iajs-2288 167 5 coro coro X iajs-2288 167 6 . . PUNCT iajs-2288 167 7 of of ADP iajs-2288 167 8 theo theo PROPN iajs-2288 167 9 . . PUNCT iajs-2288 168 1 9 9 NUM iajs-2288 168 2 ] ] PUNCT iajs-2288 168 3 . . PUNCT iajs-2288 169 1 let let VERB iajs-2288 169 2 𝑇 𝑇 PROPN iajs-2288 169 3 be be AUX iajs-2288 169 4 a a DET iajs-2288 169 5 finitely finitely ADV iajs-2288 169 6 generated generate VERB iajs-2288 169 7 multiplication multiplication NOUN iajs-2288 169 8 𝑅-module 𝑅-module PROPN iajs-2288 169 9 and and CCONJ iajs-2288 169 10 𝐼 𝐼 PROPN iajs-2288 169 11 , , PUNCT iajs-2288 169 12 𝐽 𝐽 PROPN iajs-2288 169 13 are be AUX iajs-2288 169 14 ideals ideal NOUN iajs-2288 169 15 of of ADP iajs-2288 169 16 a a DET iajs-2288 169 17 ring ring NOUN iajs-2288 169 18 𝑅. 𝑅. ADV iajs-2288 169 19 then then ADV iajs-2288 169 20 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 169 21 βŠ† βŠ† NUM iajs-2288 169 22 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 169 23 if if SCONJ iajs-2288 169 24 and and CCONJ iajs-2288 169 25 only only ADV iajs-2288 169 26 if if SCONJ iajs-2288 169 27 𝐼 𝐼 PROPN iajs-2288 169 28 βŠ† βŠ† NUM iajs-2288 169 29 𝐽 𝐽 NOUN iajs-2288 169 30 + + CCONJ iajs-2288 169 31 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 NUM iajs-2288 169 32 ) ) PUNCT iajs-2288 169 33 . . PUNCT iajs-2288 170 1 proposition proposition NOUN iajs-2288 170 2 ( ( PUNCT iajs-2288 170 3 24 24 NUM iajs-2288 170 4 ) ) PUNCT iajs-2288 170 5 let let VERB iajs-2288 170 6 𝑇 𝑇 PROPN iajs-2288 170 7 be be AUX iajs-2288 170 8 a a DET iajs-2288 170 9 faithful faithful ADJ iajs-2288 170 10 finitely finitely ADV iajs-2288 170 11 generated generate VERB iajs-2288 170 12 multiplication multiplication NOUN iajs-2288 170 13 𝑅-module 𝑅-module PROPN iajs-2288 170 14 and and CCONJ iajs-2288 170 15 𝐽 𝐽 PROPN iajs-2288 170 16 be be AUX iajs-2288 170 17 an an DET iajs-2288 170 18 app app ADJ iajs-2288 170 19 - - PUNCT iajs-2288 170 20 semiprime semiprime NOUN iajs-2288 170 21 ideal ideal NOUN iajs-2288 170 22 of of ADP iajs-2288 170 23 𝑅. 𝑅. NOUN iajs-2288 170 24 then then ADV iajs-2288 170 25 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 170 26 is be AUX iajs-2288 170 27 an an DET iajs-2288 170 28 app app ADJ iajs-2288 170 29 - - PUNCT iajs-2288 170 30 semi semi ADJ iajs-2288 170 31 - - ADJ iajs-2288 170 32 prime prime ADJ iajs-2288 170 33 submodule submodule NOUN iajs-2288 170 34 of of ADP iajs-2288 170 35 𝑇. 𝑇. PROPN iajs-2288 170 36 proof proof NOUN iajs-2288 170 37 let let VERB iajs-2288 170 38 π‘Žπ‘›πΈ π‘Žπ‘›πΈ X iajs-2288 170 39 βŠ† βŠ† NUM iajs-2288 170 40 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 170 41 , , PUNCT iajs-2288 170 42 where where SCONJ iajs-2288 170 43 π‘Ž π‘Ž PRON iajs-2288 170 44 ∈ ∈ PROPN iajs-2288 170 45 𝑅 𝑅 PROPN iajs-2288 170 46 , , PUNCT iajs-2288 170 47 𝐸 𝐸 PROPN iajs-2288 170 48 be be VERB iajs-2288 170 49 a a DET iajs-2288 170 50 submodule submodule NOUN iajs-2288 170 51 of of ADP iajs-2288 170 52 𝑇 𝑇 PROPN iajs-2288 170 53 and and CCONJ iajs-2288 170 54 𝑛 𝑛 DET iajs-2288 170 55 ∈ ∈ PROPN iajs-2288 170 56 𝑍+ 𝑍+ NOUN iajs-2288 170 57 . . PUNCT iajs-2288 171 1 since since SCONJ iajs-2288 171 2 𝑇 𝑇 PROPN iajs-2288 171 3 is be AUX iajs-2288 171 4 a a DET iajs-2288 171 5 multiplication multiplication NOUN iajs-2288 171 6 , , PUNCT iajs-2288 171 7 then then ADV iajs-2288 171 8 𝐸 𝐸 PROPN iajs-2288 171 9 = = PUNCT iajs-2288 171 10 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 171 11 for for ADP iajs-2288 171 12 some some DET iajs-2288 171 13 ideal ideal NOUN iajs-2288 171 14 𝐼 𝐼 ADP iajs-2288 171 15 of of ADP iajs-2288 171 16 𝑅. 𝑅. NOUN iajs-2288 171 17 that that PRON iajs-2288 171 18 is be AUX iajs-2288 171 19 π‘Žπ‘›πΌπ‘‡ π‘Žπ‘›πΌπ‘‡ NUM iajs-2288 171 20 βŠ† βŠ† NUM iajs-2288 171 21 𝐽𝑇. 𝐽𝑇. NOUN iajs-2288 171 22 but but CCONJ iajs-2288 171 23 𝑇 𝑇 PROPN iajs-2288 171 24 is be AUX iajs-2288 171 25 a a DET iajs-2288 171 26 finitely finitely ADV iajs-2288 171 27 generated generate VERB iajs-2288 171 28 , , PUNCT iajs-2288 171 29 so so ADV iajs-2288 171 30 by by ADP iajs-2288 171 31 lemma lemma PROPN iajs-2288 171 32 ( ( PUNCT iajs-2288 171 33 2.23 2.23 NUM iajs-2288 171 34 ) ) PUNCT iajs-2288 171 35 we we PRON iajs-2288 171 36 have have AUX iajs-2288 171 37 π‘Žπ‘›πΌ π‘Žπ‘›πΌ VERB iajs-2288 171 38 βŠ† βŠ† NUM iajs-2288 171 39 𝐽 𝐽 PROPN iajs-2288 171 40 + + CCONJ iajs-2288 171 41 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 NUM iajs-2288 171 42 ) ) PUNCT iajs-2288 171 43 , , PUNCT iajs-2288 171 44 but but CCONJ iajs-2288 171 45 𝑇 𝑇 PROPN iajs-2288 171 46 is be AUX iajs-2288 171 47 faithful faithful ADJ iajs-2288 171 48 , , PUNCT iajs-2288 171 49 then then ADV iajs-2288 171 50 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 NOUN iajs-2288 171 51 ) ) PUNCT iajs-2288 171 52 = = SYM iajs-2288 171 53 ( ( PUNCT iajs-2288 171 54 0 0 NUM iajs-2288 171 55 ) ) PUNCT iajs-2288 171 56 , , PUNCT iajs-2288 171 57 hence hence ADV iajs-2288 171 58 π‘Žπ‘›πΌ π‘Žπ‘›πΌ VERB iajs-2288 171 59 βŠ† βŠ† NUM iajs-2288 171 60 𝐽 𝐽 PROPN iajs-2288 171 61 , , PUNCT iajs-2288 171 62 but but CCONJ iajs-2288 171 63 𝐽 𝐽 PRON iajs-2288 171 64 an an DET iajs-2288 171 65 app app ADJ iajs-2288 171 66 - - PUNCT iajs-2288 171 67 semi semi ADJ iajs-2288 171 68 - - ADJ iajs-2288 171 69 prime prime ADJ iajs-2288 171 70 ideal ideal NOUN iajs-2288 171 71 of of ADP iajs-2288 171 72 𝑅 𝑅 PROPN iajs-2288 171 73 , , PUNCT iajs-2288 171 74 then then ADV iajs-2288 171 75 by by ADP iajs-2288 171 76 proposition proposition NOUN iajs-2288 171 77 ( ( PUNCT iajs-2288 171 78 2.3 2.3 NUM iajs-2288 171 79 ) ) PUNCT iajs-2288 171 80 π‘ŽπΌ π‘ŽπΌ PROPN iajs-2288 171 81 βŠ† βŠ† NUM iajs-2288 171 82 𝐽 𝐽 NOUN iajs-2288 171 83 + + CCONJ iajs-2288 171 84 π‘ π‘œπ‘(𝑅 π‘ π‘œπ‘(𝑅 PROPN iajs-2288 171 85 ) ) PUNCT iajs-2288 171 86 . . PUNCT iajs-2288 172 1 that that PRON iajs-2288 172 2 is be AUX iajs-2288 172 3 π‘ŽπΌπ‘‡ π‘ŽπΌπ‘‡ NOUN iajs-2288 172 4 βŠ† βŠ† NUM iajs-2288 172 5 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 172 6 + + CCONJ iajs-2288 172 7 π‘ π‘œπ‘(𝑅)𝑇 π‘ π‘œπ‘(𝑅)𝑇 NOUN iajs-2288 172 8 , , PUNCT iajs-2288 172 9 then then ADV iajs-2288 172 10 by by ADP iajs-2288 172 11 lemma lemma PROPN iajs-2288 172 12 ( ( PUNCT iajs-2288 172 13 2.19 2.19 NUM iajs-2288 172 14 ) ) PUNCT iajs-2288 172 15 we we PRON iajs-2288 172 16 get get VERB iajs-2288 172 17 π‘ŽπΈ π‘ŽπΈ NOUN iajs-2288 172 18 βŠ† βŠ† NUM iajs-2288 172 19 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 172 20 + + CCONJ iajs-2288 172 21 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 172 22 ) ) PUNCT iajs-2288 172 23 . . PUNCT iajs-2288 173 1 hence hence ADV iajs-2288 173 2 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 173 3 is be AUX iajs-2288 173 4 an an DET iajs-2288 173 5 app app ADJ iajs-2288 173 6 - - PUNCT iajs-2288 173 7 semi semi ADJ iajs-2288 173 8 - - ADJ iajs-2288 173 9 prime prime ADJ iajs-2288 173 10 submodule submodule NOUN iajs-2288 173 11 of of ADP iajs-2288 173 12 𝑇. 𝑇. PROPN iajs-2288 173 13 124 124 NUM iajs-2288 173 14 ibn ibn PROPN iajs-2288 173 15 al al PROPN iajs-2288 173 16 - - PUNCT iajs-2288 173 17 haitham haitham PROPN iajs-2288 173 18 jour jour X iajs-2288 173 19 . . PROPN iajs-2288 173 20 for for ADP iajs-2288 173 21 pure pure ADJ iajs-2288 173 22 & & CCONJ iajs-2288 173 23 appl appl PROPN iajs-2288 173 24 . . PUNCT iajs-2288 174 1 sci sci PROPN iajs-2288 174 2 . . PROPN iajs-2288 174 3 32 32 NUM iajs-2288 174 4 ( ( PUNCT iajs-2288 174 5 3 3 NUM iajs-2288 174 6 ) ) SYM iajs-2288 174 7 2019 2019 NUM iajs-2288 174 8 proposition proposition NOUN iajs-2288 174 9 ( ( PUNCT iajs-2288 174 10 25 25 NUM iajs-2288 174 11 ) ) PUNCT iajs-2288 174 12 let let VERB iajs-2288 174 13 𝑇 𝑇 PROPN iajs-2288 174 14 be be AUX iajs-2288 174 15 a a DET iajs-2288 174 16 finitely finitely ADV iajs-2288 174 17 generated generate VERB iajs-2288 174 18 multiplication multiplication NOUN iajs-2288 174 19 non non ADJ iajs-2288 174 20 - - ADJ iajs-2288 174 21 singular singular ADJ iajs-2288 174 22 𝑅-module 𝑅-module PROPN iajs-2288 174 23 and and CCONJ iajs-2288 174 24 𝐽 𝐽 PROPN iajs-2288 174 25 be be AUX iajs-2288 174 26 an an DET iajs-2288 174 27 app app ADJ iajs-2288 174 28 - - PUNCT iajs-2288 174 29 semiprime semiprime NOUN iajs-2288 174 30 ideal ideal NOUN iajs-2288 174 31 of of ADP iajs-2288 174 32 𝑅 𝑅 PROPN iajs-2288 174 33 with with ADP iajs-2288 174 34 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 NOUN iajs-2288 174 35 ) ) PUNCT iajs-2288 174 36 βŠ† βŠ† X iajs-2288 174 37 𝐽. 𝐽. PROPN iajs-2288 174 38 then then ADV iajs-2288 174 39 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 174 40 is be AUX iajs-2288 174 41 an an DET iajs-2288 174 42 app app ADJ iajs-2288 174 43 - - PUNCT iajs-2288 174 44 semi semi ADJ iajs-2288 174 45 - - ADJ iajs-2288 174 46 prime prime ADJ iajs-2288 174 47 submodule submodule NOUN iajs-2288 174 48 of of ADP iajs-2288 174 49 𝑇. 𝑇. PROPN iajs-2288 174 50 proof proof NOUN iajs-2288 174 51 suppose suppose VERB iajs-2288 174 52 that that SCONJ iajs-2288 174 53 π‘Žπ‘›πΏ π‘Žπ‘›πΏ PROPN iajs-2288 174 54 βŠ† βŠ† NUM iajs-2288 174 55 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 174 56 , , PUNCT iajs-2288 174 57 where where SCONJ iajs-2288 174 58 π‘Ž π‘Ž PRON iajs-2288 174 59 ∈ ∈ PROPN iajs-2288 174 60 𝑅 𝑅 PROPN iajs-2288 174 61 , , PUNCT iajs-2288 174 62 𝐿 𝐿 PROPN iajs-2288 174 63 be be VERB iajs-2288 174 64 a a DET iajs-2288 174 65 submodule submodule NOUN iajs-2288 174 66 of of ADP iajs-2288 174 67 𝑇 𝑇 PROPN iajs-2288 174 68 and and CCONJ iajs-2288 174 69 𝑛 𝑛 DET iajs-2288 174 70 ∈ ∈ PROPN iajs-2288 174 71 𝑍+ 𝑍+ NOUN iajs-2288 174 72 . . PUNCT iajs-2288 175 1 since since SCONJ iajs-2288 175 2 𝑇 𝑇 PROPN iajs-2288 175 3 is be AUX iajs-2288 175 4 a a DET iajs-2288 175 5 multiplication multiplication NOUN iajs-2288 175 6 , , PUNCT iajs-2288 175 7 then then ADV iajs-2288 175 8 𝐿 𝐿 PROPN iajs-2288 175 9 = = PROPN iajs-2288 175 10 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 175 11 for for ADP iajs-2288 175 12 some some DET iajs-2288 175 13 ideal ideal NOUN iajs-2288 175 14 𝐼 𝐼 ADP iajs-2288 175 15 of of ADP iajs-2288 175 16 𝑅. 𝑅. NOUN iajs-2288 175 17 that that PRON iajs-2288 175 18 is be AUX iajs-2288 175 19 π‘Žπ‘›πΌπ‘‡ π‘Žπ‘›πΌπ‘‡ NUM iajs-2288 175 20 βŠ† βŠ† NUM iajs-2288 175 21 𝐽𝑇. 𝐽𝑇. NOUN iajs-2288 175 22 but but CCONJ iajs-2288 175 23 𝑇 𝑇 PROPN iajs-2288 175 24 is be AUX iajs-2288 175 25 a a DET iajs-2288 175 26 finitely finitely ADV iajs-2288 175 27 generated generate VERB iajs-2288 175 28 , , PUNCT iajs-2288 175 29 so so ADV iajs-2288 175 30 by by ADP iajs-2288 175 31 lemma lemma PROPN iajs-2288 175 32 ( ( PUNCT iajs-2288 175 33 2.23 2.23 NUM iajs-2288 175 34 ) ) PUNCT iajs-2288 175 35 we we PRON iajs-2288 175 36 have have AUX iajs-2288 175 37 π‘Žπ‘›πΌ π‘Žπ‘›πΌ VERB iajs-2288 175 38 βŠ† βŠ† NUM iajs-2288 175 39 𝐽 𝐽 PROPN iajs-2288 175 40 + + CCONJ iajs-2288 175 41 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 NUM iajs-2288 175 42 ) ) PUNCT iajs-2288 175 43 , , PUNCT iajs-2288 175 44 but but CCONJ iajs-2288 175 45 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 NUM iajs-2288 175 46 ) ) PUNCT iajs-2288 175 47 βŠ† βŠ† NUM iajs-2288 175 48 𝐽 𝐽 PROPN iajs-2288 175 49 , , PUNCT iajs-2288 175 50 implies imply VERB iajs-2288 175 51 that that SCONJ iajs-2288 175 52 𝐽 𝐽 PROPN iajs-2288 175 53 + + CCONJ iajs-2288 175 54 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 NUM iajs-2288 175 55 ) ) PUNCT iajs-2288 175 56 = = SYM iajs-2288 175 57 𝐽 𝐽 PROPN iajs-2288 175 58 , , PUNCT iajs-2288 175 59 so so ADV iajs-2288 175 60 π‘Žπ‘›πΌ π‘Žπ‘›πΌ VERB iajs-2288 175 61 βŠ† βŠ† NUM iajs-2288 175 62 𝐽 𝐽 PROPN iajs-2288 175 63 , , PUNCT iajs-2288 175 64 but but CCONJ iajs-2288 175 65 𝐽 𝐽 PRON iajs-2288 175 66 an an DET iajs-2288 175 67 app app ADJ iajs-2288 175 68 - - PUNCT iajs-2288 175 69 semi semi ADJ iajs-2288 175 70 - - ADJ iajs-2288 175 71 prime prime ADJ iajs-2288 175 72 ideal ideal NOUN iajs-2288 175 73 of of ADP iajs-2288 175 74 𝑅 𝑅 PROPN iajs-2288 175 75 , , PUNCT iajs-2288 175 76 we we PRON iajs-2288 175 77 have have VERB iajs-2288 175 78 π‘ŽπΌ π‘ŽπΌ PROPN iajs-2288 175 79 βŠ† βŠ† NUM iajs-2288 175 80 𝐽 𝐽 NOUN iajs-2288 175 81 + + CCONJ iajs-2288 175 82 π‘ π‘œπ‘(𝑅 π‘ π‘œπ‘(𝑅 PROPN iajs-2288 175 83 ) ) PUNCT iajs-2288 175 84 . . PUNCT iajs-2288 176 1 that that PRON iajs-2288 176 2 is be AUX iajs-2288 176 3 π‘ŽπΌπ‘‡ π‘ŽπΌπ‘‡ NOUN iajs-2288 176 4 βŠ† βŠ† NUM iajs-2288 176 5 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 176 6 + + CCONJ iajs-2288 176 7 π‘ π‘œπ‘(𝑅)𝑇 π‘ π‘œπ‘(𝑅)𝑇 NOUN iajs-2288 176 8 , , PUNCT iajs-2288 176 9 then then ADV iajs-2288 176 10 by by ADP iajs-2288 176 11 lemma lemma PROPN iajs-2288 176 12 ( ( PUNCT iajs-2288 176 13 2.19 2.19 NUM iajs-2288 176 14 ) ) PUNCT iajs-2288 176 15 we we PRON iajs-2288 176 16 get get VERB iajs-2288 176 17 π‘ŽπΏ π‘ŽπΏ PROPN iajs-2288 176 18 βŠ† βŠ† NUM iajs-2288 176 19 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 176 20 + + CCONJ iajs-2288 176 21 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 176 22 ) ) PUNCT iajs-2288 176 23 . . PUNCT iajs-2288 177 1 hence hence ADV iajs-2288 177 2 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 177 3 is be AUX iajs-2288 177 4 an an DET iajs-2288 177 5 app app ADJ iajs-2288 177 6 - - PUNCT iajs-2288 177 7 semi semi ADJ iajs-2288 177 8 - - ADJ iajs-2288 177 9 prime prime ADJ iajs-2288 177 10 submodule submodule NOUN iajs-2288 177 11 of of ADP iajs-2288 177 12 𝑇. 𝑇. PROPN iajs-2288 177 13 theorem theorem NOUN iajs-2288 177 14 ( ( PUNCT iajs-2288 177 15 26 26 NUM iajs-2288 177 16 ) ) PUNCT iajs-2288 177 17 let let VERB iajs-2288 177 18 𝑇 𝑇 PROPN iajs-2288 177 19 be be AUX iajs-2288 177 20 a a DET iajs-2288 177 21 faithful faithful ADJ iajs-2288 177 22 finitely finitely ADV iajs-2288 177 23 generated generate VERB iajs-2288 177 24 multiplication multiplication NOUN iajs-2288 177 25 𝑅-module 𝑅-module PROPN iajs-2288 177 26 and and CCONJ iajs-2288 177 27 𝐿 𝐿 PROPN iajs-2288 177 28 be be AUX iajs-2288 177 29 a a DET iajs-2288 177 30 proper proper ADJ iajs-2288 177 31 submodule submodule NOUN iajs-2288 177 32 of of ADP iajs-2288 177 33 𝑇. 𝑇. PROPN iajs-2288 177 34 then then ADV iajs-2288 177 35 the the DET iajs-2288 177 36 following follow VERB iajs-2288 177 37 statements statement NOUN iajs-2288 177 38 are be AUX iajs-2288 177 39 equivalent equivalent ADJ iajs-2288 177 40 . . PUNCT iajs-2288 178 1 1 1 X iajs-2288 178 2 ) ) PUNCT iajs-2288 178 3 𝐿 𝐿 PROPN iajs-2288 178 4 is be AUX iajs-2288 178 5 an an DET iajs-2288 178 6 app app ADJ iajs-2288 178 7 - - PUNCT iajs-2288 178 8 semi semi ADJ iajs-2288 178 9 - - ADJ iajs-2288 178 10 prime prime ADJ iajs-2288 178 11 submodule submodule NOUN iajs-2288 178 12 of of ADP iajs-2288 178 13 𝑇. 𝑇. PROPN iajs-2288 178 14 2 2 NUM iajs-2288 178 15 ) ) PUNCT iajs-2288 179 1 [ [ X iajs-2288 179 2 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 179 3 𝑇 𝑇 PROPN iajs-2288 179 4 ] ] PUNCT iajs-2288 179 5 is be AUX iajs-2288 179 6 an an DET iajs-2288 179 7 app app ADJ iajs-2288 179 8 - - PUNCT iajs-2288 179 9 semi semi ADJ iajs-2288 179 10 - - ADJ iajs-2288 179 11 prime prime ADJ iajs-2288 179 12 ideal ideal NOUN iajs-2288 179 13 of of ADP iajs-2288 179 14 𝑅. 𝑅. NOUN iajs-2288 179 15 3 3 NUM iajs-2288 179 16 ) ) PUNCT iajs-2288 179 17 𝐿 𝐿 PROPN iajs-2288 179 18 = = SYM iajs-2288 179 19 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 179 20 for for ADP iajs-2288 179 21 some some DET iajs-2288 179 22 app app ADJ iajs-2288 179 23 - - PUNCT iajs-2288 179 24 semi semi ADJ iajs-2288 179 25 - - ADJ iajs-2288 179 26 prime prime ADJ iajs-2288 179 27 ideal ideal NOUN iajs-2288 179 28 𝐼 𝐼 ADP iajs-2288 179 29 of of ADP iajs-2288 179 30 𝑅. 𝑅. ADJ iajs-2288 179 31 proof proof NOUN iajs-2288 179 32 ( ( PUNCT iajs-2288 179 33 1 1 X iajs-2288 179 34 ) ) PUNCT iajs-2288 179 35 ⇐ ⇐ NOUN iajs-2288 179 36 ( ( PUNCT iajs-2288 179 37 2 2 X iajs-2288 179 38 ) ) PUNCT iajs-2288 179 39 follows follow VERB iajs-2288 179 40 by by ADP iajs-2288 179 41 proposition proposition NOUN iajs-2288 179 42 ( ( PUNCT iajs-2288 179 43 2.20 2.20 NUM iajs-2288 179 44 ) ) PUNCT iajs-2288 179 45 . . PUNCT iajs-2288 180 1 ( ( PUNCT iajs-2288 180 2 2 2 X iajs-2288 180 3 ) ) PUNCT iajs-2288 180 4 β‡’ β‡’ NOUN iajs-2288 180 5 ( ( PUNCT iajs-2288 180 6 3 3 NUM iajs-2288 180 7 ) ) PUNCT iajs-2288 180 8 since since SCONJ iajs-2288 180 9 [ [ X iajs-2288 180 10 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 180 11 𝑇 𝑇 PROPN iajs-2288 180 12 ] ] PUNCT iajs-2288 180 13 is be AUX iajs-2288 180 14 an an DET iajs-2288 180 15 app app ADJ iajs-2288 180 16 - - PUNCT iajs-2288 180 17 semi semi ADJ iajs-2288 180 18 - - ADJ iajs-2288 180 19 prime prime ADJ iajs-2288 180 20 ideal ideal NOUN iajs-2288 180 21 of of ADP iajs-2288 180 22 𝑅 𝑅 PROPN iajs-2288 180 23 , , PUNCT iajs-2288 180 24 and and CCONJ iajs-2288 180 25 𝐿 𝐿 PROPN iajs-2288 180 26 = = SYM iajs-2288 180 27 [ [ X iajs-2288 180 28 𝐿:𝑅 𝐿:𝑅 X iajs-2288 180 29 𝑇]𝑇 𝑇]𝑇 PROPN iajs-2288 180 30 for for ADP iajs-2288 180 31 𝑇 𝑇 PROPN iajs-2288 180 32 is be AUX iajs-2288 180 33 a a DET iajs-2288 180 34 multiplication multiplication NOUN iajs-2288 180 35 , , PUNCT iajs-2288 180 36 implies imply VERB iajs-2288 180 37 that that SCONJ iajs-2288 180 38 𝐿 𝐿 PROPN iajs-2288 180 39 = = PROPN iajs-2288 180 40 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 180 41 where where SCONJ iajs-2288 180 42 [ [ X iajs-2288 180 43 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 180 44 𝑇 𝑇 PROPN iajs-2288 180 45 ] ] PUNCT iajs-2288 180 46 is be AUX iajs-2288 180 47 an an DET iajs-2288 180 48 app app ADJ iajs-2288 180 49 - - PUNCT iajs-2288 180 50 semi semi ADJ iajs-2288 180 51 - - ADJ iajs-2288 180 52 prime prime ADJ iajs-2288 180 53 ideal ideal NOUN iajs-2288 180 54 of of ADP iajs-2288 180 55 𝑅. 𝑅. NOUN iajs-2288 180 56 ( ( PUNCT iajs-2288 180 57 3 3 X iajs-2288 180 58 ) ) PUNCT iajs-2288 180 59 β‡’ β‡’ NOUN iajs-2288 180 60 ( ( PUNCT iajs-2288 180 61 2 2 X iajs-2288 180 62 ) ) PUNCT iajs-2288 180 63 suppose suppose VERB iajs-2288 180 64 that that SCONJ iajs-2288 180 65 𝐿 𝐿 PROPN iajs-2288 180 66 = = PUNCT iajs-2288 180 67 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 180 68 for for ADP iajs-2288 180 69 some some DET iajs-2288 180 70 app app ADJ iajs-2288 180 71 - - PUNCT iajs-2288 180 72 semi semi ADJ iajs-2288 180 73 - - ADJ iajs-2288 180 74 prime prime ADJ iajs-2288 180 75 ideal ideal NOUN iajs-2288 180 76 𝐼 𝐼 ADP iajs-2288 180 77 of of ADP iajs-2288 180 78 𝑅. 𝑅. NOUN iajs-2288 180 79 but but CCONJ iajs-2288 180 80 𝑇 𝑇 PROPN iajs-2288 180 81 is be AUX iajs-2288 180 82 a a DET iajs-2288 180 83 faithful faithful ADJ iajs-2288 180 84 finitely finitely ADV iajs-2288 180 85 generated generate VERB iajs-2288 180 86 multiplication multiplication NOUN iajs-2288 180 87 , , PUNCT iajs-2288 180 88 then then ADV iajs-2288 180 89 by by ADP iajs-2288 180 90 lemma lemma PROPN iajs-2288 180 91 ( ( PUNCT iajs-2288 180 92 2.23 2.23 NUM iajs-2288 180 93 ) ) PUNCT iajs-2288 181 1 𝐼 𝐼 NOUN iajs-2288 181 2 = = PUNCT iajs-2288 182 1 [ [ X iajs-2288 182 2 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 182 3 𝑇 𝑇 PROPN iajs-2288 182 4 ] ] PUNCT iajs-2288 182 5 , , PUNCT iajs-2288 182 6 hence hence ADV iajs-2288 182 7 [ [ X iajs-2288 182 8 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 182 9 𝑇 𝑇 PROPN iajs-2288 182 10 ] ] PUNCT iajs-2288 182 11 is be AUX iajs-2288 182 12 an an DET iajs-2288 182 13 appsemi appsemi ADJ iajs-2288 182 14 - - ADJ iajs-2288 182 15 prime prime ADJ iajs-2288 182 16 ideal ideal NOUN iajs-2288 182 17 of of ADP iajs-2288 182 18 𝑅. 𝑅. NOUN iajs-2288 182 19 theorem theorem NOUN iajs-2288 182 20 ( ( PUNCT iajs-2288 182 21 27 27 NUM iajs-2288 182 22 ) ) PUNCT iajs-2288 182 23 let let VERB iajs-2288 182 24 𝑇 𝑇 PROPN iajs-2288 182 25 be be AUX iajs-2288 182 26 non non ADJ iajs-2288 182 27 - - ADJ iajs-2288 182 28 singular singular ADJ iajs-2288 182 29 finitely finitely ADV iajs-2288 182 30 generated generate VERB iajs-2288 182 31 multiplication multiplication NOUN iajs-2288 182 32 𝑅-module 𝑅-module PROPN iajs-2288 182 33 and and CCONJ iajs-2288 182 34 𝐿 𝐿 PROPN iajs-2288 182 35 be be AUX iajs-2288 182 36 a a DET iajs-2288 182 37 proper proper ADJ iajs-2288 182 38 submodule submodule NOUN iajs-2288 182 39 of of ADP iajs-2288 182 40 𝑇. 𝑇. PROPN iajs-2288 182 41 then then ADV iajs-2288 182 42 the the DET iajs-2288 182 43 following follow VERB iajs-2288 182 44 statements statement NOUN iajs-2288 182 45 are be AUX iajs-2288 182 46 equivalent equivalent ADJ iajs-2288 182 47 . . PUNCT iajs-2288 183 1 1 1 X iajs-2288 183 2 ) ) PUNCT iajs-2288 183 3 𝐿 𝐿 PROPN iajs-2288 183 4 is be AUX iajs-2288 183 5 an an DET iajs-2288 183 6 app app ADJ iajs-2288 183 7 - - PUNCT iajs-2288 183 8 semi semi ADJ iajs-2288 183 9 - - ADJ iajs-2288 183 10 prime prime ADJ iajs-2288 183 11 submodule submodule NOUN iajs-2288 183 12 of of ADP iajs-2288 183 13 𝑇. 𝑇. PROPN iajs-2288 183 14 2 2 NUM iajs-2288 183 15 ) ) PUNCT iajs-2288 184 1 [ [ X iajs-2288 184 2 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 184 3 𝑇 𝑇 PROPN iajs-2288 184 4 ] ] PUNCT iajs-2288 184 5 is be AUX iajs-2288 184 6 an an DET iajs-2288 184 7 app app ADJ iajs-2288 184 8 - - PUNCT iajs-2288 184 9 semi semi ADJ iajs-2288 184 10 - - ADJ iajs-2288 184 11 prime prime ADJ iajs-2288 184 12 ideal ideal NOUN iajs-2288 184 13 of of ADP iajs-2288 184 14 𝑅. 𝑅. NOUN iajs-2288 184 15 3 3 NUM iajs-2288 184 16 ) ) PUNCT iajs-2288 184 17 𝐿 𝐿 PROPN iajs-2288 184 18 = = SYM iajs-2288 184 19 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 184 20 for for ADP iajs-2288 184 21 some some DET iajs-2288 184 22 app app ADJ iajs-2288 184 23 - - PUNCT iajs-2288 184 24 semi semi ADJ iajs-2288 184 25 - - ADJ iajs-2288 184 26 prime prime ADJ iajs-2288 184 27 ideal ideal NOUN iajs-2288 184 28 𝐼 𝐼 PROPN iajs-2288 184 29 of of ADP iajs-2288 184 30 𝑅 𝑅 PROPN iajs-2288 184 31 with with ADP iajs-2288 184 32 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 NOUN iajs-2288 184 33 ) ) PUNCT iajs-2288 184 34 βŠ† βŠ† X iajs-2288 184 35 𝐼. 𝐼. PROPN iajs-2288 184 36 proof proof NOUN iajs-2288 184 37 ( ( PUNCT iajs-2288 184 38 1 1 X iajs-2288 184 39 ) ) PUNCT iajs-2288 184 40 ⇐ ⇐ NOUN iajs-2288 184 41 ( ( PUNCT iajs-2288 184 42 2 2 X iajs-2288 184 43 ) ) PUNCT iajs-2288 184 44 follows follow VERB iajs-2288 184 45 by by ADP iajs-2288 184 46 proposition proposition NOUN iajs-2288 184 47 ( ( PUNCT iajs-2288 184 48 2.22 2.22 NUM iajs-2288 184 49 ) ) PUNCT iajs-2288 184 50 . . PUNCT iajs-2288 185 1 ( ( PUNCT iajs-2288 185 2 2 2 X iajs-2288 185 3 ) ) PUNCT iajs-2288 185 4 β‡’ β‡’ NOUN iajs-2288 185 5 ( ( PUNCT iajs-2288 185 6 3 3 NUM iajs-2288 185 7 ) ) PUNCT iajs-2288 185 8 since since SCONJ iajs-2288 185 9 [ [ X iajs-2288 185 10 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 185 11 𝑇 𝑇 PROPN iajs-2288 185 12 ] ] PUNCT iajs-2288 185 13 is be AUX iajs-2288 185 14 an an DET iajs-2288 185 15 app app ADJ iajs-2288 185 16 - - PUNCT iajs-2288 185 17 semi semi ADJ iajs-2288 185 18 - - ADJ iajs-2288 185 19 prime prime ADJ iajs-2288 185 20 ideal ideal NOUN iajs-2288 185 21 of of ADP iajs-2288 185 22 𝑅 𝑅 PROPN iajs-2288 185 23 , , PUNCT iajs-2288 185 24 and and CCONJ iajs-2288 185 25 𝐿 𝐿 PROPN iajs-2288 185 26 = = SYM iajs-2288 185 27 [ [ X iajs-2288 185 28 𝐿:𝑅 𝐿:𝑅 X iajs-2288 185 29 𝑇]𝑇 𝑇]𝑇 PROPN iajs-2288 185 30 for for ADP iajs-2288 185 31 𝑇 𝑇 PROPN iajs-2288 185 32 is be AUX iajs-2288 185 33 a a DET iajs-2288 185 34 multiplication multiplication NOUN iajs-2288 185 35 , , PUNCT iajs-2288 185 36 then then ADV iajs-2288 185 37 𝐿 𝐿 PROPN iajs-2288 185 38 = = PROPN iajs-2288 185 39 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 185 40 and and CCONJ iajs-2288 185 41 𝐼 𝐼 PROPN iajs-2288 185 42 = = SYM iajs-2288 186 1 [ [ X iajs-2288 186 2 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 186 3 𝑇 𝑇 PROPN iajs-2288 186 4 ] ] PUNCT iajs-2288 186 5 is be AUX iajs-2288 186 6 an an DET iajs-2288 186 7 app app ADJ iajs-2288 186 8 - - PUNCT iajs-2288 186 9 semi semi ADJ iajs-2288 186 10 - - ADJ iajs-2288 186 11 prime prime ADJ iajs-2288 186 12 ideal ideal NOUN iajs-2288 186 13 of of ADP iajs-2288 186 14 𝑅 𝑅 PROPN iajs-2288 186 15 with with ADP iajs-2288 186 16 [ [ X iajs-2288 186 17 ( ( PUNCT iajs-2288 186 18 0):𝑅 0):𝑅 ADJ iajs-2288 186 19 𝑇 𝑇 PROPN iajs-2288 186 20 ] ] X iajs-2288 186 21 = = PUNCT iajs-2288 186 22 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 NUM iajs-2288 186 23 ) ) PUNCT iajs-2288 186 24 βŠ† βŠ† NUM iajs-2288 186 25 [ [ X iajs-2288 186 26 𝐿:𝑅 𝐿:𝑅 X iajs-2288 186 27 𝑇 𝑇 PROPN iajs-2288 186 28 ] ] PUNCT iajs-2288 186 29 . . PUNCT iajs-2288 187 1 ( ( PUNCT iajs-2288 187 2 3 3 X iajs-2288 187 3 ) ) PUNCT iajs-2288 187 4 β‡’ β‡’ NOUN iajs-2288 187 5 ( ( PUNCT iajs-2288 187 6 2 2 X iajs-2288 187 7 ) ) PUNCT iajs-2288 187 8 suppose suppose VERB iajs-2288 187 9 that that SCONJ iajs-2288 187 10 𝐿 𝐿 PROPN iajs-2288 187 11 = = PUNCT iajs-2288 187 12 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 187 13 for for ADP iajs-2288 187 14 some some DET iajs-2288 187 15 app app ADJ iajs-2288 187 16 - - PUNCT iajs-2288 187 17 semi semi ADJ iajs-2288 187 18 - - ADJ iajs-2288 187 19 prime prime ADJ iajs-2288 187 20 ideal ideal NOUN iajs-2288 187 21 𝐼 𝐼 PROPN iajs-2288 187 22 of of ADP iajs-2288 187 23 𝑅 𝑅 PROPN iajs-2288 187 24 with with ADP iajs-2288 187 25 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 NOUN iajs-2288 187 26 ) ) PUNCT iajs-2288 187 27 βŠ† βŠ† NUM iajs-2288 187 28 𝐼. 𝐼. NOUN iajs-2288 187 29 but but CCONJ iajs-2288 187 30 𝑇 𝑇 PROPN iajs-2288 187 31 is be AUX iajs-2288 187 32 a a DET iajs-2288 187 33 multiplication multiplication NOUN iajs-2288 187 34 , , PUNCT iajs-2288 187 35 we we PRON iajs-2288 187 36 have have VERB iajs-2288 187 37 𝐼 𝐼 NOUN iajs-2288 187 38 = = SYM iajs-2288 187 39 [ [ X iajs-2288 187 40 𝐿:𝑅 𝐿:𝑅 X iajs-2288 187 41 𝑇]𝑇 𝑇]𝑇 PROPN iajs-2288 187 42 = = SYM iajs-2288 187 43 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 187 44 , , PUNCT iajs-2288 187 45 that that PRON iajs-2288 187 46 is be AUX iajs-2288 187 47 [ [ X iajs-2288 187 48 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 187 49 𝑇 𝑇 PROPN iajs-2288 187 50 ] ] X iajs-2288 187 51 = = PUNCT iajs-2288 187 52 𝐼 𝐼 PROPN iajs-2288 187 53 + + CCONJ iajs-2288 187 54 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 PROPN iajs-2288 187 55 ) ) PUNCT iajs-2288 187 56 = = SYM iajs-2288 187 57 𝐼 𝐼 PROPN iajs-2288 187 58 , , PUNCT iajs-2288 187 59 it it PRON iajs-2288 187 60 follows follow VERB iajs-2288 187 61 that that SCONJ iajs-2288 187 62 [ [ X iajs-2288 187 63 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 187 64 𝑇 𝑇 PROPN iajs-2288 187 65 ] ] PUNCT iajs-2288 187 66 is be AUX iajs-2288 187 67 an an DET iajs-2288 187 68 app app ADJ iajs-2288 187 69 - - PUNCT iajs-2288 187 70 semi semi ADJ iajs-2288 187 71 - - ADJ iajs-2288 187 72 prime prime ADJ iajs-2288 187 73 ideal ideal NOUN iajs-2288 187 74 of of ADP iajs-2288 187 75 𝑅. 𝑅. NOUN iajs-2288 187 76 recall recall NOUN iajs-2288 187 77 that that SCONJ iajs-2288 187 78 an an DET iajs-2288 187 79 envelope envelope NOUN iajs-2288 187 80 of of ADP iajs-2288 187 81 a a DET iajs-2288 187 82 submodule submodule NOUN iajs-2288 187 83 𝐿 𝐿 PROPN iajs-2288 187 84 of of ADP iajs-2288 187 85 an an DET iajs-2288 187 86 𝑅-module 𝑅-module PROPN iajs-2288 187 87 denoted denote VERB iajs-2288 187 88 by by ADP iajs-2288 187 89 𝐸𝑇(𝐿 𝐸𝑇(𝐿 PROPN iajs-2288 187 90 ) ) PUNCT iajs-2288 187 91 defined define VERB iajs-2288 187 92 by by ADP iajs-2288 187 93 𝐸𝑇(𝐿 𝐸𝑇(𝐿 PROPN iajs-2288 187 94 ) ) PUNCT iajs-2288 187 95 = = PRON iajs-2288 188 1 { { PUNCT iajs-2288 188 2 π‘Žπ‘‘ π‘Žπ‘‘ INTJ iajs-2288 188 3 : : PUNCT iajs-2288 188 4 π‘Ž π‘Ž PROPN iajs-2288 188 5 ∈ ∈ PROPN iajs-2288 188 6 𝑅 𝑅 PROPN iajs-2288 188 7 , , PUNCT iajs-2288 188 8 𝑑 𝑑 PROPN iajs-2288 188 9 ∈ ∈ PROPN iajs-2288 188 10 𝑇 𝑇 PROPN iajs-2288 188 11 such such ADJ iajs-2288 188 12 that that SCONJ iajs-2288 188 13 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 188 14 ∈ ∈ PROPN iajs-2288 188 15 𝐿 𝐿 PROPN iajs-2288 188 16 , , PUNCT iajs-2288 188 17 𝑛 𝑛 PRON iajs-2288 188 18 ∈ ∈ PROPN iajs-2288 188 19 𝑍+ 𝑍+ NOUN iajs-2288 188 20 } } PUNCT iajs-2288 188 21 and and CCONJ iajs-2288 188 22 𝐿 𝐿 PROPN iajs-2288 188 23 βŠ† βŠ† NUM iajs-2288 188 24 𝐸𝑇(𝐿 𝐸𝑇(𝐿 NOUN iajs-2288 188 25 ) ) PUNCT iajs-2288 189 1 [ [ X iajs-2288 189 2 17 17 NUM iajs-2288 189 3 ] ] PUNCT iajs-2288 189 4 . . PUNCT iajs-2288 190 1 125 125 NUM iajs-2288 190 2 ibn ibn PROPN iajs-2288 190 3 al al PROPN iajs-2288 190 4 - - PUNCT iajs-2288 190 5 haitham haitham PROPN iajs-2288 190 6 jour jour X iajs-2288 190 7 . . PROPN iajs-2288 190 8 for for ADP iajs-2288 190 9 pure pure ADJ iajs-2288 190 10 & & CCONJ iajs-2288 190 11 appl appl PROPN iajs-2288 190 12 . . PUNCT iajs-2288 191 1 sci sci PROPN iajs-2288 191 2 . . PROPN iajs-2288 191 3 32 32 NUM iajs-2288 191 4 ( ( PUNCT iajs-2288 191 5 3 3 NUM iajs-2288 191 6 ) ) SYM iajs-2288 191 7 2019 2019 NUM iajs-2288 191 8 proposition proposition NOUN iajs-2288 191 9 ( ( PUNCT iajs-2288 191 10 28 28 NUM iajs-2288 191 11 ) ) PUNCT iajs-2288 191 12 let let VERB iajs-2288 191 13 𝐿 𝐿 PROPN iajs-2288 191 14 be be AUX iajs-2288 191 15 a a DET iajs-2288 191 16 proper proper ADJ iajs-2288 191 17 submodule submodule NOUN iajs-2288 191 18 of of ADP iajs-2288 191 19 an an DET iajs-2288 191 20 𝑅-module 𝑅-module PROPN iajs-2288 191 21 𝑇. 𝑇. PROPN iajs-2288 191 22 then then ADV iajs-2288 191 23 𝐿 𝐿 PROPN iajs-2288 191 24 is be AUX iajs-2288 191 25 an an DET iajs-2288 191 26 app app ADJ iajs-2288 191 27 - - PUNCT iajs-2288 191 28 semi semi ADJ iajs-2288 191 29 - - ADJ iajs-2288 191 30 prime prime ADJ iajs-2288 191 31 submodule submodule NOUN iajs-2288 191 32 of of ADP iajs-2288 191 33 𝑇 𝑇 PROPN iajs-2288 192 1 if if SCONJ iajs-2288 192 2 and and CCONJ iajs-2288 192 3 only only ADV iajs-2288 192 4 if if SCONJ iajs-2288 192 5 𝐸𝑇(𝐿 𝐸𝑇(𝐿 PROPN iajs-2288 192 6 ) ) PUNCT iajs-2288 192 7 βŠ† βŠ† NUM iajs-2288 192 8 𝐿 𝐿 PROPN iajs-2288 192 9 + + NOUN iajs-2288 192 10 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 192 11 ) ) PUNCT iajs-2288 192 12 . . PUNCT iajs-2288 193 1 proof proof NOUN iajs-2288 193 2 ( ( PUNCT iajs-2288 193 3 β‡’ β‡’ PROPN iajs-2288 193 4 ) ) PUNCT iajs-2288 193 5 let let VERB iajs-2288 193 6 ∈ ∈ PROPN iajs-2288 193 7 𝐸𝑇(𝐿 𝐸𝑇(𝐿 PROPN iajs-2288 193 8 ) ) PUNCT iajs-2288 193 9 , , PUNCT iajs-2288 193 10 implies imply VERB iajs-2288 193 11 that that SCONJ iajs-2288 193 12 𝑦 𝑦 NOUN iajs-2288 193 13 = = X iajs-2288 193 14 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 193 15 , , PUNCT iajs-2288 193 16 where where SCONJ iajs-2288 193 17 π‘Ž π‘Ž PROPN iajs-2288 193 18 ∈ ∈ PROPN iajs-2288 193 19 𝑅 𝑅 PROPN iajs-2288 193 20 , , PUNCT iajs-2288 193 21 𝑑 𝑑 PROPN iajs-2288 193 22 ∈ ∈ PROPN iajs-2288 193 23 𝑇 𝑇 PROPN iajs-2288 193 24 such such ADJ iajs-2288 193 25 that that SCONJ iajs-2288 193 26 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 193 27 ∈ ∈ PROPN iajs-2288 193 28 𝐿 𝐿 PROPN iajs-2288 193 29 for for ADP iajs-2288 193 30 some some DET iajs-2288 193 31 𝑛 𝑛 DET iajs-2288 193 32 ∈ ∈ PROPN iajs-2288 193 33 𝑍+ 𝑍+ NOUN iajs-2288 193 34 . . PUNCT iajs-2288 194 1 but but CCONJ iajs-2288 194 2 𝐿 𝐿 PROPN iajs-2288 194 3 is be AUX iajs-2288 194 4 an an DET iajs-2288 194 5 app app ADJ iajs-2288 194 6 - - PUNCT iajs-2288 194 7 semi semi ADJ iajs-2288 194 8 - - ADJ iajs-2288 194 9 prime prime ADJ iajs-2288 194 10 submodule submodule NOUN iajs-2288 194 11 of of ADP iajs-2288 194 12 𝑇 𝑇 PROPN iajs-2288 194 13 , , PUNCT iajs-2288 194 14 then then ADV iajs-2288 194 15 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 194 16 ∈ ∈ PROPN iajs-2288 194 17 𝐿 𝐿 PROPN iajs-2288 194 18 + + NOUN iajs-2288 194 19 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 194 20 ) ) PUNCT iajs-2288 194 21 , , PUNCT iajs-2288 194 22 that that PRON iajs-2288 194 23 is be AUX iajs-2288 194 24 𝑦 𝑦 NUM iajs-2288 194 25 ∈ ∈ PROPN iajs-2288 194 26 𝐿 𝐿 PROPN iajs-2288 194 27 + + NOUN iajs-2288 194 28 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 194 29 ) ) PUNCT iajs-2288 194 30 so so ADV iajs-2288 194 31 𝐸𝑇(𝐿 𝐸𝑇(𝐿 PROPN iajs-2288 194 32 ) ) PUNCT iajs-2288 194 33 βŠ† βŠ† NUM iajs-2288 194 34 𝐿 𝐿 PROPN iajs-2288 194 35 + + NOUN iajs-2288 194 36 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 194 37 ) ) PUNCT iajs-2288 194 38 . . PUNCT iajs-2288 195 1 ( ( PUNCT iajs-2288 195 2 ⇐ ⇐ PROPN iajs-2288 195 3 ) ) PUNCT iajs-2288 195 4 suppose suppose VERB iajs-2288 195 5 that that SCONJ iajs-2288 195 6 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 195 7 ∈ ∈ PROPN iajs-2288 195 8 𝐿 𝐿 PROPN iajs-2288 195 9 , , PUNCT iajs-2288 195 10 where where SCONJ iajs-2288 195 11 π‘Ž π‘Ž PROPN iajs-2288 195 12 ∈ ∈ PROPN iajs-2288 195 13 𝑅 𝑅 PROPN iajs-2288 195 14 , , PUNCT iajs-2288 195 15 𝑑 𝑑 PROPN iajs-2288 195 16 ∈ ∈ PROPN iajs-2288 195 17 𝑇 𝑇 PROPN iajs-2288 195 18 and and CCONJ iajs-2288 195 19 𝑛 𝑛 PRON iajs-2288 195 20 ∈ ∈ PROPN iajs-2288 195 21 𝑍+ 𝑍+ NOUN iajs-2288 195 22 . . PUNCT iajs-2288 196 1 since since SCONJ iajs-2288 196 2 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN iajs-2288 196 3 ∈ ∈ PROPN iajs-2288 196 4 𝐿 𝐿 PROPN iajs-2288 196 5 then then ADV iajs-2288 196 6 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 196 7 ∈ ∈ PROPN iajs-2288 196 8 𝐸𝑇(𝐿 𝐸𝑇(𝐿 PROPN iajs-2288 196 9 ) ) PUNCT iajs-2288 196 10 βŠ† βŠ† NUM iajs-2288 196 11 𝐿 𝐿 PROPN iajs-2288 196 12 + + NOUN iajs-2288 196 13 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 196 14 ) ) PUNCT iajs-2288 196 15 by by ADP iajs-2288 196 16 hypothesis hypothesis NOUN iajs-2288 196 17 . . PUNCT iajs-2288 197 1 it it PRON iajs-2288 197 2 follows follow VERB iajs-2288 197 3 that that SCONJ iajs-2288 197 4 π‘Ÿπ‘‘ π‘Ÿπ‘‘ PROPN iajs-2288 197 5 ∈ ∈ PROPN iajs-2288 197 6 𝐿 𝐿 PROPN iajs-2288 197 7 + + NOUN iajs-2288 197 8 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 197 9 ) ) PUNCT iajs-2288 197 10 . . PUNCT iajs-2288 198 1 hence hence ADV iajs-2288 198 2 𝐿 𝐿 PROPN iajs-2288 198 3 is be AUX iajs-2288 198 4 an an DET iajs-2288 198 5 appsemi appsemi ADJ iajs-2288 198 6 - - ADJ iajs-2288 198 7 prime prime ADJ iajs-2288 198 8 submodule submodule NOUN iajs-2288 198 9 of of ADP iajs-2288 198 10 𝑇. 𝑇. PROPN iajs-2288 198 11 proposition proposition NOUN iajs-2288 198 12 ( ( PUNCT iajs-2288 198 13 29 29 NUM iajs-2288 198 14 ) ) PUNCT iajs-2288 198 15 let let VERB iajs-2288 198 16 𝐿 𝐿 PROPN iajs-2288 198 17 be be AUX iajs-2288 198 18 a a DET iajs-2288 198 19 proper proper ADJ iajs-2288 198 20 submodule submodule NOUN iajs-2288 198 21 of of ADP iajs-2288 198 22 an an DET iajs-2288 198 23 𝑅-module 𝑅-module PROPN iajs-2288 198 24 𝑇 𝑇 PROPN iajs-2288 198 25 with with ADP iajs-2288 198 26 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 198 27 ) ) PUNCT iajs-2288 198 28 βŠ† βŠ† NUM iajs-2288 198 29 𝐿. 𝐿. VERB iajs-2288 198 30 then then ADV iajs-2288 198 31 𝐿 𝐿 PROPN iajs-2288 198 32 is be AUX iajs-2288 198 33 an an DET iajs-2288 198 34 app app ADJ iajs-2288 198 35 - - PUNCT iajs-2288 198 36 semiprime semiprime NOUN iajs-2288 198 37 submodule submodule NOUN iajs-2288 198 38 of of ADP iajs-2288 198 39 𝑇 𝑇 PROPN iajs-2288 198 40 if if SCONJ iajs-2288 198 41 and and CCONJ iajs-2288 198 42 only only ADV iajs-2288 198 43 if if SCONJ iajs-2288 198 44 [ [ X iajs-2288 198 45 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 198 46 𝑇 𝑇 PROPN iajs-2288 198 47 ] ] PUNCT iajs-2288 198 48 is be AUX iajs-2288 198 49 a a DET iajs-2288 198 50 semi semi ADJ iajs-2288 198 51 - - ADJ iajs-2288 198 52 prime prime ADJ iajs-2288 198 53 ideal ideal NOUN iajs-2288 198 54 of of ADP iajs-2288 198 55 𝑅. 𝑅. ADJ iajs-2288 198 56 proof proof NOUN iajs-2288 198 57 ( ( PUNCT iajs-2288 198 58 β‡’ β‡’ PROPN iajs-2288 198 59 ) ) PUNCT iajs-2288 198 60 let let VERB iajs-2288 198 61 π‘Ž π‘Ž PROPN iajs-2288 198 62 ∈ ∈ PROPN iajs-2288 198 63 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2288 198 64 𝑇 𝑇 PROPN iajs-2288 198 65 ] ] PUNCT iajs-2288 198 66 , , PUNCT iajs-2288 198 67 implies imply VERB iajs-2288 198 68 that that SCONJ iajs-2288 198 69 π‘Žπ‘› π‘Žπ‘› PRON iajs-2288 198 70 ∈ ∈ PROPN iajs-2288 198 71 [ [ X iajs-2288 198 72 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 198 73 𝑇 𝑇 PROPN iajs-2288 198 74 ] ] PUNCT iajs-2288 198 75 for for ADP iajs-2288 198 76 some some DET iajs-2288 198 77 𝑛 𝑛 PRON iajs-2288 198 78 ∈ ∈ PROPN iajs-2288 198 79 𝑍+ 𝑍+ NOUN iajs-2288 198 80 , , PUNCT iajs-2288 198 81 it it PRON iajs-2288 198 82 follows follow VERB iajs-2288 198 83 that that SCONJ iajs-2288 198 84 π‘Žπ‘›π‘‡ π‘Žπ‘›π‘‡ PROPN iajs-2288 198 85 βŠ† βŠ† NUM iajs-2288 198 86 𝐿 𝐿 PROPN iajs-2288 198 87 , , PUNCT iajs-2288 198 88 that that PRON iajs-2288 198 89 is is ADV iajs-2288 198 90 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN iajs-2288 198 91 ∈ ∈ PROPN iajs-2288 198 92 𝐿 𝐿 PROPN iajs-2288 198 93 for for ADP iajs-2288 198 94 all all DET iajs-2288 198 95 𝑑 𝑑 ADP iajs-2288 198 96 ∈ ∈ PROPN iajs-2288 198 97 𝑇. 𝑇. PROPN iajs-2288 198 98 but but CCONJ iajs-2288 198 99 𝐿 𝐿 PROPN iajs-2288 198 100 is be AUX iajs-2288 198 101 an an DET iajs-2288 198 102 app app ADJ iajs-2288 198 103 - - PUNCT iajs-2288 198 104 semi semi ADJ iajs-2288 198 105 - - ADJ iajs-2288 198 106 prime prime ADJ iajs-2288 198 107 submodule submodule NOUN iajs-2288 198 108 of of ADP iajs-2288 198 109 𝑇 𝑇 PROPN iajs-2288 198 110 , , PUNCT iajs-2288 198 111 then then ADV iajs-2288 198 112 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 198 113 ∈ ∈ PROPN iajs-2288 198 114 𝐿 𝐿 PROPN iajs-2288 198 115 + + NOUN iajs-2288 198 116 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 198 117 ) ) PUNCT iajs-2288 198 118 for for ADP iajs-2288 198 119 all all DET iajs-2288 198 120 𝑑 𝑑 PROPN iajs-2288 198 121 ∈ ∈ PROPN iajs-2288 198 122 𝑇. 𝑇. PROPN iajs-2288 198 123 that that PRON iajs-2288 198 124 is be AUX iajs-2288 198 125 π‘Žπ‘‡ π‘Žπ‘‡ NOUN iajs-2288 198 126 βŠ† βŠ† NUM iajs-2288 198 127 𝐿 𝐿 PROPN iajs-2288 198 128 + + NOUN iajs-2288 198 129 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 198 130 ) ) PUNCT iajs-2288 198 131 . . PUNCT iajs-2288 199 1 since since SCONJ iajs-2288 199 2 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 199 3 ) ) PUNCT iajs-2288 199 4 βŠ† βŠ† NUM iajs-2288 199 5 𝐿 𝐿 PROPN iajs-2288 199 6 we we PRON iajs-2288 199 7 get get VERB iajs-2288 199 8 π‘Žπ‘‡ π‘Žπ‘‡ NOUN iajs-2288 199 9 βŠ† βŠ† NUM iajs-2288 199 10 𝐿 𝐿 PROPN iajs-2288 199 11 since since SCONJ iajs-2288 199 12 𝐿 𝐿 PROPN iajs-2288 199 13 + + NOUN iajs-2288 199 14 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 199 15 ) ) PUNCT iajs-2288 199 16 = = PUNCT iajs-2288 199 17 𝐿. 𝐿. VERB iajs-2288 199 18 that that PRON iajs-2288 199 19 is be AUX iajs-2288 199 20 π‘Ž π‘Ž PRON iajs-2288 199 21 ∈ ∈ NOUN iajs-2288 199 22 [ [ X iajs-2288 199 23 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 199 24 𝑇 𝑇 PROPN iajs-2288 199 25 ] ] PUNCT iajs-2288 199 26 , , PUNCT iajs-2288 199 27 hence hence ADV iajs-2288 199 28 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2288 199 29 𝑇 𝑇 PROPN iajs-2288 199 30 ] ] PUNCT iajs-2288 199 31 βŠ† βŠ† NUM iajs-2288 199 32 [ [ X iajs-2288 199 33 𝐿:𝑅 𝐿:𝑅 X iajs-2288 199 34 𝑇 𝑇 PROPN iajs-2288 199 35 ] ] PUNCT iajs-2288 199 36 . . PUNCT iajs-2288 200 1 but but CCONJ iajs-2288 200 2 [ [ X iajs-2288 200 3 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 200 4 𝑇 𝑇 PROPN iajs-2288 200 5 ] ] PUNCT iajs-2288 200 6 βŠ† βŠ† NUM iajs-2288 200 7 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2288 200 8 𝑇 𝑇 PROPN iajs-2288 200 9 ] ] PUNCT iajs-2288 200 10 , , PUNCT iajs-2288 200 11 it it PRON iajs-2288 200 12 follows follow VERB iajs-2288 200 13 that that SCONJ iajs-2288 200 14 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2288 200 15 𝑇 𝑇 PROPN iajs-2288 200 16 ] ] PUNCT iajs-2288 200 17 = = PUNCT iajs-2288 201 1 [ [ X iajs-2288 201 2 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 201 3 𝑇 𝑇 PROPN iajs-2288 201 4 ] ] PUNCT iajs-2288 201 5 , , PUNCT iajs-2288 201 6 hence hence ADV iajs-2288 201 7 [ [ X iajs-2288 201 8 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 201 9 𝑇 𝑇 PROPN iajs-2288 201 10 ] ] PUNCT iajs-2288 201 11 is be AUX iajs-2288 201 12 a a DET iajs-2288 201 13 semi semi ADJ iajs-2288 201 14 - - ADJ iajs-2288 201 15 prime prime ADJ iajs-2288 201 16 ideal ideal NOUN iajs-2288 201 17 of of ADP iajs-2288 201 18 𝑅. 𝑅. NOUN iajs-2288 201 19 ( ( PUNCT iajs-2288 201 20 ⇐ ⇐ PROPN iajs-2288 201 21 ) ) PUNCT iajs-2288 201 22 suppose suppose VERB iajs-2288 201 23 that that SCONJ iajs-2288 201 24 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 201 25 ∈ ∈ PROPN iajs-2288 201 26 𝐿 𝐿 PROPN iajs-2288 201 27 , , PUNCT iajs-2288 201 28 where where SCONJ iajs-2288 201 29 π‘Ž π‘Ž PROPN iajs-2288 201 30 ∈ ∈ PROPN iajs-2288 201 31 𝑅 𝑅 PROPN iajs-2288 201 32 , , PUNCT iajs-2288 201 33 𝑑 𝑑 PROPN iajs-2288 201 34 ∈ ∈ PROPN iajs-2288 201 35 𝑇 𝑇 PROPN iajs-2288 201 36 and and CCONJ iajs-2288 201 37 𝑛 𝑛 PRON iajs-2288 201 38 ∈ ∈ PROPN iajs-2288 201 39 𝑍+ 𝑍+ NOUN iajs-2288 201 40 , , PUNCT iajs-2288 201 41 it it PRON iajs-2288 201 42 follows follow VERB iajs-2288 201 43 that that SCONJ iajs-2288 201 44 π‘Žπ‘› π‘Žπ‘› PRON iajs-2288 201 45 ∈ ∈ PROPN iajs-2288 202 1 [ [ X iajs-2288 202 2 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 202 3 𝑇 𝑇 PROPN iajs-2288 202 4 ] ] PUNCT iajs-2288 202 5 , , PUNCT iajs-2288 202 6 implies imply VERB iajs-2288 202 7 that that SCONJ iajs-2288 202 8 π‘Ž π‘Ž PROPN iajs-2288 202 9 ∈ ∈ PROPN iajs-2288 202 10 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2288 202 11 𝑇 𝑇 PROPN iajs-2288 202 12 ] ] PUNCT iajs-2288 202 13 . . PUNCT iajs-2288 203 1 but but CCONJ iajs-2288 203 2 [ [ X iajs-2288 203 3 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 203 4 𝑇 𝑇 PROPN iajs-2288 203 5 ] ] PUNCT iajs-2288 203 6 is be AUX iajs-2288 203 7 a a DET iajs-2288 203 8 semi semi ADJ iajs-2288 203 9 - - ADJ iajs-2288 203 10 prime prime ADJ iajs-2288 203 11 ideal ideal NOUN iajs-2288 203 12 of of ADP iajs-2288 203 13 𝑅 𝑅 PROPN iajs-2288 203 14 , , PUNCT iajs-2288 203 15 implies imply VERB iajs-2288 203 16 that that SCONJ iajs-2288 203 17 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2288 203 18 𝑇 𝑇 PROPN iajs-2288 203 19 ] ] PUNCT iajs-2288 203 20 = = PUNCT iajs-2288 204 1 [ [ X iajs-2288 204 2 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 204 3 𝑇 𝑇 PROPN iajs-2288 204 4 ] ] PUNCT iajs-2288 204 5 , , PUNCT iajs-2288 204 6 hence hence ADV iajs-2288 204 7 π‘Ž π‘Ž PROPN iajs-2288 204 8 ∈ ∈ NOUN iajs-2288 204 9 [ [ X iajs-2288 204 10 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 204 11 𝑇 𝑇 PROPN iajs-2288 204 12 ] ] PUNCT iajs-2288 204 13 , , PUNCT iajs-2288 204 14 it it PRON iajs-2288 204 15 follows follow VERB iajs-2288 204 16 that that SCONJ iajs-2288 204 17 π‘Žπ‘‡ π‘Žπ‘‡ VERB iajs-2288 204 18 βŠ† βŠ† NUM iajs-2288 204 19 𝐿 𝐿 PROPN iajs-2288 204 20 = = SYM iajs-2288 204 21 𝐿 𝐿 PROPN iajs-2288 204 22 + + NOUN iajs-2288 204 23 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 204 24 ) ) PUNCT iajs-2288 204 25 . . PUNCT iajs-2288 205 1 thus thus ADV iajs-2288 205 2 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 205 3 ∈ ∈ PROPN iajs-2288 205 4 𝐿 𝐿 PROPN iajs-2288 205 5 + + NOUN iajs-2288 205 6 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 205 7 ) ) PUNCT iajs-2288 205 8 for for ADP iajs-2288 205 9 all all DET iajs-2288 205 10 𝑑 𝑑 PROPN iajs-2288 205 11 ∈ ∈ PROPN iajs-2288 205 12 𝑇. 𝑇. PROPN iajs-2288 205 13 thus thus ADV iajs-2288 205 14 𝐿 𝐿 PROPN iajs-2288 205 15 is be AUX iajs-2288 205 16 an an DET iajs-2288 205 17 app app ADJ iajs-2288 205 18 - - PUNCT iajs-2288 205 19 semi semi ADJ iajs-2288 205 20 - - ADJ iajs-2288 205 21 prime prime ADJ iajs-2288 205 22 submodule submodule NOUN iajs-2288 205 23 of of ADP iajs-2288 205 24 𝑇. 𝑇. PROPN iajs-2288 205 25 proposition proposition NOUN iajs-2288 205 26 ( ( PUNCT iajs-2288 205 27 30 30 NUM iajs-2288 205 28 ) ) PUNCT iajs-2288 205 29 let let VERB iajs-2288 205 30 𝑇 𝑇 PROPN iajs-2288 205 31 = = PRON iajs-2288 205 32 𝑇1 𝑇1 NOUN iajs-2288 205 33 βŠ•π‘‡2 βŠ•π‘‡2 NOUN iajs-2288 205 34 be be AUX iajs-2288 205 35 an an DET iajs-2288 205 36 𝑅-module 𝑅-module PROPN iajs-2288 205 37 , , PUNCT iajs-2288 205 38 where where SCONJ iajs-2288 205 39 𝑇1 𝑇1 NOUN iajs-2288 205 40 and and CCONJ iajs-2288 205 41 𝑇2 𝑇2 NOUN iajs-2288 205 42 𝑅-modules 𝑅-modules PROPN iajs-2288 205 43 , , PUNCT iajs-2288 205 44 and and CCONJ iajs-2288 205 45 𝐿 𝐿 PROPN iajs-2288 205 46 = = PUNCT iajs-2288 205 47 𝐿1 𝐿1 X iajs-2288 205 48 βŠ•πΏ2 βŠ•πΏ2 PRON iajs-2288 205 49 be be VERB iajs-2288 205 50 a a DET iajs-2288 205 51 submodule submodule NOUN iajs-2288 205 52 of of ADP iajs-2288 205 53 𝑇 𝑇 PROPN iajs-2288 205 54 , , PUNCT iajs-2288 205 55 where where SCONJ iajs-2288 205 56 𝐿1 𝐿1 PROPN iajs-2288 205 57 is be AUX iajs-2288 205 58 a a DET iajs-2288 205 59 submodule submodule NOUN iajs-2288 205 60 of of ADP iajs-2288 205 61 𝑇1 𝑇1 NOUN iajs-2288 205 62 and and CCONJ iajs-2288 205 63 𝐿2 𝐿2 NOUN iajs-2288 205 64 is be AUX iajs-2288 205 65 a a DET iajs-2288 205 66 submodule submodule NOUN iajs-2288 205 67 of of ADP iajs-2288 205 68 𝑇2 𝑇2 NOUN iajs-2288 205 69 with with ADP iajs-2288 205 70 𝐿 𝐿 PROPN iajs-2288 205 71 βŠ† βŠ† NUM iajs-2288 205 72 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 205 73 ) ) PUNCT iajs-2288 205 74 = = SYM iajs-2288 205 75 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 205 76 ) ) PUNCT iajs-2288 205 77 βŠ• βŠ• PROPN iajs-2288 205 78 π‘ π‘œπ‘(𝑇2 π‘ π‘œπ‘(𝑇2 PROPN iajs-2288 205 79 ) ) PUNCT iajs-2288 205 80 . . PUNCT iajs-2288 206 1 if if SCONJ iajs-2288 206 2 𝐿 𝐿 PROPN iajs-2288 206 3 is be AUX iajs-2288 206 4 an an DET iajs-2288 206 5 app app ADJ iajs-2288 206 6 - - PUNCT iajs-2288 206 7 semi semi ADJ iajs-2288 206 8 - - ADJ iajs-2288 206 9 prime prime ADJ iajs-2288 206 10 submodule submodule NOUN iajs-2288 206 11 of of ADP iajs-2288 206 12 𝑇 𝑇 PROPN iajs-2288 206 13 , , PUNCT iajs-2288 206 14 then then ADV iajs-2288 206 15 𝐿1 𝐿1 PROPN iajs-2288 206 16 is be AUX iajs-2288 206 17 an an DET iajs-2288 206 18 appsemi appsemi ADJ iajs-2288 206 19 - - ADJ iajs-2288 206 20 prime prime ADJ iajs-2288 206 21 submodule submodule NOUN iajs-2288 206 22 of of ADP iajs-2288 206 23 𝑇1 𝑇1 NOUN iajs-2288 206 24 and and CCONJ iajs-2288 206 25 𝐿2 𝐿2 NOUN iajs-2288 206 26 is be AUX iajs-2288 206 27 an an DET iajs-2288 206 28 app app ADJ iajs-2288 206 29 - - PUNCT iajs-2288 206 30 semi semi ADJ iajs-2288 206 31 - - ADJ iajs-2288 206 32 prime prime ADJ iajs-2288 206 33 submodule submodule NOUN iajs-2288 206 34 of of ADP iajs-2288 206 35 𝑇2 𝑇2 PROPN iajs-2288 206 36 . . PUNCT iajs-2288 207 1 proof proof NOUN iajs-2288 207 2 let let VERB iajs-2288 207 3 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN iajs-2288 207 4 ∈ ∈ PROPN iajs-2288 207 5 𝐿1 𝐿1 PROPN iajs-2288 207 6 , , PUNCT iajs-2288 207 7 where where SCONJ iajs-2288 207 8 π‘Ž π‘Ž PROPN iajs-2288 207 9 ∈ ∈ PROPN iajs-2288 207 10 𝑅 𝑅 PROPN iajs-2288 207 11 , , PUNCT iajs-2288 207 12 𝑑 𝑑 PROPN iajs-2288 207 13 ∈ ∈ NOUN iajs-2288 207 14 𝑇1 𝑇1 NOUN iajs-2288 207 15 and and CCONJ iajs-2288 207 16 𝑛 𝑛 PRON iajs-2288 207 17 ∈ ∈ PROPN iajs-2288 207 18 𝑍+ 𝑍+ NOUN iajs-2288 207 19 , , PUNCT iajs-2288 207 20 it it PRON iajs-2288 207 21 follows follow VERB iajs-2288 207 22 that that SCONJ iajs-2288 207 23 π‘Žπ‘›(𝑑 π‘Žπ‘›(𝑑 PROPN iajs-2288 207 24 , , PUNCT iajs-2288 207 25 0 0 NUM iajs-2288 207 26 ) ) PUNCT iajs-2288 207 27 ∈ ∈ PROPN iajs-2288 207 28 𝐿. 𝐿. NOUN iajs-2288 207 29 but but CCONJ iajs-2288 207 30 𝐿 𝐿 PROPN iajs-2288 207 31 is be AUX iajs-2288 207 32 an an DET iajs-2288 207 33 app app ADJ iajs-2288 207 34 - - PUNCT iajs-2288 207 35 semi semi ADJ iajs-2288 207 36 - - ADJ iajs-2288 207 37 prime prime ADJ iajs-2288 207 38 submodule submodule NOUN iajs-2288 207 39 of of ADP iajs-2288 207 40 𝑇 𝑇 PROPN iajs-2288 207 41 , , PUNCT iajs-2288 207 42 then then ADV iajs-2288 207 43 π‘Ž(𝑑 π‘Ž(𝑑 PROPN iajs-2288 207 44 , , PUNCT iajs-2288 207 45 0 0 NUM iajs-2288 207 46 ) ) PUNCT iajs-2288 207 47 ∈ ∈ PROPN iajs-2288 207 48 𝐿 𝐿 PROPN iajs-2288 207 49 + + NOUN iajs-2288 207 50 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 207 51 ) ) PUNCT iajs-2288 207 52 . . PUNCT iajs-2288 208 1 but but CCONJ iajs-2288 208 2 𝐿 𝐿 PROPN iajs-2288 208 3 βŠ† βŠ† NUM iajs-2288 208 4 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 208 5 ) ) PUNCT iajs-2288 208 6 , , PUNCT iajs-2288 208 7 implies imply VERB iajs-2288 208 8 that that SCONJ iajs-2288 208 9 𝐿 𝐿 PROPN iajs-2288 208 10 + + NOUN iajs-2288 208 11 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 208 12 ) ) PUNCT iajs-2288 208 13 = = SYM iajs-2288 208 14 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 208 15 ) ) PUNCT iajs-2288 208 16 , , PUNCT iajs-2288 208 17 thus thus ADV iajs-2288 208 18 π‘Ž(𝑑 π‘Ž(𝑑 NOUN iajs-2288 208 19 , , PUNCT iajs-2288 208 20 0 0 NUM iajs-2288 208 21 ) ) PUNCT iajs-2288 208 22 ∈ ∈ NOUN iajs-2288 208 23 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 208 24 ) ) PUNCT iajs-2288 208 25 = = SYM iajs-2288 208 26 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 208 27 ) ) PUNCT iajs-2288 208 28 βŠ• βŠ• PROPN iajs-2288 208 29 π‘ π‘œπ‘(𝑇2 π‘ π‘œπ‘(𝑇2 PROPN iajs-2288 208 30 ) ) PUNCT iajs-2288 208 31 , , PUNCT iajs-2288 208 32 it it PRON iajs-2288 208 33 follows follow VERB iajs-2288 208 34 that that SCONJ iajs-2288 208 35 π‘Žπ‘‘ π‘Žπ‘‘ PRON iajs-2288 208 36 ∈ ∈ PROPN iajs-2288 208 37 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 208 38 ) ) PUNCT iajs-2288 208 39 βŠ† βŠ† NUM iajs-2288 208 40 𝐿1 𝐿1 NOUN iajs-2288 208 41 + + CCONJ iajs-2288 208 42 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 NOUN iajs-2288 208 43 ) ) PUNCT iajs-2288 208 44 . . PUNCT iajs-2288 209 1 thus thus ADV iajs-2288 209 2 𝐿1 𝐿1 PROPN iajs-2288 209 3 is be AUX iajs-2288 209 4 an an DET iajs-2288 209 5 app app ADJ iajs-2288 209 6 - - PUNCT iajs-2288 209 7 semi semi ADJ iajs-2288 209 8 - - ADJ iajs-2288 209 9 prime prime ADJ iajs-2288 209 10 submodule submodule NOUN iajs-2288 209 11 of of ADP iajs-2288 209 12 𝑇1 𝑇1 NOUN iajs-2288 209 13 . . PUNCT iajs-2288 210 1 similarly similarly ADV iajs-2288 210 2 we we PRON iajs-2288 210 3 can can AUX iajs-2288 210 4 prove prove VERB iajs-2288 210 5 𝐿2 𝐿2 NOUN iajs-2288 210 6 is be AUX iajs-2288 210 7 an an DET iajs-2288 210 8 app app ADJ iajs-2288 210 9 - - PUNCT iajs-2288 210 10 semi semi ADJ iajs-2288 210 11 - - ADJ iajs-2288 210 12 prime prime ADJ iajs-2288 210 13 submodule submodule NOUN iajs-2288 210 14 of of ADP iajs-2288 210 15 𝑇2 𝑇2 PROPN iajs-2288 210 16 . . PUNCT iajs-2288 211 1 proposition proposition NOUN iajs-2288 211 2 ( ( PUNCT iajs-2288 211 3 31 31 NUM iajs-2288 211 4 ) ) PUNCT iajs-2288 211 5 let let VERB iajs-2288 211 6 𝑇 𝑇 PROPN iajs-2288 211 7 = = PRON iajs-2288 211 8 𝑇1 𝑇1 NOUN iajs-2288 211 9 βŠ•π‘‡2 βŠ•π‘‡2 NOUN iajs-2288 211 10 be be AUX iajs-2288 211 11 an an DET iajs-2288 211 12 𝑅-module 𝑅-module PROPN iajs-2288 211 13 , , PUNCT iajs-2288 211 14 where where SCONJ iajs-2288 211 15 each each PRON iajs-2288 211 16 of of ADP iajs-2288 211 17 𝑇1 𝑇1 NOUN iajs-2288 211 18 and and CCONJ iajs-2288 211 19 𝑇2 𝑇2 NOUN iajs-2288 211 20 𝑅-module 𝑅-module PROPN iajs-2288 211 21 . . PUNCT iajs-2288 212 1 then then ADV iajs-2288 212 2 the the DET iajs-2288 212 3 following follow VERB iajs-2288 212 4 statements statement NOUN iajs-2288 212 5 are be AUX iajs-2288 212 6 satisfy satisfy ADJ iajs-2288 212 7 : : PUNCT iajs-2288 212 8 1 1 X iajs-2288 212 9 ) ) PUNCT iajs-2288 212 10 𝐿1 𝐿1 PROPN iajs-2288 212 11 is be AUX iajs-2288 212 12 an an DET iajs-2288 212 13 app app ADJ iajs-2288 212 14 - - PUNCT iajs-2288 212 15 semi semi ADJ iajs-2288 212 16 - - ADJ iajs-2288 212 17 prime prime ADJ iajs-2288 212 18 submodule submodule NOUN iajs-2288 212 19 of of ADP iajs-2288 212 20 𝑇1 𝑇1 NOUN iajs-2288 212 21 such such ADJ iajs-2288 212 22 that that SCONJ iajs-2288 212 23 𝐿1 𝐿1 VERB iajs-2288 212 24 βŠ† βŠ† NUM iajs-2288 212 25 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 ADJ iajs-2288 212 26 ) ) PUNCT iajs-2288 212 27 and and CCONJ iajs-2288 212 28 𝑇2 𝑇2 PROPN iajs-2288 212 29 = = SYM iajs-2288 212 30 π‘ π‘œπ‘(𝑇2 π‘ π‘œπ‘(𝑇2 PROPN iajs-2288 212 31 ) ) PUNCT iajs-2288 212 32 if if SCONJ iajs-2288 212 33 and and CCONJ iajs-2288 212 34 only only ADV iajs-2288 212 35 if if SCONJ iajs-2288 212 36 𝐿1 𝐿1 PROPN iajs-2288 212 37 βŠ•π‘‡2 βŠ•π‘‡2 NOUN iajs-2288 212 38 is be AUX iajs-2288 212 39 an an DET iajs-2288 212 40 app app ADJ iajs-2288 212 41 - - PUNCT iajs-2288 212 42 semi semi ADJ iajs-2288 212 43 - - ADJ iajs-2288 212 44 prime prime ADJ iajs-2288 212 45 submodule submodule NOUN iajs-2288 212 46 of of ADP iajs-2288 212 47 𝑇. 𝑇. PROPN iajs-2288 212 48 2 2 NUM iajs-2288 212 49 ) ) PUNCT iajs-2288 212 50 𝐿2 𝐿2 NOUN iajs-2288 212 51 is be AUX iajs-2288 212 52 an an DET iajs-2288 212 53 app app ADJ iajs-2288 212 54 - - PUNCT iajs-2288 212 55 semi semi ADJ iajs-2288 212 56 - - ADJ iajs-2288 212 57 prime prime ADJ iajs-2288 212 58 submodule submodule NOUN iajs-2288 212 59 of of ADP iajs-2288 212 60 𝑇2 𝑇2 NOUN iajs-2288 212 61 such such ADJ iajs-2288 212 62 that that SCONJ iajs-2288 212 63 𝐿2 𝐿2 NOUN iajs-2288 212 64 βŠ† βŠ† NUM iajs-2288 212 65 π‘ π‘œπ‘(𝑇2 π‘ π‘œπ‘(𝑇2 PROPN iajs-2288 212 66 ) ) PUNCT iajs-2288 212 67 and and CCONJ iajs-2288 212 68 𝑇1 𝑇1 NOUN iajs-2288 212 69 = = SYM iajs-2288 212 70 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 212 71 ) ) PUNCT iajs-2288 212 72 if if SCONJ iajs-2288 212 73 and and CCONJ iajs-2288 212 74 only only ADV iajs-2288 212 75 if if SCONJ iajs-2288 212 76 𝑇1 𝑇1 NOUN iajs-2288 212 77 βŠ•πΏ2 βŠ•πΏ2 NOUN iajs-2288 212 78 is be AUX iajs-2288 212 79 an an DET iajs-2288 212 80 app app ADJ iajs-2288 212 81 - - PUNCT iajs-2288 212 82 semi semi ADJ iajs-2288 212 83 - - ADJ iajs-2288 212 84 prime prime ADJ iajs-2288 212 85 submodule submodule NOUN iajs-2288 212 86 of of ADP iajs-2288 212 87 𝑇. 𝑇. PROPN iajs-2288 212 88 126 126 NUM iajs-2288 212 89 ibn ibn PROPN iajs-2288 212 90 al al PROPN iajs-2288 212 91 - - PUNCT iajs-2288 212 92 haitham haitham PROPN iajs-2288 212 93 jour jour X iajs-2288 212 94 . . PROPN iajs-2288 213 1 for for ADP iajs-2288 213 2 pure pure ADJ iajs-2288 213 3 & & CCONJ iajs-2288 213 4 appl appl PROPN iajs-2288 213 5 . . PUNCT iajs-2288 214 1 sci sci PROPN iajs-2288 214 2 . . PROPN iajs-2288 214 3 32 32 NUM iajs-2288 214 4 ( ( PUNCT iajs-2288 214 5 3 3 NUM iajs-2288 214 6 ) ) PUNCT iajs-2288 214 7 2019 2019 NUM iajs-2288 214 8 proof proof NOUN iajs-2288 214 9 1 1 NUM iajs-2288 214 10 ) ) PUNCT iajs-2288 214 11 ( ( PUNCT iajs-2288 214 12 β‡’ β‡’ NOUN iajs-2288 214 13 ) ) PUNCT iajs-2288 214 14 let let VERB iajs-2288 214 15 π‘Žπ‘›(𝑑1 π‘Žπ‘›(𝑑1 VERB iajs-2288 214 16 , , PUNCT iajs-2288 214 17 𝑑2 𝑑2 ADJ iajs-2288 214 18 ) ) PUNCT iajs-2288 214 19 ∈ ∈ PROPN iajs-2288 214 20 𝐿1 𝐿1 PROPN iajs-2288 215 1 βŠ•π‘‡2 βŠ•π‘‡2 NOUN iajs-2288 215 2 , , PUNCT iajs-2288 215 3 where where SCONJ iajs-2288 215 4 π‘Ž π‘Ž PROPN iajs-2288 215 5 ∈ ∈ PROPN iajs-2288 215 6 𝑅 𝑅 PROPN iajs-2288 215 7 , , PUNCT iajs-2288 215 8 ( ( PUNCT iajs-2288 215 9 𝑑1 𝑑1 NOUN iajs-2288 215 10 , , PUNCT iajs-2288 215 11 𝑑2 𝑑2 ADJ iajs-2288 215 12 ) ) PUNCT iajs-2288 215 13 ∈ ∈ PROPN iajs-2288 215 14 𝑇 𝑇 PROPN iajs-2288 215 15 and and CCONJ iajs-2288 215 16 𝑛 𝑛 PRON iajs-2288 215 17 ∈ ∈ PROPN iajs-2288 215 18 𝑍+ 𝑍+ NOUN iajs-2288 215 19 , , PUNCT iajs-2288 215 20 then then ADV iajs-2288 215 21 π‘Žπ‘›π‘‘1 π‘Žπ‘›π‘‘1 PROPN iajs-2288 215 22 ∈ ∈ PROPN iajs-2288 215 23 𝐿1 𝐿1 PROPN iajs-2288 215 24 . . PUNCT iajs-2288 216 1 but but CCONJ iajs-2288 216 2 𝐿1 𝐿1 PROPN iajs-2288 216 3 is be AUX iajs-2288 216 4 an an DET iajs-2288 216 5 app app ADJ iajs-2288 216 6 - - PUNCT iajs-2288 216 7 semi semi ADJ iajs-2288 216 8 - - ADJ iajs-2288 216 9 prime prime ADJ iajs-2288 216 10 submodule submodule NOUN iajs-2288 216 11 of of ADP iajs-2288 216 12 𝑇1 𝑇1 NOUN iajs-2288 216 13 and and CCONJ iajs-2288 216 14 𝐿1 𝐿1 VERB iajs-2288 216 15 βŠ† βŠ† NUM iajs-2288 216 16 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 NOUN iajs-2288 216 17 ) ) PUNCT iajs-2288 216 18 , , PUNCT iajs-2288 216 19 then then ADV iajs-2288 216 20 π‘Žπ‘‘1 π‘Žπ‘‘1 VERB iajs-2288 216 21 ∈ ∈ PROPN iajs-2288 216 22 𝐿1 𝐿1 NOUN iajs-2288 217 1 + + CCONJ iajs-2288 217 2 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 217 3 ) ) PUNCT iajs-2288 217 4 = = SYM iajs-2288 217 5 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 217 6 ) ) PUNCT iajs-2288 217 7 . . PUNCT iajs-2288 218 1 now now ADV iajs-2288 218 2 , , PUNCT iajs-2288 218 3 we we PRON iajs-2288 218 4 have have VERB iajs-2288 218 5 𝑇2 𝑇2 NOUN iajs-2288 218 6 = = SYM iajs-2288 218 7 π‘ π‘œπ‘(𝑇2 π‘ π‘œπ‘(𝑇2 PROPN iajs-2288 218 8 ) ) PUNCT iajs-2288 218 9 then then ADV iajs-2288 218 10 π‘Ž(𝑑1 π‘Ž(𝑑1 PROPN iajs-2288 218 11 , , PUNCT iajs-2288 218 12 𝑑2 𝑑2 PROPN iajs-2288 218 13 ) ) PUNCT iajs-2288 218 14 ∈ ∈ PROPN iajs-2288 218 15 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 218 16 ) ) PUNCT iajs-2288 218 17 βŠ• βŠ• NOUN iajs-2288 218 18 π‘ π‘œπ‘(𝑇2 π‘ π‘œπ‘(𝑇2 PROPN iajs-2288 218 19 ) ) PUNCT iajs-2288 218 20 = = PUNCT iajs-2288 219 1 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 219 2 βŠ• βŠ• PROPN iajs-2288 219 3 𝑇2 𝑇2 PROPN iajs-2288 219 4 ) ) PUNCT iajs-2288 219 5 βŠ† βŠ† NUM iajs-2288 219 6 𝐿1 𝐿1 NOUN iajs-2288 219 7 βŠ•π‘‡2 βŠ•π‘‡2 NOUN iajs-2288 219 8 + + NUM iajs-2288 219 9 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 219 10 βŠ• βŠ• PROPN iajs-2288 219 11 𝑇2 𝑇2 PROPN iajs-2288 219 12 ) ) PUNCT iajs-2288 219 13 . . PUNCT iajs-2288 220 1 thus thus ADV iajs-2288 220 2 𝐿1 𝐿1 X iajs-2288 220 3 βŠ•π‘‡2 βŠ•π‘‡2 PROPN iajs-2288 220 4 is be AUX iajs-2288 220 5 an an DET iajs-2288 220 6 app app ADJ iajs-2288 220 7 - - PUNCT iajs-2288 220 8 semi semi ADJ iajs-2288 220 9 - - ADJ iajs-2288 220 10 prime prime ADJ iajs-2288 220 11 submodule submodule NOUN iajs-2288 220 12 of of ADP iajs-2288 220 13 𝑇. 𝑇. PROPN iajs-2288 220 14 ( ( PUNCT iajs-2288 220 15 ⇐ ⇐ PROPN iajs-2288 220 16 ) ) PUNCT iajs-2288 220 17 suppose suppose VERB iajs-2288 220 18 that that SCONJ iajs-2288 220 19 π‘Žπ‘›π‘‘1 π‘Žπ‘›π‘‘1 PROPN iajs-2288 220 20 ∈ ∈ PROPN iajs-2288 220 21 𝐿1 𝐿1 PROPN iajs-2288 220 22 , , PUNCT iajs-2288 220 23 where where SCONJ iajs-2288 220 24 π‘Ž π‘Ž PROPN iajs-2288 220 25 ∈ ∈ PROPN iajs-2288 220 26 𝑅 𝑅 PROPN iajs-2288 220 27 , , PUNCT iajs-2288 220 28 𝑑1 𝑑1 PROPN iajs-2288 220 29 ∈ ∈ NOUN iajs-2288 220 30 𝑇1 𝑇1 NOUN iajs-2288 220 31 and and CCONJ iajs-2288 220 32 𝑛 𝑛 PRON iajs-2288 220 33 ∈ ∈ PROPN iajs-2288 220 34 𝑍+ 𝑍+ NOUN iajs-2288 220 35 . . PUNCT iajs-2288 221 1 then then ADV iajs-2288 221 2 for for SCONJ iajs-2288 221 3 each each DET iajs-2288 221 4 𝑑2 𝑑2 PROPN iajs-2288 221 5 ∈ ∈ PROPN iajs-2288 221 6 𝑇2 𝑇2 PROPN iajs-2288 221 7 π‘Žπ‘›(𝑑1 π‘Žπ‘›(𝑑1 PROPN iajs-2288 221 8 , , PUNCT iajs-2288 221 9 𝑑2 𝑑2 ADJ iajs-2288 221 10 ) ) PUNCT iajs-2288 221 11 ∈ ∈ PROPN iajs-2288 221 12 𝐿1 𝐿1 PROPN iajs-2288 221 13 βŠ•π‘‡2 βŠ•π‘‡2 NOUN iajs-2288 221 14 , , PUNCT iajs-2288 221 15 but but CCONJ iajs-2288 221 16 𝐿1 𝐿1 PROPN iajs-2288 221 17 βŠ•π‘‡2 βŠ•π‘‡2 NOUN iajs-2288 221 18 is be AUX iajs-2288 221 19 an an DET iajs-2288 221 20 app app ADJ iajs-2288 221 21 - - PUNCT iajs-2288 221 22 semi semi ADJ iajs-2288 221 23 - - ADJ iajs-2288 221 24 prime prime ADJ iajs-2288 221 25 submodule submodule NOUN iajs-2288 221 26 of of ADP iajs-2288 221 27 𝑇 𝑇 PROPN iajs-2288 221 28 , , PUNCT iajs-2288 221 29 so so ADV iajs-2288 221 30 π‘Ž(𝑑1 π‘Ž(𝑑1 PROPN iajs-2288 221 31 , , PUNCT iajs-2288 221 32 𝑑2 𝑑2 ADJ iajs-2288 221 33 ) ) PUNCT iajs-2288 221 34 ∈ ∈ PROPN iajs-2288 222 1 𝐿1 𝐿1 PROPN iajs-2288 223 1 βŠ• βŠ• PROPN iajs-2288 223 2 𝑇2 𝑇2 NOUN iajs-2288 223 3 + + CCONJ iajs-2288 223 4 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 223 5 βŠ• βŠ• PROPN iajs-2288 223 6 𝑇2 𝑇2 NOUN iajs-2288 223 7 ) ) PUNCT iajs-2288 224 1 = = PRON iajs-2288 224 2 ( ( PUNCT iajs-2288 224 3 𝐿1βŠ•π‘‡2 𝐿1βŠ•π‘‡2 NOUN iajs-2288 224 4 ) ) PUNCT iajs-2288 225 1 + + CCONJ iajs-2288 225 2 ( ( PUNCT iajs-2288 225 3 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 ADJ iajs-2288 225 4 ) ) PUNCT iajs-2288 225 5 βŠ• βŠ• PROPN iajs-2288 225 6 π‘ π‘œπ‘(𝑇2 π‘ π‘œπ‘(𝑇2 PROPN iajs-2288 225 7 ) ) PUNCT iajs-2288 225 8 ) ) PUNCT iajs-2288 225 9 . . PUNCT iajs-2288 226 1 since since SCONJ iajs-2288 226 2 𝐿1 𝐿1 ADP iajs-2288 226 3 βŠ† βŠ† NUM iajs-2288 226 4 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 NUM iajs-2288 226 5 ) ) PUNCT iajs-2288 226 6 and and CCONJ iajs-2288 226 7 𝑇2 𝑇2 NOUN iajs-2288 226 8 = = SYM iajs-2288 226 9 π‘ π‘œπ‘(𝑇2 π‘ π‘œπ‘(𝑇2 PROPN iajs-2288 226 10 ) ) PUNCT iajs-2288 226 11 , , PUNCT iajs-2288 226 12 it it PRON iajs-2288 226 13 follows follow VERB iajs-2288 226 14 that that PRON iajs-2288 226 15 π‘Ž(𝑑1 π‘Ž(𝑑1 PROPN iajs-2288 226 16 , , PUNCT iajs-2288 226 17 𝑑2 𝑑2 ADJ iajs-2288 226 18 ) ) PUNCT iajs-2288 226 19 ∈ ∈ PROPN iajs-2288 226 20 ( ( PUNCT iajs-2288 226 21 𝐿1 𝐿1 PROPN iajs-2288 226 22 βŠ•π‘‡2 βŠ•π‘‡2 NUM iajs-2288 226 23 ) ) PUNCT iajs-2288 227 1 + + CCONJ iajs-2288 227 2 ( ( PUNCT iajs-2288 227 3 ( ( PUNCT iajs-2288 227 4 𝐿1 𝐿1 PROPN iajs-2288 227 5 + + CCONJ iajs-2288 227 6 π‘ π‘œπ‘(𝑇1))βŠ• π‘ π‘œπ‘(𝑇1))βŠ• ADJ iajs-2288 227 7 𝑇2 𝑇2 NOUN iajs-2288 227 8 ) ) PUNCT iajs-2288 227 9 , , PUNCT iajs-2288 227 10 implies imply VERB iajs-2288 227 11 that that PRON iajs-2288 227 12 π‘Ž(𝑑1 π‘Ž(𝑑1 PROPN iajs-2288 227 13 , , PUNCT iajs-2288 227 14 𝑑2 𝑑2 ADJ iajs-2288 227 15 ) ) PUNCT iajs-2288 227 16 ∈ ∈ PROPN iajs-2288 227 17 ( ( PUNCT iajs-2288 227 18 𝐿1 𝐿1 PROPN iajs-2288 227 19 + + CCONJ iajs-2288 227 20 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 NOUN iajs-2288 227 21 ) ) PUNCT iajs-2288 227 22 ) ) PUNCT iajs-2288 228 1 βŠ• βŠ• PROPN iajs-2288 228 2 𝑇2 𝑇2 PROPN iajs-2288 229 1 [ [ X iajs-2288 229 2 because because SCONJ iajs-2288 229 3 𝐿1βŠ•π‘‡2 𝐿1βŠ•π‘‡2 PROPN iajs-2288 229 4 βŠ† βŠ† NUM iajs-2288 229 5 ( ( PUNCT iajs-2288 229 6 𝐿1 𝐿1 PROPN iajs-2288 229 7 + + CCONJ iajs-2288 229 8 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 NOUN iajs-2288 229 9 ) ) PUNCT iajs-2288 229 10 ) ) PUNCT iajs-2288 229 11 βŠ• βŠ• PROPN iajs-2288 229 12 𝑇2 𝑇2 PROPN iajs-2288 229 13 implies imply VERB iajs-2288 229 14 that that SCONJ iajs-2288 229 15 ( ( PUNCT iajs-2288 229 16 𝐿1βŠ•π‘‡2 𝐿1βŠ•π‘‡2 NOUN iajs-2288 229 17 ) ) PUNCT iajs-2288 230 1 + + CCONJ iajs-2288 230 2 ( ( PUNCT iajs-2288 230 3 ( ( PUNCT iajs-2288 230 4 𝐿1 𝐿1 PROPN iajs-2288 230 5 + + CCONJ iajs-2288 230 6 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 NOUN iajs-2288 230 7 ) ) PUNCT iajs-2288 230 8 ) ) PUNCT iajs-2288 231 1 βŠ• βŠ• PROPN iajs-2288 231 2 𝑇2 𝑇2 PROPN iajs-2288 231 3 ) ) PUNCT iajs-2288 232 1 = = PRON iajs-2288 232 2 ( ( PUNCT iajs-2288 232 3 𝐿1 𝐿1 PROPN iajs-2288 232 4 + + CCONJ iajs-2288 233 1 π‘ π‘œπ‘(𝑇1))βŠ• π‘ π‘œπ‘(𝑇1))βŠ• ADJ iajs-2288 233 2 𝑇2 𝑇2 NOUN iajs-2288 233 3 ] ] PUNCT iajs-2288 233 4 . . PUNCT iajs-2288 234 1 thus thus ADV iajs-2288 234 2 π‘Žπ‘‘1 π‘Žπ‘‘1 VERB iajs-2288 234 3 ∈ ∈ PROPN iajs-2288 234 4 𝐿1 𝐿1 NOUN iajs-2288 235 1 + + CCONJ iajs-2288 235 2 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 NOUN iajs-2288 235 3 ) ) PUNCT iajs-2288 235 4 . . PUNCT iajs-2288 236 1 hence hence ADV iajs-2288 236 2 𝐿1 𝐿1 PROPN iajs-2288 236 3 is be AUX iajs-2288 236 4 an an DET iajs-2288 236 5 app app ADJ iajs-2288 236 6 - - PUNCT iajs-2288 236 7 semi semi ADJ iajs-2288 236 8 - - ADJ iajs-2288 236 9 prime prime ADJ iajs-2288 236 10 submodule submodule NOUN iajs-2288 236 11 of of ADP iajs-2288 236 12 𝑇1 𝑇1 NOUN iajs-2288 236 13 . . PUNCT iajs-2288 237 1 2 2 NUM iajs-2288 237 2 ) ) PUNCT iajs-2288 237 3 similarly similarly ADV iajs-2288 237 4 we we PRON iajs-2288 237 5 can can AUX iajs-2288 237 6 prove prove VERB iajs-2288 237 7 ( ( PUNCT iajs-2288 237 8 2 2 NUM iajs-2288 237 9 ) ) PUNCT iajs-2288 237 10 . . PUNCT iajs-2288 238 1 proposition proposition NOUN iajs-2288 238 2 ( ( PUNCT iajs-2288 238 3 32 32 NUM iajs-2288 238 4 ) ) PUNCT iajs-2288 238 5 let let VERB iajs-2288 238 6 𝑓 𝑓 PRON iajs-2288 238 7 : : PUNCT iajs-2288 238 8 𝑇 𝑇 PROPN iajs-2288 238 9 β†’ β†’ SYM iajs-2288 238 10 𝑇′ 𝑇′ NOUN iajs-2288 238 11 be be AUX iajs-2288 238 12 an an DET iajs-2288 238 13 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2288 238 14 and and CCONJ iajs-2288 238 15 𝐿 𝐿 PROPN iajs-2288 238 16 is be AUX iajs-2288 238 17 an an DET iajs-2288 238 18 app app ADJ iajs-2288 238 19 - - PUNCT iajs-2288 238 20 semi semi ADJ iajs-2288 238 21 - - ADJ iajs-2288 238 22 prime prime ADJ iajs-2288 238 23 submodule submodule NOUN iajs-2288 238 24 of of ADP iajs-2288 238 25 𝑇′. 𝑇′. NOUN iajs-2288 238 26 then then ADV iajs-2288 238 27 π‘“βˆ’1(𝐿 π‘“βˆ’1(𝐿 NOUN iajs-2288 238 28 ) ) PUNCT iajs-2288 238 29 is be AUX iajs-2288 238 30 an an DET iajs-2288 238 31 app app ADJ iajs-2288 238 32 - - PUNCT iajs-2288 238 33 semi semi ADJ iajs-2288 238 34 - - ADJ iajs-2288 238 35 prime prime ADJ iajs-2288 238 36 submodule submodule NOUN iajs-2288 238 37 of of ADP iajs-2288 238 38 𝑇. 𝑇. PROPN iajs-2288 238 39 proof proof NOUN iajs-2288 238 40 it it PRON iajs-2288 238 41 is be AUX iajs-2288 238 42 clear clear ADJ iajs-2288 238 43 that that SCONJ iajs-2288 238 44 π‘“βˆ’1(𝐿 π‘“βˆ’1(𝐿 X iajs-2288 238 45 ) ) PUNCT iajs-2288 238 46 is be AUX iajs-2288 238 47 a a DET iajs-2288 238 48 proper proper ADJ iajs-2288 238 49 submodule submodule NOUN iajs-2288 238 50 of of ADP iajs-2288 238 51 𝑇. 𝑇. PROPN iajs-2288 238 52 now now ADV iajs-2288 238 53 , , PUNCT iajs-2288 238 54 let let VERB iajs-2288 238 55 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ NOUN iajs-2288 238 56 ∈ ∈ PROPN iajs-2288 238 57 π‘“βˆ’1(𝐿 π‘“βˆ’1(𝐿 NOUN iajs-2288 238 58 ) ) PUNCT iajs-2288 238 59 , , PUNCT iajs-2288 238 60 where where SCONJ iajs-2288 238 61 π‘Ž π‘Ž PROPN iajs-2288 238 62 ∈ ∈ PROPN iajs-2288 238 63 𝑅 𝑅 PROPN iajs-2288 238 64 , , PUNCT iajs-2288 238 65 𝑑 𝑑 PROPN iajs-2288 238 66 ∈ ∈ PROPN iajs-2288 238 67 𝑇 𝑇 PROPN iajs-2288 238 68 and and CCONJ iajs-2288 238 69 𝑛 𝑛 DET iajs-2288 238 70 ∈ ∈ PROPN iajs-2288 238 71 𝑍+ 𝑍+ NOUN iajs-2288 238 72 , , PUNCT iajs-2288 238 73 implies imply VERB iajs-2288 238 74 that that SCONJ iajs-2288 238 75 π‘Žπ‘›π‘“(𝑑 π‘Žπ‘›π‘“(𝑑 PROPN iajs-2288 238 76 ) ) PUNCT iajs-2288 238 77 ∈ ∈ PROPN iajs-2288 238 78 𝐿. 𝐿. NOUN iajs-2288 238 79 but but CCONJ iajs-2288 238 80 l l NOUN iajs-2288 238 81 is be AUX iajs-2288 238 82 an an DET iajs-2288 238 83 app app ADJ iajs-2288 238 84 - - PUNCT iajs-2288 238 85 semi semi ADJ iajs-2288 238 86 - - ADJ iajs-2288 238 87 prime prime ADJ iajs-2288 238 88 submodule submodule NOUN iajs-2288 238 89 of of ADP iajs-2288 238 90 𝑇′ 𝑇′ NUM iajs-2288 238 91 , , PUNCT iajs-2288 238 92 implies imply VERB iajs-2288 238 93 that that SCONJ iajs-2288 238 94 π‘Žπ‘“(𝑑 π‘Žπ‘“(𝑑 NUM iajs-2288 238 95 ) ) PUNCT iajs-2288 238 96 ∈ ∈ PROPN iajs-2288 238 97 𝐿 𝐿 PROPN iajs-2288 238 98 + + PROPN iajs-2288 238 99 π‘ π‘œπ‘(𝑇′ π‘ π‘œπ‘(𝑇′ PROPN iajs-2288 238 100 ) ) PUNCT iajs-2288 238 101 , , PUNCT iajs-2288 238 102 it it PRON iajs-2288 238 103 follows follow VERB iajs-2288 238 104 that that SCONJ iajs-2288 238 105 π‘Žπ‘‘ π‘Žπ‘‘ PROPN iajs-2288 238 106 ∈ ∈ PROPN iajs-2288 238 107 π‘“βˆ’1(𝐿 π‘“βˆ’1(𝐿 NOUN iajs-2288 238 108 ) ) PUNCT iajs-2288 238 109 + + CCONJ iajs-2288 239 1 π‘“βˆ’1(π‘ π‘œπ‘(𝑇′ π‘“βˆ’1(π‘ π‘œπ‘(𝑇′ NUM iajs-2288 239 2 ) ) PUNCT iajs-2288 239 3 ) ) PUNCT iajs-2288 240 1 βŠ† βŠ† NUM iajs-2288 240 2 π‘“βˆ’1(𝐿 π‘“βˆ’1(𝐿 NOUN iajs-2288 240 3 ) ) PUNCT iajs-2288 240 4 + + NUM iajs-2288 240 5 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 240 6 ) ) PUNCT iajs-2288 240 7 . . PUNCT iajs-2288 241 1 thus thus ADV iajs-2288 241 2 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 241 3 ∈ ∈ PROPN iajs-2288 241 4 π‘“βˆ’1(𝐿 π‘“βˆ’1(𝐿 X iajs-2288 241 5 ) ) PUNCT iajs-2288 241 6 + + NUM iajs-2288 241 7 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 241 8 ) ) PUNCT iajs-2288 241 9 . . PUNCT iajs-2288 242 1 therefore therefore ADV iajs-2288 242 2 π‘“βˆ’1(𝐿 π‘“βˆ’1(𝐿 X iajs-2288 242 3 ) ) PUNCT iajs-2288 242 4 is be AUX iajs-2288 242 5 an an DET iajs-2288 242 6 app app ADJ iajs-2288 242 7 - - PUNCT iajs-2288 242 8 semi semi ADJ iajs-2288 242 9 - - ADJ iajs-2288 242 10 prime prime ADJ iajs-2288 242 11 submodule submodule NOUN iajs-2288 242 12 of of ADP iajs-2288 242 13 𝑇. 𝑇. PROPN iajs-2288 242 14 proposition proposition NOUN iajs-2288 242 15 ( ( PUNCT iajs-2288 242 16 33 33 NUM iajs-2288 242 17 ) ) PUNCT iajs-2288 242 18 let let VERB iajs-2288 242 19 𝑓 𝑓 PRON iajs-2288 242 20 : : PUNCT iajs-2288 242 21 𝑇 𝑇 PROPN iajs-2288 242 22 β†’ β†’ SYM iajs-2288 242 23 𝑇′ 𝑇′ NOUN iajs-2288 242 24 be be AUX iajs-2288 242 25 an an DET iajs-2288 242 26 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2288 242 27 and and CCONJ iajs-2288 242 28 𝐾 𝐾 PROPN iajs-2288 242 29 be be VERB iajs-2288 242 30 an an DET iajs-2288 242 31 app app ADJ iajs-2288 242 32 - - PUNCT iajs-2288 242 33 semi semi ADJ iajs-2288 242 34 - - ADJ iajs-2288 242 35 prime prime ADJ iajs-2288 242 36 submodule submodule NOUN iajs-2288 242 37 of of ADP iajs-2288 242 38 𝑇 𝑇 PROPN iajs-2288 242 39 with with ADP iajs-2288 242 40 πΎπ‘’π‘Ÿ πΎπ‘’π‘Ÿ PROPN iajs-2288 242 41 𝑓 𝑓 PRON iajs-2288 242 42 βŠ† βŠ† NUM iajs-2288 242 43 𝐾. 𝐾. PROPN iajs-2288 242 44 then then ADV iajs-2288 242 45 𝑓(𝐾 𝑓(𝐾 VERB iajs-2288 242 46 ) ) PUNCT iajs-2288 242 47 is be AUX iajs-2288 242 48 an an DET iajs-2288 242 49 app app ADJ iajs-2288 242 50 - - PUNCT iajs-2288 242 51 quasi quasi ADJ iajs-2288 242 52 - - ADJ iajs-2288 242 53 prime prime ADJ iajs-2288 242 54 submodule submodule NOUN iajs-2288 242 55 of of ADP iajs-2288 242 56 𝑇′. 𝑇′. NOUN iajs-2288 242 57 proof proof NOUN iajs-2288 242 58 𝑓(𝐾 𝑓(𝐾 VERB iajs-2288 242 59 ) ) PUNCT iajs-2288 243 1 is be AUX iajs-2288 243 2 a a DET iajs-2288 243 3 proper proper ADJ iajs-2288 243 4 submodule submodule NOUN iajs-2288 243 5 of of ADP iajs-2288 243 6 𝑇′. 𝑇′. NOUN iajs-2288 243 7 if if SCONJ iajs-2288 243 8 not not PART iajs-2288 243 9 , , PUNCT iajs-2288 243 10 𝑓(𝐾 𝑓(𝐾 ADJ iajs-2288 243 11 ) ) PUNCT iajs-2288 244 1 = = SYM iajs-2288 244 2 𝑇′ 𝑇′ NOUN iajs-2288 244 3 , , PUNCT iajs-2288 244 4 that that PRON iajs-2288 244 5 is be AUX iajs-2288 244 6 𝑑 𝑑 PROPN iajs-2288 244 7 ∈ ∈ PROPN iajs-2288 244 8 𝑇 𝑇 PROPN iajs-2288 244 9 , , PUNCT iajs-2288 244 10 then then ADV iajs-2288 244 11 𝑓(𝑑 𝑓(𝑑 PROPN iajs-2288 244 12 ) ) PUNCT iajs-2288 244 13 ∈ ∈ PROPN iajs-2288 244 14 𝑇′ 𝑇′ NOUN iajs-2288 244 15 = = SYM iajs-2288 244 16 𝑓(𝐾 𝑓(𝐾 PROPN iajs-2288 244 17 ) ) PUNCT iajs-2288 244 18 , , PUNCT iajs-2288 244 19 implies imply VERB iajs-2288 244 20 that that SCONJ iajs-2288 244 21 𝑓(𝑑 𝑓(𝑑 NOUN iajs-2288 244 22 ) ) PUNCT iajs-2288 244 23 = = PUNCT iajs-2288 244 24 𝑓(π‘˜ 𝑓(π‘˜ PROPN iajs-2288 244 25 ) ) PUNCT iajs-2288 244 26 for for ADP iajs-2288 244 27 some some DET iajs-2288 244 28 π‘˜ π‘˜ PRON iajs-2288 244 29 ∈ ∈ PROPN iajs-2288 244 30 𝐾 𝐾 PROPN iajs-2288 244 31 , , PUNCT iajs-2288 244 32 that that PRON iajs-2288 244 33 is is ADV iajs-2288 244 34 𝑓(𝑑 𝑓(𝑑 PROPN iajs-2288 244 35 βˆ’ βˆ’ PROPN iajs-2288 244 36 π‘˜ π‘˜ NOUN iajs-2288 244 37 ) ) PUNCT iajs-2288 244 38 = = SYM iajs-2288 244 39 0 0 NUM iajs-2288 244 40 , , PUNCT iajs-2288 244 41 thus thus ADV iajs-2288 244 42 𝑑 𝑑 VERB iajs-2288 244 43 βˆ’ βˆ’ PROPN iajs-2288 244 44 π‘˜ π‘˜ PRON iajs-2288 244 45 ∈ ∈ NOUN iajs-2288 245 1 πΎπ‘’π‘Ÿ πΎπ‘’π‘Ÿ NOUN iajs-2288 245 2 𝑓 𝑓 DET iajs-2288 245 3 βŠ† βŠ† NUM iajs-2288 245 4 𝐾 𝐾 PROPN iajs-2288 245 5 , , PUNCT iajs-2288 245 6 it it PRON iajs-2288 245 7 follows follow VERB iajs-2288 245 8 that that SCONJ iajs-2288 245 9 𝑑 𝑑 PROPN iajs-2288 245 10 ∈ ∈ PROPN iajs-2288 245 11 𝐾 𝐾 PROPN iajs-2288 245 12 , , PUNCT iajs-2288 245 13 that that PRON iajs-2288 245 14 is be AUX iajs-2288 245 15 𝑇 𝑇 PROPN iajs-2288 245 16 βŠ† βŠ† NUM iajs-2288 245 17 𝐾 𝐾 PROPN iajs-2288 245 18 , , PUNCT iajs-2288 245 19 but but CCONJ iajs-2288 245 20 𝐾 𝐾 PROPN iajs-2288 245 21 βŠ† βŠ† NUM iajs-2288 245 22 𝑇 𝑇 PROPN iajs-2288 245 23 , , PUNCT iajs-2288 245 24 so so ADV iajs-2288 245 25 𝑇 𝑇 PROPN iajs-2288 245 26 = = SYM iajs-2288 245 27 𝐾 𝐾 PROPN iajs-2288 245 28 contradiction contradiction NOUN iajs-2288 245 29 . . PUNCT iajs-2288 246 1 now now ADV iajs-2288 246 2 let let VERB iajs-2288 246 3 π‘Žπ‘›π‘‘β€² π‘Žπ‘›π‘‘β€² PROPN iajs-2288 246 4 ∈ ∈ PROPN iajs-2288 246 5 𝑓(𝐾 𝑓(𝐾 NOUN iajs-2288 246 6 ) ) PUNCT iajs-2288 246 7 , , PUNCT iajs-2288 246 8 where where SCONJ iajs-2288 246 9 π‘Ž π‘Ž PROPN iajs-2288 246 10 ∈ ∈ PROPN iajs-2288 246 11 𝑅 𝑅 PROPN iajs-2288 246 12 , , PUNCT iajs-2288 246 13 𝑑′ 𝑑′ NUM iajs-2288 246 14 ∈ ∈ PROPN iajs-2288 246 15 𝑇′ 𝑇′ NOUN iajs-2288 246 16 and and CCONJ iajs-2288 246 17 𝑛 𝑛 PRON iajs-2288 246 18 ∈ ∈ PROPN iajs-2288 246 19 𝑍+ 𝑍+ NOUN iajs-2288 246 20 . . PUNCT iajs-2288 247 1 but but CCONJ iajs-2288 247 2 𝑓 𝑓 PRON iajs-2288 247 3 is be AUX iajs-2288 247 4 an an DET iajs-2288 247 5 epimorphism epimorphism NOUN iajs-2288 247 6 , , PUNCT iajs-2288 247 7 then then ADV iajs-2288 247 8 𝑓(𝑑 𝑓(𝑑 PROPN iajs-2288 247 9 ) ) PUNCT iajs-2288 247 10 = = SYM iajs-2288 247 11 𝑑′ 𝑑′ NOUN iajs-2288 247 12 for for ADP iajs-2288 247 13 some some DET iajs-2288 247 14 𝑑 𝑑 PROPN iajs-2288 247 15 ∈ ∈ PROPN iajs-2288 247 16 𝑇. 𝑇. PROPN iajs-2288 247 17 that that PRON iajs-2288 247 18 is be AUX iajs-2288 247 19 π‘Žπ‘›π‘“(𝑑 π‘Žπ‘›π‘“(𝑑 PROPN iajs-2288 247 20 ) ) PUNCT iajs-2288 247 21 ∈ ∈ NOUN iajs-2288 247 22 𝑓(𝐾 𝑓(𝐾 NOUN iajs-2288 247 23 ) ) PUNCT iajs-2288 247 24 , , PUNCT iajs-2288 247 25 implies imply VERB iajs-2288 247 26 that that SCONJ iajs-2288 247 27 π‘Žπ‘›π‘“(𝑑 π‘Žπ‘›π‘“(𝑑 X iajs-2288 247 28 ) ) PUNCT iajs-2288 247 29 = = PUNCT iajs-2288 247 30 𝑓(π‘˜ 𝑓(π‘˜ PROPN iajs-2288 247 31 ) ) PUNCT iajs-2288 247 32 for for ADP iajs-2288 247 33 some some PRON iajs-2288 247 34 π‘˜ π‘˜ PRON iajs-2288 247 35 ∈ ∈ ADJ iajs-2288 248 1 𝐾. 𝐾. NOUN iajs-2288 248 2 that that PRON iajs-2288 248 3 is be AUX iajs-2288 248 4 𝑓(π‘Žπ‘›π‘‘ 𝑓(π‘Žπ‘›π‘‘ NOUN iajs-2288 248 5 βˆ’ βˆ’ PROPN iajs-2288 248 6 π‘˜ π‘˜ NOUN iajs-2288 248 7 ) ) PUNCT iajs-2288 248 8 = = SYM iajs-2288 248 9 0 0 NUM iajs-2288 248 10 , , PUNCT iajs-2288 248 11 it it PRON iajs-2288 248 12 follows follow VERB iajs-2288 248 13 that that SCONJ iajs-2288 248 14 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 248 15 βˆ’ βˆ’ PROPN iajs-2288 248 16 π‘˜ π‘˜ PRON iajs-2288 248 17 ∈ ∈ NOUN iajs-2288 249 1 πΎπ‘’π‘Ÿ πΎπ‘’π‘Ÿ NOUN iajs-2288 249 2 𝑓 𝑓 DET iajs-2288 249 3 βŠ† βŠ† NUM iajs-2288 249 4 𝐾 𝐾 PROPN iajs-2288 249 5 , , PUNCT iajs-2288 249 6 hence hence ADV iajs-2288 249 7 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ VERB iajs-2288 249 8 ∈ ∈ PROPN iajs-2288 249 9 𝐾. 𝐾. PROPN iajs-2288 249 10 but but CCONJ iajs-2288 249 11 𝐾 𝐾 PROPN iajs-2288 249 12 is be AUX iajs-2288 249 13 an an DET iajs-2288 249 14 app app ADJ iajs-2288 249 15 - - PUNCT iajs-2288 249 16 semi semi ADJ iajs-2288 249 17 - - ADJ iajs-2288 249 18 prime prime ADJ iajs-2288 249 19 submodule submodule NOUN iajs-2288 249 20 of of ADP iajs-2288 249 21 𝑇 𝑇 PROPN iajs-2288 249 22 , , PUNCT iajs-2288 249 23 then then ADV iajs-2288 249 24 π‘Žπ‘‘ π‘Žπ‘‘ ADP iajs-2288 249 25 ∈ ∈ PROPN iajs-2288 249 26 𝐾 𝐾 PROPN iajs-2288 249 27 + + CCONJ iajs-2288 249 28 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 249 29 ) ) PUNCT iajs-2288 249 30 . . PUNCT iajs-2288 250 1 thus thus ADV iajs-2288 250 2 ( ( PUNCT iajs-2288 250 3 𝑑 𝑑 NOUN iajs-2288 250 4 ) ) PUNCT iajs-2288 250 5 ∈ ∈ NOUN iajs-2288 250 6 𝑓(𝐾 𝑓(𝐾 NOUN iajs-2288 250 7 ) ) PUNCT iajs-2288 251 1 + + CCONJ iajs-2288 251 2 𝑓(π‘ π‘œπ‘(𝑇 𝑓(π‘ π‘œπ‘(𝑇 X iajs-2288 251 3 ) ) PUNCT iajs-2288 251 4 ) ) PUNCT iajs-2288 252 1 βŠ† βŠ† NUM iajs-2288 252 2 𝑓(𝐾 𝑓(𝐾 NOUN iajs-2288 252 3 ) ) PUNCT iajs-2288 253 1 + + PROPN iajs-2288 253 2 π‘ π‘œπ‘(𝑇′ π‘ π‘œπ‘(𝑇′ PROPN iajs-2288 253 3 ) ) PUNCT iajs-2288 253 4 . . PUNCT iajs-2288 254 1 that that PRON iajs-2288 254 2 is be AUX iajs-2288 254 3 π‘Žπ‘‘β€² π‘Žπ‘‘β€² ADP iajs-2288 254 4 ∈ ∈ PROPN iajs-2288 254 5 𝑓(𝐾 𝑓(𝐾 VERB iajs-2288 254 6 ) ) PUNCT iajs-2288 255 1 + + PUNCT iajs-2288 256 1 π‘ π‘œπ‘(𝑇′ π‘ π‘œπ‘(𝑇′ PROPN iajs-2288 256 2 ) ) PUNCT iajs-2288 256 3 . . PUNCT iajs-2288 257 1 hence hence ADV iajs-2288 257 2 𝑓(𝐾 𝑓(𝐾 NUM iajs-2288 257 3 ) ) PUNCT iajs-2288 257 4 is be AUX iajs-2288 257 5 an an DET iajs-2288 257 6 app app ADJ iajs-2288 257 7 - - PUNCT iajs-2288 257 8 semi semi ADJ iajs-2288 257 9 - - ADJ iajs-2288 257 10 prime prime ADJ iajs-2288 257 11 submodule submodule NOUN iajs-2288 257 12 of of ADP iajs-2288 257 13 𝑇′. 𝑇′. NOUN iajs-2288 257 14 3 3 NUM iajs-2288 257 15 . . X iajs-2288 257 16 conclusion conclusion NOUN iajs-2288 257 17 in in ADP iajs-2288 257 18 this this DET iajs-2288 257 19 paper paper NOUN iajs-2288 257 20 we we PRON iajs-2288 257 21 define define VERB iajs-2288 257 22 the the DET iajs-2288 257 23 concept concept NOUN iajs-2288 257 24 of of ADP iajs-2288 257 25 approximaitly approximaitly ADV iajs-2288 257 26 semi semi ADJ iajs-2288 257 27 - - ADJ iajs-2288 257 28 prime prime ADJ iajs-2288 257 29 ( ( PUNCT iajs-2288 257 30 for for ADP iajs-2288 257 31 short short ADJ iajs-2288 257 32 app app ADJ iajs-2288 257 33 - - PUNCT iajs-2288 257 34 semiprime semiprime NOUN iajs-2288 257 35 ) ) PUNCT iajs-2288 257 36 submodules submodule NOUN iajs-2288 257 37 , , PUNCT iajs-2288 257 38 and and CCONJ iajs-2288 257 39 we we PRON iajs-2288 257 40 introduced introduce VERB iajs-2288 257 41 several several ADJ iajs-2288 257 42 properties property NOUN iajs-2288 257 43 , , PUNCT iajs-2288 257 44 characterizations characterization NOUN iajs-2288 257 45 of of ADP iajs-2288 257 46 it it PRON iajs-2288 257 47 . . PUNCT iajs-2288 258 1 also also ADV iajs-2288 258 2 , , PUNCT iajs-2288 258 3 we we PRON iajs-2288 258 4 investigate investigate VERB iajs-2288 258 5 the the DET iajs-2288 258 6 relationships relationship NOUN iajs-2288 258 7 of of ADP iajs-2288 258 8 app app ADJ iajs-2288 258 9 - - PUNCT iajs-2288 258 10 semi semi ADJ iajs-2288 258 11 - - ADJ iajs-2288 258 12 prime prime ADJ iajs-2288 258 13 submodules submodule NOUN iajs-2288 258 14 with with ADP iajs-2288 258 15 prime prime ADJ iajs-2288 258 16 submodules submodule NOUN iajs-2288 258 17 , , PUNCT iajs-2288 258 18 semi semi ADV iajs-2288 258 19 127 127 NUM iajs-2288 258 20 ibn ibn PROPN iajs-2288 258 21 al al PROPN iajs-2288 258 22 - - PUNCT iajs-2288 258 23 haitham haitham PROPN iajs-2288 258 24 jour jour X iajs-2288 258 25 . . PROPN iajs-2288 259 1 for for ADP iajs-2288 259 2 pure pure ADJ iajs-2288 259 3 & & CCONJ iajs-2288 259 4 appl appl PROPN iajs-2288 259 5 . . PUNCT iajs-2288 260 1 sci sci PROPN iajs-2288 260 2 . . PROPN iajs-2288 260 3 32 32 NUM iajs-2288 260 4 ( ( PUNCT iajs-2288 260 5 3 3 NUM iajs-2288 260 6 ) ) SYM iajs-2288 260 7 2019 2019 NUM iajs-2288 260 8 prime prime ADJ iajs-2288 260 9 submodules submodule NOUN iajs-2288 260 10 , , PUNCT iajs-2288 260 11 quasi quasi ADJ iajs-2288 260 12 - - ADJ iajs-2288 260 13 prime prime ADJ iajs-2288 260 14 submodules submodule NOUN iajs-2288 260 15 and and CCONJ iajs-2288 260 16 approximaitly approximaitly ADV iajs-2288 260 17 prime prime ADJ iajs-2288 260 18 submodules submodule NOUN iajs-2288 260 19 , , PUNCT iajs-2288 260 20 we we PRON iajs-2288 260 21 proved prove VERB iajs-2288 260 22 that that SCONJ iajs-2288 260 23 app app NOUN iajs-2288 260 24 - - PUNCT iajs-2288 260 25 semi semi ADJ iajs-2288 260 26 - - ADJ iajs-2288 260 27 prime prime ADJ iajs-2288 260 28 submodules submodule NOUN iajs-2288 260 29 are be AUX iajs-2288 260 30 generalizations generalization NOUN iajs-2288 260 31 of of ADP iajs-2288 260 32 above above ADJ iajs-2288 260 33 concepts concept NOUN iajs-2288 260 34 , , PUNCT iajs-2288 260 35 and and CCONJ iajs-2288 260 36 we we PRON iajs-2288 260 37 illistright illistright VERB iajs-2288 260 38 the the DET iajs-2288 260 39 convers conver NOUN iajs-2288 260 40 by by ADP iajs-2288 260 41 examples example NOUN iajs-2288 260 42 . . PUNCT iajs-2288 261 1 also also ADV iajs-2288 261 2 , , PUNCT iajs-2288 261 3 we we PRON iajs-2288 261 4 show show VERB iajs-2288 261 5 by by ADP iajs-2288 261 6 example example NOUN iajs-2288 261 7 that that SCONJ iajs-2288 261 8 the the DET iajs-2288 261 9 resudule resudule NOUN iajs-2288 261 10 of of ADP iajs-2288 261 11 an an DET iajs-2288 261 12 app app ADJ iajs-2288 261 13 - - PUNCT iajs-2288 261 14 semi semi ADJ iajs-2288 261 15 - - ADJ iajs-2288 261 16 prime prime ADJ iajs-2288 261 17 submodule submodule NOUN iajs-2288 261 18 is be AUX iajs-2288 261 19 not not PART iajs-2288 261 20 necessary necessary ADJ iajs-2288 261 21 app app ADJ iajs-2288 261 22 - - PUNCT iajs-2288 261 23 semi semi ADJ iajs-2288 261 24 - - ADJ iajs-2288 261 25 prime prime ADJ iajs-2288 261 26 ideal ideal NOUN iajs-2288 261 27 of of ADP iajs-2288 261 28 𝑅 𝑅 PROPN iajs-2288 261 29 , , PUNCT iajs-2288 261 30 but but CCONJ iajs-2288 261 31 we we PRON iajs-2288 261 32 prove prove VERB iajs-2288 261 33 under under ADP iajs-2288 261 34 certain certain ADJ iajs-2288 261 35 conditions condition NOUN iajs-2288 261 36 they they PRON iajs-2288 261 37 are be AUX iajs-2288 261 38 equivalents equivalent NOUN iajs-2288 261 39 . . PUNCT iajs-2288 262 1 a a DET iajs-2288 262 2 mange mange NOUN iajs-2288 262 3 the the DET iajs-2288 262 4 main main ADJ iajs-2288 262 5 results result NOUN iajs-2288 262 6 we we PRON iajs-2288 262 7 get get VERB iajs-2288 262 8 are be AUX iajs-2288 262 9 the the DET iajs-2288 262 10 following following NOUN iajs-2288 262 11 . . PUNCT iajs-2288 263 1 1 1 X iajs-2288 263 2 ) ) PUNCT iajs-2288 263 3 let let VERB iajs-2288 263 4 𝐿 𝐿 PROPN iajs-2288 263 5 be be AUX iajs-2288 263 6 a a DET iajs-2288 263 7 proper proper ADJ iajs-2288 263 8 submodule submodule NOUN iajs-2288 263 9 of of ADP iajs-2288 263 10 an an DET iajs-2288 263 11 𝑅-module 𝑅-module PROPN iajs-2288 263 12 𝑇. 𝑇. PROPN iajs-2288 263 13 then then ADV iajs-2288 263 14 𝐿 𝐿 PROPN iajs-2288 263 15 + + CCONJ iajs-2288 263 16 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 263 17 ) ) PUNCT iajs-2288 263 18 is be AUX iajs-2288 263 19 an an DET iajs-2288 263 20 app app ADJ iajs-2288 263 21 - - PUNCT iajs-2288 263 22 semi semi ADJ iajs-2288 263 23 - - ADJ iajs-2288 263 24 prime prime ADJ iajs-2288 263 25 submodule submodule NOUN iajs-2288 263 26 of of ADP iajs-2288 263 27 𝑇 𝑇 PROPN iajs-2288 263 28 if if SCONJ iajs-2288 263 29 and and CCONJ iajs-2288 263 30 only only ADV iajs-2288 263 31 if if SCONJ iajs-2288 263 32 [ [ X iajs-2288 263 33 𝐿 𝐿 PROPN iajs-2288 263 34 + + NOUN iajs-2288 263 35 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 263 36 ): ): PUNCT iajs-2288 263 37 𝑇 𝑇 PROPN iajs-2288 263 38 ] ] PUNCT iajs-2288 263 39 is be AUX iajs-2288 263 40 a a DET iajs-2288 263 41 semi semi ADJ iajs-2288 263 42 - - ADJ iajs-2288 263 43 prime prime ADJ iajs-2288 263 44 ideal ideal NOUN iajs-2288 263 45 of of ADP iajs-2288 263 46 𝑅 𝑅 PROPN iajs-2288 263 47 ( ( PUNCT iajs-2288 263 48 hence hence ADV iajs-2288 263 49 an an DET iajs-2288 263 50 app app ADJ iajs-2288 263 51 - - PUNCT iajs-2288 263 52 semiprime semiprime NOUN iajs-2288 263 53 ) ) PUNCT iajs-2288 263 54 . . PUNCT iajs-2288 264 1 2 2 X iajs-2288 264 2 ) ) PUNCT iajs-2288 264 3 let let VERB iajs-2288 264 4 𝐿 𝐿 PROPN iajs-2288 264 5 be be AUX iajs-2288 264 6 a a DET iajs-2288 264 7 proper proper ADJ iajs-2288 264 8 submodule submodule NOUN iajs-2288 264 9 of of ADP iajs-2288 264 10 an an DET iajs-2288 264 11 𝑅-module 𝑅-module PROPN iajs-2288 264 12 𝑇 𝑇 PROPN iajs-2288 264 13 with with ADP iajs-2288 264 14 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 264 15 ) ) PUNCT iajs-2288 264 16 βŠ† βŠ† NUM iajs-2288 264 17 𝐿. 𝐿. VERB iajs-2288 264 18 then then ADV iajs-2288 264 19 𝐿 𝐿 PROPN iajs-2288 264 20 is be AUX iajs-2288 264 21 an an DET iajs-2288 264 22 app app ADJ iajs-2288 264 23 - - PUNCT iajs-2288 264 24 semiprime semiprime NOUN iajs-2288 264 25 submodule submodule NOUN iajs-2288 264 26 of of ADP iajs-2288 264 27 𝑇 𝑇 PROPN iajs-2288 264 28 if if SCONJ iajs-2288 264 29 and and CCONJ iajs-2288 264 30 only only ADV iajs-2288 264 31 if if SCONJ iajs-2288 264 32 [ [ X iajs-2288 264 33 𝐿:𝑇 𝐿:𝑇 X iajs-2288 264 34 π‘Ž π‘Ž PART iajs-2288 264 35 𝑛 𝑛 NOUN iajs-2288 264 36 ] ] PUNCT iajs-2288 264 37 = = SYM iajs-2288 265 1 [ [ X iajs-2288 265 2 𝐿:𝑇 𝐿:𝑇 X iajs-2288 265 3 π‘Ž π‘Ž X iajs-2288 265 4 ] ] X iajs-2288 265 5 for for ADP iajs-2288 265 6 π‘Ž π‘Ž PROPN iajs-2288 265 7 ∈ ∈ PROPN iajs-2288 265 8 𝑅 𝑅 PROPN iajs-2288 265 9 and and CCONJ iajs-2288 265 10 some some DET iajs-2288 265 11 𝑛 𝑛 DET iajs-2288 265 12 ∈ ∈ PROPN iajs-2288 265 13 𝑍+ 𝑍+ NOUN iajs-2288 265 14 . . NOUN iajs-2288 265 15 3 3 X iajs-2288 265 16 ) ) PUNCT iajs-2288 265 17 let let VERB iajs-2288 265 18 𝐿 𝐿 PROPN iajs-2288 265 19 and and CCONJ iajs-2288 265 20 𝐸 𝐸 PROPN iajs-2288 265 21 are be AUX iajs-2288 265 22 two two NUM iajs-2288 265 23 app app ADJ iajs-2288 265 24 - - PUNCT iajs-2288 265 25 semi semi ADJ iajs-2288 265 26 - - ADJ iajs-2288 265 27 prime prime ADJ iajs-2288 265 28 submodules submodule NOUN iajs-2288 265 29 of of ADP iajs-2288 265 30 an an DET iajs-2288 265 31 𝑅-module 𝑅-module PROPN iajs-2288 265 32 𝑇 𝑇 PROPN iajs-2288 265 33 with with ADP iajs-2288 265 34 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 265 35 ) ) PUNCT iajs-2288 265 36 βŠ† βŠ† NUM iajs-2288 265 37 𝐿 𝐿 PROPN iajs-2288 265 38 or or CCONJ iajs-2288 265 39 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 265 40 ) ) PUNCT iajs-2288 265 41 βŠ† βŠ† NUM iajs-2288 265 42 𝐸. 𝐸. PROPN iajs-2288 265 43 then then ADV iajs-2288 265 44 𝐿 𝐿 PROPN iajs-2288 265 45 ∩ ∩ NOUN iajs-2288 265 46 𝐸 𝐸 PROPN iajs-2288 265 47 is be AUX iajs-2288 265 48 an an DET iajs-2288 265 49 app app ADJ iajs-2288 265 50 - - PUNCT iajs-2288 265 51 semi semi ADJ iajs-2288 265 52 - - ADJ iajs-2288 265 53 prime prime ADJ iajs-2288 265 54 submodule submodule NOUN iajs-2288 265 55 of of ADP iajs-2288 265 56 𝑇. 𝑇. PROPN iajs-2288 265 57 4 4 NUM iajs-2288 265 58 ) ) PUNCT iajs-2288 265 59 let let VERB iajs-2288 265 60 𝐿 𝐿 PROPN iajs-2288 265 61 be be AUX iajs-2288 265 62 a a DET iajs-2288 265 63 proper proper ADJ iajs-2288 265 64 submodule submodule NOUN iajs-2288 265 65 of of ADP iajs-2288 265 66 a a DET iajs-2288 265 67 multiplication multiplication NOUN iajs-2288 265 68 𝑅-module 𝑅-module PROPN iajs-2288 265 69 𝑇. 𝑇. PROPN iajs-2288 265 70 then then ADV iajs-2288 265 71 𝐿 𝐿 PROPN iajs-2288 265 72 is be AUX iajs-2288 265 73 an an DET iajs-2288 265 74 app app ADJ iajs-2288 265 75 - - PUNCT iajs-2288 265 76 semi semi ADJ iajs-2288 265 77 - - ADJ iajs-2288 265 78 prime prime ADJ iajs-2288 265 79 submodule submodule NOUN iajs-2288 265 80 of of ADP iajs-2288 265 81 𝑇 𝑇 PROPN iajs-2288 266 1 if if SCONJ iajs-2288 266 2 and and CCONJ iajs-2288 266 3 only only ADV iajs-2288 266 4 if if SCONJ iajs-2288 266 5 𝐾𝑛𝐹 𝐾𝑛𝐹 PROPN iajs-2288 266 6 βŠ† βŠ† NUM iajs-2288 266 7 𝐿 𝐿 PROPN iajs-2288 266 8 implies imply VERB iajs-2288 266 9 that that SCONJ iajs-2288 266 10 𝐾𝐹 𝐾𝐹 PROPN iajs-2288 266 11 βŠ† βŠ† NUM iajs-2288 266 12 𝐿 𝐿 PROPN iajs-2288 266 13 + + NOUN iajs-2288 266 14 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 266 15 ) ) PUNCT iajs-2288 266 16 , , PUNCT iajs-2288 266 17 where where SCONJ iajs-2288 266 18 𝐾 𝐾 PROPN iajs-2288 266 19 , , PUNCT iajs-2288 266 20 𝐹 𝐹 PROPN iajs-2288 266 21 are be AUX iajs-2288 266 22 submodules submodule NOUN iajs-2288 266 23 of of ADP iajs-2288 266 24 𝑇 𝑇 PROPN iajs-2288 266 25 , , PUNCT iajs-2288 266 26 𝑛 𝑛 PRON iajs-2288 266 27 ∈ ∈ PROPN iajs-2288 266 28 𝑍+ 𝑍+ NOUN iajs-2288 266 29 . . NOUN iajs-2288 266 30 5 5 NUM iajs-2288 266 31 ) ) PUNCT iajs-2288 266 32 let let VERB iajs-2288 266 33 𝐿 𝐿 PROPN iajs-2288 266 34 be be AUX iajs-2288 266 35 a a DET iajs-2288 266 36 proper proper ADJ iajs-2288 266 37 submodule submodule NOUN iajs-2288 266 38 of of ADP iajs-2288 266 39 a a DET iajs-2288 266 40 multiplication multiplication NOUN iajs-2288 266 41 𝑅-module 𝑅-module PROPN iajs-2288 266 42 𝑇. 𝑇. PROPN iajs-2288 266 43 then then ADV iajs-2288 266 44 the the DET iajs-2288 266 45 following follow VERB iajs-2288 266 46 statements statement NOUN iajs-2288 266 47 are be AUX iajs-2288 266 48 equivalent equivalent ADJ iajs-2288 266 49 : : PUNCT iajs-2288 266 50 1 1 X iajs-2288 266 51 ) ) PUNCT iajs-2288 266 52 𝐿 𝐿 PROPN iajs-2288 266 53 is be AUX iajs-2288 266 54 an an DET iajs-2288 266 55 app app ADJ iajs-2288 266 56 - - PUNCT iajs-2288 266 57 semi semi ADJ iajs-2288 266 58 - - ADJ iajs-2288 266 59 prime prime ADJ iajs-2288 266 60 submodule submodule NOUN iajs-2288 266 61 of of ADP iajs-2288 266 62 𝑇. 𝑇. PROPN iajs-2288 266 63 2 2 NUM iajs-2288 266 64 ) ) PUNCT iajs-2288 266 65 𝑑𝑛 𝑑𝑛 VERB iajs-2288 266 66 ∈ ∈ PROPN iajs-2288 266 67 𝐿 𝐿 PROPN iajs-2288 266 68 implies imply VERB iajs-2288 266 69 that that SCONJ iajs-2288 266 70 𝑑 𝑑 PROPN iajs-2288 266 71 ∈ ∈ PROPN iajs-2288 266 72 𝐿 𝐿 PROPN iajs-2288 266 73 + + NOUN iajs-2288 266 74 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 266 75 ) ) PUNCT iajs-2288 266 76 for for ADP iajs-2288 266 77 every every DET iajs-2288 266 78 𝑑 𝑑 PROPN iajs-2288 266 79 ∈ ∈ PROPN iajs-2288 266 80 𝑇. 𝑇. PROPN iajs-2288 266 81 3 3 NUM iajs-2288 266 82 ) ) PUNCT iajs-2288 266 83 √𝐿 √𝐿 NOUN iajs-2288 266 84 βŠ† βŠ† NUM iajs-2288 266 85 𝐿 𝐿 PROPN iajs-2288 266 86 + + NOUN iajs-2288 266 87 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 266 88 ) ) PUNCT iajs-2288 266 89 . . PUNCT iajs-2288 267 1 4 4 X iajs-2288 267 2 ) ) PUNCT iajs-2288 267 3 𝐹1𝐹2 𝐹1𝐹2 NOUN iajs-2288 267 4 … … SYM iajs-2288 267 5 … … PUNCT iajs-2288 267 6 𝐹𝑗 𝐹𝑗 PROPN iajs-2288 267 7 βŠ† βŠ† NUM iajs-2288 267 8 𝐿 𝐿 PROPN iajs-2288 267 9 , , PUNCT iajs-2288 267 10 implies imply VERB iajs-2288 267 11 that that SCONJ iajs-2288 267 12 𝐹1 𝐹1 PROPN iajs-2288 267 13 ∩ ∩ PROPN iajs-2288 267 14 𝐹2 𝐹2 PROPN iajs-2288 267 15 ∩ ∩ NOUN iajs-2288 267 16 … … PUNCT iajs-2288 267 17 … … SYM iajs-2288 267 18 ∩ ∩ X iajs-2288 267 19 𝐹𝑗 𝐹𝑗 PROPN iajs-2288 267 20 βŠ† βŠ† NUM iajs-2288 267 21 𝐿 𝐿 PROPN iajs-2288 267 22 + + NOUN iajs-2288 267 23 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 267 24 ) ) PUNCT iajs-2288 267 25 for for ADP iajs-2288 267 26 every every DET iajs-2288 267 27 submodules submodule NOUN iajs-2288 267 28 𝐹1 𝐹1 NOUN iajs-2288 267 29 , , PUNCT iajs-2288 267 30 𝐹2 𝐹2 NOUN iajs-2288 267 31 , , PUNCT iajs-2288 267 32 … … PUNCT iajs-2288 267 33 … … PUNCT iajs-2288 267 34 , , PUNCT iajs-2288 267 35 𝐹𝑗 𝐹𝑗 PROPN iajs-2288 267 36 of of ADP iajs-2288 267 37 𝑇 𝑇 PROPN iajs-2288 267 38 and and CCONJ iajs-2288 267 39 𝑗 𝑗 PRON iajs-2288 267 40 ∈ ∈ PROPN iajs-2288 267 41 𝑍+ 𝑍+ NOUN iajs-2288 267 42 . . NOUN iajs-2288 267 43 6 6 NUM iajs-2288 267 44 ) ) PUNCT iajs-2288 267 45 let let VERB iajs-2288 267 46 𝑇 𝑇 PROPN iajs-2288 267 47 be be AUX iajs-2288 267 48 a a DET iajs-2288 267 49 faithful faithful ADJ iajs-2288 267 50 multiplication multiplication NOUN iajs-2288 267 51 𝑅-module 𝑅-module PROPN iajs-2288 267 52 and and CCONJ iajs-2288 267 53 𝐿 𝐿 PROPN iajs-2288 267 54 be be AUX iajs-2288 267 55 a a DET iajs-2288 267 56 proper proper ADJ iajs-2288 267 57 submodule submodule NOUN iajs-2288 267 58 of of ADP iajs-2288 267 59 𝑇. 𝑇. PROPN iajs-2288 267 60 then then ADV iajs-2288 267 61 𝐿 𝐿 PROPN iajs-2288 267 62 is be AUX iajs-2288 267 63 an an DET iajs-2288 267 64 app app ADJ iajs-2288 267 65 - - PUNCT iajs-2288 267 66 semi semi ADJ iajs-2288 267 67 - - ADJ iajs-2288 267 68 prime prime ADJ iajs-2288 267 69 submodule submodule NOUN iajs-2288 267 70 of of ADP iajs-2288 267 71 𝑇 𝑇 PROPN iajs-2288 268 1 if if SCONJ iajs-2288 268 2 and and CCONJ iajs-2288 268 3 only only ADV iajs-2288 268 4 if if SCONJ iajs-2288 268 5 [ [ X iajs-2288 268 6 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 268 7 𝑇 𝑇 PROPN iajs-2288 268 8 ] ] PUNCT iajs-2288 268 9 is be AUX iajs-2288 268 10 an an DET iajs-2288 268 11 app app ADJ iajs-2288 268 12 - - PUNCT iajs-2288 268 13 semi semi ADJ iajs-2288 268 14 - - ADJ iajs-2288 268 15 prime prime ADJ iajs-2288 268 16 ideal ideal NOUN iajs-2288 268 17 of of ADP iajs-2288 268 18 𝑅. 𝑅. NOUN iajs-2288 268 19 7 7 NUM iajs-2288 268 20 ) ) PUNCT iajs-2288 268 21 let let VERB iajs-2288 268 22 𝑇 𝑇 PROPN iajs-2288 268 23 be be AUX iajs-2288 268 24 a a DET iajs-2288 268 25 faithful faithful ADJ iajs-2288 268 26 finitely finitely ADV iajs-2288 268 27 generated generate VERB iajs-2288 268 28 multiplication multiplication NOUN iajs-2288 268 29 𝑅-module 𝑅-module PROPN iajs-2288 268 30 and and CCONJ iajs-2288 268 31 𝐽 𝐽 PROPN iajs-2288 268 32 be be AUX iajs-2288 268 33 an an DET iajs-2288 268 34 app app ADJ iajs-2288 268 35 - - PUNCT iajs-2288 268 36 semi semi ADJ iajs-2288 268 37 - - ADJ iajs-2288 268 38 prime prime ADJ iajs-2288 268 39 ideal ideal NOUN iajs-2288 268 40 of of ADP iajs-2288 268 41 𝑅. 𝑅. NOUN iajs-2288 268 42 then then ADV iajs-2288 268 43 𝐽𝑇 𝐽𝑇 PROPN iajs-2288 268 44 is be AUX iajs-2288 268 45 an an DET iajs-2288 268 46 app app ADJ iajs-2288 268 47 - - PUNCT iajs-2288 268 48 semi semi ADJ iajs-2288 268 49 - - ADJ iajs-2288 268 50 prime prime ADJ iajs-2288 268 51 submodule submodule NOUN iajs-2288 268 52 of of ADP iajs-2288 268 53 𝑇. 𝑇. PROPN iajs-2288 268 54 8) 8) NUM iajs-2288 268 55 let let VERB iajs-2288 268 56 𝑇 𝑇 PROPN iajs-2288 268 57 be be AUX iajs-2288 268 58 a a DET iajs-2288 268 59 faithful faithful ADJ iajs-2288 268 60 finitely finitely ADV iajs-2288 268 61 generated generate VERB iajs-2288 268 62 multiplication multiplication NOUN iajs-2288 268 63 𝑅-module 𝑅-module PROPN iajs-2288 268 64 and and CCONJ iajs-2288 268 65 𝐿 𝐿 PROPN iajs-2288 268 66 be be AUX iajs-2288 268 67 a a DET iajs-2288 268 68 proper proper ADJ iajs-2288 268 69 submodule submodule NOUN iajs-2288 268 70 of of ADP iajs-2288 268 71 𝑇. 𝑇. PROPN iajs-2288 268 72 then then ADV iajs-2288 268 73 the the DET iajs-2288 268 74 following follow VERB iajs-2288 268 75 statements statement NOUN iajs-2288 268 76 are be AUX iajs-2288 268 77 equivalent equivalent ADJ iajs-2288 268 78 . . PUNCT iajs-2288 269 1 1 1 X iajs-2288 269 2 ) ) PUNCT iajs-2288 269 3 𝐿 𝐿 PROPN iajs-2288 269 4 is be AUX iajs-2288 269 5 an an DET iajs-2288 269 6 app app ADJ iajs-2288 269 7 - - PUNCT iajs-2288 269 8 semi semi ADJ iajs-2288 269 9 - - ADJ iajs-2288 269 10 prime prime ADJ iajs-2288 269 11 submodule submodule NOUN iajs-2288 269 12 of of ADP iajs-2288 269 13 𝑇. 𝑇. PROPN iajs-2288 269 14 2 2 NUM iajs-2288 269 15 ) ) PUNCT iajs-2288 270 1 [ [ X iajs-2288 270 2 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 270 3 𝑇 𝑇 PROPN iajs-2288 270 4 ] ] PUNCT iajs-2288 270 5 is be AUX iajs-2288 270 6 an an DET iajs-2288 270 7 app app ADJ iajs-2288 270 8 - - PUNCT iajs-2288 270 9 semi semi ADJ iajs-2288 270 10 - - ADJ iajs-2288 270 11 prime prime ADJ iajs-2288 270 12 ideal ideal NOUN iajs-2288 270 13 of of ADP iajs-2288 270 14 𝑅. 𝑅. NOUN iajs-2288 270 15 3 3 NUM iajs-2288 270 16 ) ) PUNCT iajs-2288 270 17 𝐿 𝐿 PROPN iajs-2288 270 18 = = SYM iajs-2288 270 19 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 270 20 for for ADP iajs-2288 270 21 some some DET iajs-2288 270 22 app app ADJ iajs-2288 270 23 - - PUNCT iajs-2288 270 24 semi semi ADJ iajs-2288 270 25 - - ADJ iajs-2288 270 26 prime prime ADJ iajs-2288 270 27 ideal ideal NOUN iajs-2288 270 28 𝐼 𝐼 ADP iajs-2288 270 29 of of ADP iajs-2288 270 30 𝑅. 𝑅. NOUN iajs-2288 270 31 9 9 NUM iajs-2288 270 32 ) ) PUNCT iajs-2288 270 33 let let VERB iajs-2288 270 34 𝑇 𝑇 PROPN iajs-2288 270 35 be be AUX iajs-2288 270 36 non non ADJ iajs-2288 270 37 - - ADJ iajs-2288 270 38 singular singular ADJ iajs-2288 270 39 finitely finitely ADV iajs-2288 270 40 generated generate VERB iajs-2288 270 41 multiplication multiplication NOUN iajs-2288 270 42 𝑅-module 𝑅-module PROPN iajs-2288 270 43 and and CCONJ iajs-2288 270 44 𝐿 𝐿 PROPN iajs-2288 270 45 be be AUX iajs-2288 270 46 a a DET iajs-2288 270 47 proper proper ADJ iajs-2288 270 48 submodule submodule NOUN iajs-2288 270 49 of of ADP iajs-2288 270 50 𝑇. 𝑇. PROPN iajs-2288 270 51 then then ADV iajs-2288 270 52 the the DET iajs-2288 270 53 following follow VERB iajs-2288 270 54 statements statement NOUN iajs-2288 270 55 are be AUX iajs-2288 270 56 equivalent equivalent ADJ iajs-2288 270 57 . . PUNCT iajs-2288 271 1 1 1 X iajs-2288 271 2 ) ) PUNCT iajs-2288 271 3 𝐿 𝐿 PROPN iajs-2288 271 4 is be AUX iajs-2288 271 5 an an DET iajs-2288 271 6 app app ADJ iajs-2288 271 7 - - PUNCT iajs-2288 271 8 semi semi ADJ iajs-2288 271 9 - - ADJ iajs-2288 271 10 prime prime ADJ iajs-2288 271 11 submodule submodule NOUN iajs-2288 271 12 of of ADP iajs-2288 271 13 𝑇. 𝑇. PROPN iajs-2288 271 14 2 2 NUM iajs-2288 271 15 ) ) PUNCT iajs-2288 272 1 [ [ X iajs-2288 272 2 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2288 272 3 𝑇 𝑇 PROPN iajs-2288 272 4 ] ] PUNCT iajs-2288 272 5 is be AUX iajs-2288 272 6 an an DET iajs-2288 272 7 app app ADJ iajs-2288 272 8 - - PUNCT iajs-2288 272 9 semi semi ADJ iajs-2288 272 10 - - ADJ iajs-2288 272 11 prime prime ADJ iajs-2288 272 12 ideal ideal NOUN iajs-2288 272 13 of of ADP iajs-2288 272 14 𝑅. 𝑅. NOUN iajs-2288 272 15 3 3 NUM iajs-2288 272 16 ) ) PUNCT iajs-2288 272 17 𝐿 𝐿 PROPN iajs-2288 272 18 = = SYM iajs-2288 272 19 𝐼𝑇 𝐼𝑇 PROPN iajs-2288 272 20 for for ADP iajs-2288 272 21 some some DET iajs-2288 272 22 app app ADJ iajs-2288 272 23 - - PUNCT iajs-2288 272 24 semi semi ADJ iajs-2288 272 25 - - ADJ iajs-2288 272 26 prime prime ADJ iajs-2288 272 27 ideal ideal NOUN iajs-2288 272 28 𝐼 𝐼 PROPN iajs-2288 272 29 of of ADP iajs-2288 272 30 𝑅 𝑅 PROPN iajs-2288 272 31 with with ADP iajs-2288 272 32 π‘Žπ‘›π‘›(𝑇 π‘Žπ‘›π‘›(𝑇 NOUN iajs-2288 272 33 ) ) PUNCT iajs-2288 272 34 βŠ† βŠ† NUM iajs-2288 272 35 𝐼. 𝐼. PROPN iajs-2288 272 36 10 10 NUM iajs-2288 272 37 ) ) PUNCT iajs-2288 272 38 let let VERB iajs-2288 272 39 𝑇 𝑇 PROPN iajs-2288 272 40 = = SYM iajs-2288 272 41 𝑇1βŠ•π‘‡2 𝑇1βŠ•π‘‡2 PROPN iajs-2288 272 42 be be AUX iajs-2288 272 43 an an DET iajs-2288 272 44 𝑅-module 𝑅-module PROPN iajs-2288 272 45 , , PUNCT iajs-2288 272 46 where where SCONJ iajs-2288 272 47 𝑇1 𝑇1 NOUN iajs-2288 272 48 and and CCONJ iajs-2288 272 49 𝑇2 𝑇2 NOUN iajs-2288 272 50 𝑅-modules 𝑅-modules PROPN iajs-2288 272 51 , , PUNCT iajs-2288 272 52 and and CCONJ iajs-2288 272 53 𝐿 𝐿 PROPN iajs-2288 272 54 = = PUNCT iajs-2288 272 55 𝐿1 𝐿1 X iajs-2288 272 56 βŠ•πΏ2 βŠ•πΏ2 PRON iajs-2288 272 57 be be VERB iajs-2288 272 58 a a DET iajs-2288 272 59 submodule submodule NOUN iajs-2288 272 60 of of ADP iajs-2288 272 61 𝑇 𝑇 PROPN iajs-2288 272 62 , , PUNCT iajs-2288 272 63 where where SCONJ iajs-2288 272 64 𝐿1 𝐿1 PROPN iajs-2288 272 65 is be AUX iajs-2288 272 66 a a DET iajs-2288 272 67 submodule submodule NOUN iajs-2288 272 68 of of ADP iajs-2288 272 69 𝑇1 𝑇1 NOUN iajs-2288 272 70 and and CCONJ iajs-2288 272 71 𝐿2 𝐿2 NOUN iajs-2288 272 72 is be AUX iajs-2288 272 73 a a DET iajs-2288 272 74 submodule submodule NOUN iajs-2288 272 75 of of ADP iajs-2288 272 76 𝑇2 𝑇2 NOUN iajs-2288 272 77 with with ADP iajs-2288 272 78 𝐿 𝐿 PROPN iajs-2288 272 79 βŠ† βŠ† NUM iajs-2288 272 80 π‘ π‘œπ‘(𝑇 π‘ π‘œπ‘(𝑇 NUM iajs-2288 272 81 ) ) PUNCT iajs-2288 272 82 = = SYM iajs-2288 272 83 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 272 84 ) ) PUNCT iajs-2288 272 85 βŠ• βŠ• PROPN iajs-2288 272 86 π‘ π‘œπ‘(𝑇2 π‘ π‘œπ‘(𝑇2 PROPN iajs-2288 272 87 ) ) PUNCT iajs-2288 272 88 . . PUNCT iajs-2288 273 1 if if SCONJ iajs-2288 273 2 𝐿 𝐿 PROPN iajs-2288 273 3 is be AUX iajs-2288 273 4 an an DET iajs-2288 273 5 app app ADJ iajs-2288 273 6 - - PUNCT iajs-2288 273 7 semi semi ADJ iajs-2288 273 8 - - ADJ iajs-2288 273 9 prime prime ADJ iajs-2288 273 10 submodule submodule NOUN iajs-2288 273 11 of of ADP iajs-2288 273 12 𝑇 𝑇 PROPN iajs-2288 273 13 , , PUNCT iajs-2288 273 14 then then ADV iajs-2288 273 15 𝐿1 𝐿1 PROPN iajs-2288 273 16 is be AUX iajs-2288 273 17 an an DET iajs-2288 273 18 appsemi appsemi ADJ iajs-2288 273 19 - - ADJ iajs-2288 273 20 prime prime ADJ iajs-2288 273 21 submodule submodule NOUN iajs-2288 273 22 of of ADP iajs-2288 273 23 𝑇1 𝑇1 NOUN iajs-2288 273 24 and and CCONJ iajs-2288 273 25 𝐿2 𝐿2 NOUN iajs-2288 273 26 is be AUX iajs-2288 273 27 an an DET iajs-2288 273 28 app app ADJ iajs-2288 273 29 - - PUNCT iajs-2288 273 30 semi semi ADJ iajs-2288 273 31 - - ADJ iajs-2288 273 32 prime prime ADJ iajs-2288 273 33 submodule submodule NOUN iajs-2288 273 34 of of ADP iajs-2288 273 35 𝑇2 𝑇2 PROPN iajs-2288 273 36 . . PROPN iajs-2288 274 1 11 11 NUM iajs-2288 274 2 ) ) PUNCT iajs-2288 274 3 let let VERB iajs-2288 274 4 𝑇 𝑇 PROPN iajs-2288 274 5 = = SYM iajs-2288 274 6 𝑇1βŠ•π‘‡2 𝑇1βŠ•π‘‡2 PROPN iajs-2288 274 7 be be AUX iajs-2288 274 8 an an DET iajs-2288 274 9 𝑅-module 𝑅-module PROPN iajs-2288 274 10 , , PUNCT iajs-2288 274 11 where where SCONJ iajs-2288 274 12 each each PRON iajs-2288 274 13 of of ADP iajs-2288 274 14 𝑇1 𝑇1 NOUN iajs-2288 274 15 and and CCONJ iajs-2288 274 16 𝑇2 𝑇2 NOUN iajs-2288 274 17 𝑅-module 𝑅-module PROPN iajs-2288 274 18 . . PUNCT iajs-2288 275 1 then then ADV iajs-2288 275 2 the the DET iajs-2288 275 3 following follow VERB iajs-2288 275 4 statements statement NOUN iajs-2288 275 5 are be AUX iajs-2288 275 6 satisfy satisfy ADJ iajs-2288 275 7 : : PUNCT iajs-2288 275 8 1 1 X iajs-2288 275 9 ) ) PUNCT iajs-2288 275 10 𝐿1 𝐿1 PROPN iajs-2288 275 11 is be AUX iajs-2288 275 12 an an DET iajs-2288 275 13 app app ADJ iajs-2288 275 14 - - PUNCT iajs-2288 275 15 semi semi ADJ iajs-2288 275 16 - - ADJ iajs-2288 275 17 prime prime ADJ iajs-2288 275 18 submodule submodule NOUN iajs-2288 275 19 of of ADP iajs-2288 275 20 𝑇1 𝑇1 NOUN iajs-2288 275 21 such such ADJ iajs-2288 275 22 that that SCONJ iajs-2288 275 23 𝐿1 𝐿1 VERB iajs-2288 275 24 βŠ† βŠ† NUM iajs-2288 275 25 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 ADJ iajs-2288 275 26 ) ) PUNCT iajs-2288 275 27 and and CCONJ iajs-2288 275 28 𝑇2 𝑇2 PROPN iajs-2288 275 29 = = SYM iajs-2288 275 30 π‘ π‘œπ‘(𝑇2 π‘ π‘œπ‘(𝑇2 PROPN iajs-2288 275 31 ) ) PUNCT iajs-2288 275 32 if if SCONJ iajs-2288 275 33 and and CCONJ iajs-2288 275 34 only only ADV iajs-2288 275 35 if if SCONJ iajs-2288 275 36 𝐿1 𝐿1 PROPN iajs-2288 275 37 βŠ•π‘‡2 βŠ•π‘‡2 NOUN iajs-2288 275 38 is be AUX iajs-2288 275 39 an an DET iajs-2288 275 40 app app ADJ iajs-2288 275 41 - - PUNCT iajs-2288 275 42 semi semi ADJ iajs-2288 275 43 - - ADJ iajs-2288 275 44 prime prime ADJ iajs-2288 275 45 submodule submodule NOUN iajs-2288 275 46 of of ADP iajs-2288 275 47 𝑇. 𝑇. PROPN iajs-2288 275 48 128 128 NUM iajs-2288 275 49 ibn ibn PROPN iajs-2288 275 50 al al PROPN iajs-2288 275 51 - - PUNCT iajs-2288 275 52 haitham haitham PROPN iajs-2288 275 53 jour jour X iajs-2288 275 54 . . PROPN iajs-2288 276 1 for for ADP iajs-2288 276 2 pure pure ADJ iajs-2288 276 3 & & CCONJ iajs-2288 276 4 appl appl PROPN iajs-2288 276 5 . . PUNCT iajs-2288 277 1 sci sci PROPN iajs-2288 277 2 . . PROPN iajs-2288 277 3 32 32 NUM iajs-2288 277 4 ( ( PUNCT iajs-2288 277 5 3 3 NUM iajs-2288 277 6 ) ) PUNCT iajs-2288 277 7 2019 2019 NUM iajs-2288 277 8 2 2 NUM iajs-2288 277 9 ) ) PUNCT iajs-2288 277 10 𝐿2 𝐿2 NOUN iajs-2288 277 11 is be AUX iajs-2288 277 12 an an DET iajs-2288 277 13 app app ADJ iajs-2288 277 14 - - PUNCT iajs-2288 277 15 semi semi ADJ iajs-2288 277 16 - - ADJ iajs-2288 277 17 prime prime ADJ iajs-2288 277 18 submodule submodule NOUN iajs-2288 277 19 of of ADP iajs-2288 277 20 𝑇2 𝑇2 NOUN iajs-2288 277 21 such such ADJ iajs-2288 277 22 that that SCONJ iajs-2288 277 23 𝐿2 𝐿2 NOUN iajs-2288 277 24 βŠ† βŠ† NUM iajs-2288 277 25 π‘ π‘œπ‘(𝑇2 π‘ π‘œπ‘(𝑇2 PROPN iajs-2288 277 26 ) ) PUNCT iajs-2288 277 27 and and CCONJ iajs-2288 277 28 𝑇1 𝑇1 NOUN iajs-2288 277 29 = = SYM iajs-2288 277 30 π‘ π‘œπ‘(𝑇1 π‘ π‘œπ‘(𝑇1 PROPN iajs-2288 277 31 ) ) PUNCT iajs-2288 277 32 if if SCONJ iajs-2288 277 33 and and CCONJ iajs-2288 277 34 only only ADV iajs-2288 277 35 if if SCONJ iajs-2288 277 36 𝑇1 𝑇1 NOUN iajs-2288 277 37 βŠ•πΏ2 βŠ•πΏ2 NOUN iajs-2288 277 38 is be AUX iajs-2288 277 39 an an DET iajs-2288 277 40 app app ADJ iajs-2288 277 41 - - PUNCT iajs-2288 277 42 semi semi ADJ iajs-2288 277 43 - - ADJ iajs-2288 277 44 prime prime ADJ iajs-2288 277 45 submodule submodule NOUN iajs-2288 277 46 of of ADP iajs-2288 277 47 𝑇. 𝑇. PROPN iajs-2288 277 48 12 12 NUM iajs-2288 277 49 ) ) PUNCT iajs-2288 277 50 let let VERB iajs-2288 277 51 𝑓 𝑓 PRON iajs-2288 277 52 : : PUNCT iajs-2288 277 53 𝑇 𝑇 PROPN iajs-2288 277 54 β†’ β†’ SYM iajs-2288 277 55 𝑇′ 𝑇′ NOUN iajs-2288 277 56 be be AUX iajs-2288 277 57 an an DET iajs-2288 277 58 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2288 277 59 and and CCONJ iajs-2288 277 60 𝐿 𝐿 PROPN iajs-2288 277 61 is be AUX iajs-2288 277 62 an an DET iajs-2288 277 63 app app ADJ iajs-2288 277 64 - - PUNCT iajs-2288 277 65 semi semi ADJ iajs-2288 277 66 - - ADJ iajs-2288 277 67 prime prime ADJ iajs-2288 277 68 submodule submodule NOUN iajs-2288 277 69 of of ADP iajs-2288 277 70 𝑇′. 𝑇′. NOUN iajs-2288 277 71 then then ADV iajs-2288 277 72 π‘“βˆ’1(𝐿 π‘“βˆ’1(𝐿 NOUN iajs-2288 277 73 ) ) PUNCT iajs-2288 277 74 is be AUX iajs-2288 277 75 an an DET iajs-2288 277 76 app app ADJ iajs-2288 277 77 - - PUNCT iajs-2288 277 78 semi semi ADJ iajs-2288 277 79 - - ADJ iajs-2288 277 80 prime prime ADJ iajs-2288 277 81 submodule submodule NOUN iajs-2288 277 82 of of ADP iajs-2288 277 83 𝑇. 𝑇. PROPN iajs-2288 277 84 13 13 NUM iajs-2288 277 85 ) ) PUNCT iajs-2288 277 86 let let VERB iajs-2288 277 87 𝑓 𝑓 PRON iajs-2288 277 88 : : PUNCT iajs-2288 277 89 𝑇 𝑇 PROPN iajs-2288 277 90 β†’ β†’ SYM iajs-2288 277 91 𝑇′ 𝑇′ NOUN iajs-2288 277 92 be be AUX iajs-2288 277 93 an an DET iajs-2288 277 94 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2288 277 95 and and CCONJ iajs-2288 277 96 𝐾 𝐾 PROPN iajs-2288 277 97 be be VERB iajs-2288 277 98 an an DET iajs-2288 277 99 app app ADJ iajs-2288 277 100 - - PUNCT iajs-2288 277 101 semi semi ADJ iajs-2288 277 102 - - ADJ iajs-2288 277 103 prime prime ADJ iajs-2288 277 104 submodule submodule NOUN iajs-2288 277 105 of of ADP iajs-2288 277 106 𝑇 𝑇 PROPN iajs-2288 277 107 with with ADP iajs-2288 277 108 πΎπ‘’π‘Ÿ πΎπ‘’π‘Ÿ PROPN iajs-2288 277 109 𝑓 𝑓 PRON iajs-2288 277 110 βŠ† βŠ† NUM iajs-2288 277 111 𝐾. 𝐾. PROPN iajs-2288 277 112 then then ADV iajs-2288 277 113 𝑓(𝐾 𝑓(𝐾 VERB iajs-2288 277 114 ) ) PUNCT iajs-2288 277 115 is be AUX iajs-2288 277 116 an an DET iajs-2288 277 117 app app ADJ iajs-2288 277 118 - - PUNCT iajs-2288 277 119 quasi quasi ADJ iajs-2288 277 120 - - ADJ iajs-2288 277 121 prime prime ADJ iajs-2288 277 122 submodule submodule NOUN iajs-2288 277 123 of of ADP iajs-2288 277 124 𝑇′. 𝑇′. NOUN iajs-2288 277 125 references reference NOUN iajs-2288 277 126 1 1 NUM iajs-2288 277 127 . . X iajs-2288 278 1 lu lu PROPN iajs-2288 278 2 , , PUNCT iajs-2288 278 3 c.p c.p PROPN iajs-2288 278 4 . . PROPN iajs-2288 278 5 prime prime ADJ iajs-2288 278 6 submodules submodule NOUN iajs-2288 278 7 of of ADP iajs-2288 278 8 modules module NOUN iajs-2288 278 9 . . PUNCT iajs-2288 279 1 comm comm NOUN iajs-2288 279 2 . . PUNCT iajs-2288 280 1 math math NOUN iajs-2288 280 2 . . PUNCT iajs-2288 281 1 university university NOUN iajs-2288 281 2 sancti sancti VERB iajs-2288 281 3 pauli.1984 pauli.1984 PROPN iajs-2288 281 4 , , PUNCT iajs-2288 281 5 33 33 NUM iajs-2288 281 6 , , PUNCT iajs-2288 281 7 61 61 NUM iajs-2288 281 8 - - SYM iajs-2288 281 9 69 69 NUM iajs-2288 281 10 . . PUNCT iajs-2288 282 1 2 2 X iajs-2288 282 2 . . X iajs-2288 282 3 haibat haibat PROPN iajs-2288 282 4 , , PUNCT iajs-2288 282 5 k.m k.m PROPN iajs-2288 282 6 . . PROPN iajs-2288 282 7 ; ; PUNCT iajs-2288 282 8 ali ali PROPN iajs-2288 282 9 , , PUNCT iajs-2288 282 10 s.h.a s.h.a PROPN iajs-2288 282 11 . . PUNCT iajs-2288 283 1 approximaitly approximaitly ADV iajs-2288 283 2 prime prime ADJ iajs-2288 283 3 submodules submodule NOUN iajs-2288 283 4 and and CCONJ iajs-2288 283 5 some some DET iajs-2288 283 6 related related ADJ iajs-2288 283 7 concepts concept NOUN iajs-2288 283 8 . . PUNCT iajs-2288 284 1 ibn ibn PROPN iajs-2288 284 2 al al PROPN iajs-2288 284 3 - - PUNCT iajs-2288 284 4 haitham haitham PROPN iajs-2288 284 5 journal journal PROPN iajs-2288 284 6 for for ADP iajs-2288 284 7 pure pure ADJ iajs-2288 284 8 and and CCONJ iajs-2288 284 9 applied apply VERB iajs-2288 284 10 science.2019 science.2019 PROPN iajs-2288 284 11 , , PUNCT iajs-2288 284 12 32 32 NUM iajs-2288 284 13 , , PUNCT iajs-2288 284 14 2 2 NUM iajs-2288 284 15 , , PUNCT iajs-2288 284 16 103 103 NUM iajs-2288 284 17 - - SYM iajs-2288 284 18 113 113 NUM iajs-2288 284 19 . . PUNCT iajs-2288 285 1 3 3 X iajs-2288 285 2 . . X iajs-2288 285 3 haibat haibat PROPN iajs-2288 285 4 , , PUNCT iajs-2288 285 5 k.m k.m PROPN iajs-2288 285 6 . . PROPN iajs-2288 285 7 ; ; PUNCT iajs-2288 285 8 saif saif PROPN iajs-2288 285 9 , , PUNCT iajs-2288 285 10 a.h.we a.h.we NOUN iajs-2288 285 11 - - PUNCT iajs-2288 285 12 prime prime NOUN iajs-2288 285 13 submodules submodule NOUN iajs-2288 285 14 and and CCONJ iajs-2288 285 15 we we PRON iajs-2288 285 16 - - PUNCT iajs-2288 285 17 semi semi ADJ iajs-2288 285 18 - - ADJ iajs-2288 285 19 prime prime ADJ iajs-2288 285 20 submodules submodule NOUN iajs-2288 285 21 . . PUNCT iajs-2288 286 1 ibnal ibnal ADJ iajs-2288 286 2 - - PUNCT iajs-2288 286 3 haitham haitham PROPN iajs-2288 286 4 journal journal NOUN iajs-2288 286 5 for for ADP iajs-2288 286 6 pure pure ADJ iajs-2288 286 7 and and CCONJ iajs-2288 286 8 applied apply VERB iajs-2288 286 9 science.2018 science.2018 PROPN iajs-2288 286 10 , , PUNCT iajs-2288 286 11 31 31 NUM iajs-2288 286 12 , , PUNCT iajs-2288 286 13 3 3 NUM iajs-2288 286 14 , , PUNCT iajs-2288 286 15 109 109 NUM iajs-2288 286 16 - - SYM iajs-2288 286 17 117 117 NUM iajs-2288 286 18 . . NOUN iajs-2288 287 1 4 4 NUM iajs-2288 287 2 . . X iajs-2288 287 3 haibat haibat PROPN iajs-2288 287 4 , , PUNCT iajs-2288 287 5 k.m k.m PROPN iajs-2288 287 6 . . PROPN iajs-2288 287 7 ; ; PUNCT iajs-2288 288 1 wissam wissam PROPN iajs-2288 288 2 , , PUNCT iajs-2288 288 3 a.h a.h PROPN iajs-2288 288 4 . . PROPN iajs-2288 288 5 wn-2 wn-2 NOUN iajs-2288 288 6 - - ADJ iajs-2288 288 7 absorbing absorbing ADJ iajs-2288 288 8 and and CCONJ iajs-2288 288 9 wes2 wes2 ADJ iajs-2288 288 10 - - PUNCT iajs-2288 288 11 absorbing absorb VERB iajs-2288 288 12 submodules submodule NOUN iajs-2288 288 13 . . PUNCT iajs-2288 289 1 ibn ibn PROPN iajs-2288 289 2 al al PROPN iajs-2288 289 3 - - PUNCT iajs-2288 289 4 haitham haitham PROPN iajs-2288 289 5 journal journal PROPN iajs-2288 289 6 for for ADP iajs-2288 289 7 pure pure ADJ iajs-2288 289 8 and and CCONJ iajs-2288 289 9 applied apply VERB iajs-2288 289 10 science.2018 science.2018 PROPN iajs-2288 289 11 , , PUNCT iajs-2288 289 12 31 31 NUM iajs-2288 289 13 , , PUNCT iajs-2288 289 14 3 3 NUM iajs-2288 289 15 , , PUNCT iajs-2288 289 16 118 118 NUM iajs-2288 289 17 - - SYM iajs-2288 289 18 125 125 NUM iajs-2288 289 19 . . PUNCT iajs-2288 290 1 5 5 NUM iajs-2288 290 2 . . 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