id sid tid token lemma pos iajs-2560 1 1 ibn ibn PROPN iajs-2560 1 2 al al PROPN iajs-2560 1 3 - - PUNCT iajs-2560 1 4 haitham haitham PROPN iajs-2560 1 5 jour jour X iajs-2560 1 6 . . PROPN iajs-2560 1 7 for for ADP iajs-2560 1 8 pure pure ADJ iajs-2560 1 9 & & CCONJ iajs-2560 1 10 appl appl PROPN iajs-2560 1 11 . . PUNCT iajs-2560 2 1 sci sci PROPN iajs-2560 2 2 . . PROPN iajs-2560 3 1 34 34 NUM iajs-2560 3 2 ( ( PUNCT iajs-2560 3 3 1 1 NUM iajs-2560 3 4 ) ) PUNCT iajs-2560 3 5 2021 2021 NUM iajs-2560 3 6 116 116 NUM iajs-2560 3 7 nearly nearly ADV iajs-2560 3 8 primary-2 primary-2 NOUN iajs-2560 3 9 - - PUNCT iajs-2560 3 10 absorbing absorbing ADJ iajs-2560 3 11 submodules submodule NOUN iajs-2560 3 12 and and CCONJ iajs-2560 3 13 other other ADJ iajs-2560 3 14 related related ADJ iajs-2560 3 15 concepts concept NOUN iajs-2560 3 16 omar omar PROPN iajs-2560 3 17 a. a. PROPN iajs-2560 3 18 abdulla abdulla PROPN iajs-2560 3 19 ali ali PROPN iajs-2560 3 20 sh sh PROPN iajs-2560 3 21 . . PROPN iajs-2560 3 22 ajeel ajeel PROPN iajs-2560 3 23 haibat haibat PROPN iajs-2560 3 24 k. k. PROPN iajs-2560 3 25 mohammadali mohammadali PROPN iajs-2560 3 26 omar.aldoori87@gmail.com omar.aldoori87@gmail.com X iajs-2560 3 27 ali.shebl@st.tu.edu.iq ali.shebl@st.tu.edu.iq PROPN iajs-2560 3 28 h.mohammadali@tu.edu.iq h.mohammadali@tu.edu.iq PROPN iajs-2560 3 29 abstract abstract NOUN iajs-2560 3 30 our our PRON iajs-2560 3 31 aim aim NOUN iajs-2560 3 32 in in ADP iajs-2560 3 33 this this DET iajs-2560 3 34 paper paper NOUN iajs-2560 3 35 is be AUX iajs-2560 3 36 to to PART iajs-2560 3 37 introduce introduce VERB iajs-2560 3 38 the the DET iajs-2560 3 39 notation notation NOUN iajs-2560 3 40 of of ADP iajs-2560 3 41 a a DET iajs-2560 3 42 nearly nearly ADV iajs-2560 3 43 primary-2 primary-2 NOUN iajs-2560 3 44 - - PUNCT iajs-2560 3 45 absorbing absorb VERB iajs-2560 3 46 submodule submodule NOUN iajs-2560 3 47 as as ADP iajs-2560 3 48 generalization generalization NOUN iajs-2560 3 49 of of ADP iajs-2560 3 50 2 2 NUM iajs-2560 3 51 - - PUNCT iajs-2560 3 52 absorbing absorb VERB iajs-2560 3 53 submodule submodule NOUN iajs-2560 3 54 where where SCONJ iajs-2560 3 55 a a DET iajs-2560 3 56 proper proper ADJ iajs-2560 3 57 submodule submodule NOUN iajs-2560 3 58 𝐾 𝐾 PROPN iajs-2560 3 59 of of ADP iajs-2560 3 60 an an DET iajs-2560 3 61 𝑅module 𝑅module PROPN iajs-2560 3 62 ∁ ∁ PROPN iajs-2560 3 63 is be AUX iajs-2560 3 64 called call VERB iajs-2560 3 65 nearly nearly ADV iajs-2560 3 66 primary-2 primary-2 NOUN iajs-2560 3 67 - - PUNCT iajs-2560 3 68 absorbing absorb VERB iajs-2560 3 69 submodule submodule NOUN iajs-2560 3 70 if if SCONJ iajs-2560 3 71 whenever whenever SCONJ iajs-2560 3 72 π‘Žπ‘π‘₯ π‘Žπ‘π‘₯ PROPN iajs-2560 3 73 ∈ ∈ PROPN iajs-2560 3 74 𝐾 𝐾 PROPN iajs-2560 3 75 , , PUNCT iajs-2560 3 76 for for ADP iajs-2560 3 77 π‘Ž π‘Ž NOUN iajs-2560 3 78 , , PUNCT iajs-2560 3 79 𝑏 𝑏 NOUN iajs-2560 3 80 , , PUNCT iajs-2560 3 81 ∈ ∈ PROPN iajs-2560 3 82 𝑅 𝑅 PROPN iajs-2560 3 83 , , PUNCT iajs-2560 3 84 π‘₯ π‘₯ PRON iajs-2560 3 85 ∈ ∈ PROPN iajs-2560 3 86 ∁ ∁ PROPN iajs-2560 3 87 implies imply VERB iajs-2560 3 88 that that SCONJ iajs-2560 3 89 either either CCONJ iajs-2560 3 90 π‘Žπ‘₯ π‘Žπ‘₯ X iajs-2560 3 91 ∈ ∈ PROPN iajs-2560 3 92 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 3 93 ) ) PUNCT iajs-2560 3 94 + + PUNCT iajs-2560 3 95 𝐽(∁ 𝐽(∁ ADJ iajs-2560 3 96 ) ) PUNCT iajs-2560 3 97 or or CCONJ iajs-2560 3 98 𝑏π‘₯ 𝑏π‘₯ ADP iajs-2560 3 99 ∈ ∈ PROPN iajs-2560 3 100 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 3 101 ) ) PUNCT iajs-2560 3 102 + + PUNCT iajs-2560 3 103 𝐽(∁ 𝐽(∁ ADJ iajs-2560 3 104 ) ) PUNCT iajs-2560 3 105 or or CCONJ iajs-2560 3 106 π‘Žπ‘ π‘Žπ‘ PRON iajs-2560 3 107 ∈ ∈ PROPN iajs-2560 4 1 [ [ X iajs-2560 4 2 𝐾 𝐾 PROPN iajs-2560 4 3 + + CCONJ iajs-2560 4 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 4 5 ∁ ∁ NOUN iajs-2560 4 6 ] ] PUNCT iajs-2560 4 7 . . PUNCT iajs-2560 5 1 we we PRON iajs-2560 5 2 got get VERB iajs-2560 5 3 many many ADJ iajs-2560 5 4 basic basic ADJ iajs-2560 5 5 properties property NOUN iajs-2560 5 6 , , PUNCT iajs-2560 5 7 examples example NOUN iajs-2560 5 8 and and CCONJ iajs-2560 5 9 characterizations characterization NOUN iajs-2560 5 10 of of ADP iajs-2560 5 11 this this DET iajs-2560 5 12 concept concept NOUN iajs-2560 5 13 . . PUNCT iajs-2560 6 1 furthermore furthermore ADV iajs-2560 6 2 , , PUNCT iajs-2560 6 3 characterizations characterization NOUN iajs-2560 6 4 of of ADP iajs-2560 6 5 nearly nearly ADV iajs-2560 6 6 primary-2 primary-2 NOUN iajs-2560 6 7 - - PUNCT iajs-2560 6 8 absorbing absorbing ADJ iajs-2560 6 9 submodules submodule NOUN iajs-2560 6 10 in in ADP iajs-2560 6 11 some some DET iajs-2560 6 12 classes class NOUN iajs-2560 6 13 of of ADP iajs-2560 6 14 modules module NOUN iajs-2560 6 15 are be AUX iajs-2560 6 16 inserted insert VERB iajs-2560 6 17 . . PUNCT iajs-2560 7 1 moreover moreover ADV iajs-2560 7 2 , , PUNCT iajs-2560 7 3 the the DET iajs-2560 7 4 behavior behavior NOUN iajs-2560 7 5 of of ADP iajs-2560 7 6 nearly nearly ADV iajs-2560 7 7 primary-2 primary-2 NOUN iajs-2560 7 8 - - PUNCT iajs-2560 7 9 absorbing absorb VERB iajs-2560 7 10 submodule submodule NOUN iajs-2560 7 11 under under ADP iajs-2560 7 12 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2560 7 13 is be AUX iajs-2560 7 14 studied study VERB iajs-2560 7 15 . . PUNCT iajs-2560 8 1 keywords keyword NOUN iajs-2560 8 2 : : PUNCT iajs-2560 8 3 primary primary ADJ iajs-2560 8 4 submodules submodule NOUN iajs-2560 8 5 , , PUNCT iajs-2560 8 6 prime prime ADJ iajs-2560 8 7 submodules submodule NOUN iajs-2560 8 8 , , PUNCT iajs-2560 8 9 multiplication multiplication NOUN iajs-2560 8 10 modules module NOUN iajs-2560 8 11 , , PUNCT iajs-2560 8 12 artirian artirian ADJ iajs-2560 8 13 ring ring NOUN iajs-2560 8 14 , , PUNCT iajs-2560 8 15 projective projective PROPN iajs-2560 8 16 𝑅-modules 𝑅-modules PROPN iajs-2560 8 17 , , PUNCT iajs-2560 8 18 jacobin jacobin NOUN iajs-2560 8 19 of of ADP iajs-2560 8 20 a a DET iajs-2560 8 21 modules module NOUN iajs-2560 8 22 . . PUNCT iajs-2560 9 1 1 1 X iajs-2560 9 2 . . X iajs-2560 9 3 introduction introduction NOUN iajs-2560 9 4 throughout throughout ADP iajs-2560 9 5 this this DET iajs-2560 9 6 paper paper NOUN iajs-2560 9 7 , , PUNCT iajs-2560 9 8 all all DET iajs-2560 9 9 rings ring NOUN iajs-2560 9 10 are be AUX iajs-2560 9 11 commutative commutative ADJ iajs-2560 9 12 with with ADP iajs-2560 9 13 identity identity NOUN iajs-2560 9 14 and and CCONJ iajs-2560 9 15 we we PRON iajs-2560 9 16 assume assume VERB iajs-2560 9 17 all all DET iajs-2560 9 18 𝑅modules 𝑅modules PROPN iajs-2560 9 19 are be AUX iajs-2560 9 20 left leave VERB iajs-2560 9 21 unitary unitary ADJ iajs-2560 9 22 . . PUNCT iajs-2560 10 1 prime prime ADJ iajs-2560 10 2 submodulse submodulse ADV iajs-2560 10 3 are be AUX iajs-2560 10 4 among among ADP iajs-2560 10 5 the the DET iajs-2560 10 6 most most ADV iajs-2560 10 7 famous famous ADJ iajs-2560 10 8 concepts concept NOUN iajs-2560 10 9 of of ADP iajs-2560 10 10 modules module NOUN iajs-2560 10 11 theory theory NOUN iajs-2560 10 12 , , PUNCT iajs-2560 10 13 where where SCONJ iajs-2560 10 14 a a DET iajs-2560 10 15 proper proper ADJ iajs-2560 10 16 submodule submodule NOUN iajs-2560 10 17 𝐻 𝐻 PROPN iajs-2560 10 18 of of ADP iajs-2560 10 19 an an DET iajs-2560 10 20 𝑅-module 𝑅-module PROPN iajs-2560 10 21 ∁ ∁ PROPN iajs-2560 10 22 is be AUX iajs-2560 10 23 said say VERB iajs-2560 10 24 to to PART iajs-2560 10 25 be be AUX iajs-2560 10 26 a a DET iajs-2560 10 27 prime prime ADJ iajs-2560 10 28 submodule submodule NOUN iajs-2560 10 29 if if SCONJ iajs-2560 10 30 whenever whenever SCONJ iajs-2560 10 31 π‘Žπ‘₯ π‘Žπ‘₯ NOUN iajs-2560 10 32 ∈ ∈ PROPN iajs-2560 10 33 𝐻 𝐻 PROPN iajs-2560 10 34 , , PUNCT iajs-2560 10 35 where where SCONJ iajs-2560 10 36 π‘Ž π‘Ž PROPN iajs-2560 10 37 ∈ ∈ PROPN iajs-2560 10 38 𝑅 𝑅 PROPN iajs-2560 10 39 , , PUNCT iajs-2560 10 40 π‘₯ π‘₯ PRON iajs-2560 10 41 ∈ ∈ PROPN iajs-2560 10 42 ∁ ∁ PROPN iajs-2560 10 43 implies imply VERB iajs-2560 10 44 that that SCONJ iajs-2560 10 45 either either CCONJ iajs-2560 10 46 π‘₯ π‘₯ DET iajs-2560 10 47 ∈ ∈ PROPN iajs-2560 10 48 𝐻 𝐻 PROPN iajs-2560 10 49 or or CCONJ iajs-2560 10 50 π‘Žβˆ π‘Žβˆ PROPN iajs-2560 11 1 βŠ† βŠ† NUM iajs-2560 11 2 𝐻[1 𝐻[1 PROPN iajs-2560 11 3 ] ] PUNCT iajs-2560 11 4 . . PUNCT iajs-2560 12 1 in in ADP iajs-2560 12 2 addition addition NOUN iajs-2560 12 3 , , PUNCT iajs-2560 12 4 primary primary ADJ iajs-2560 12 5 submodules submodule NOUN iajs-2560 12 6 is be AUX iajs-2560 12 7 introduced introduce VERB iajs-2560 12 8 in in ADP iajs-2560 12 9 [ [ X iajs-2560 12 10 2 2 NUM iajs-2560 12 11 ] ] PUNCT iajs-2560 12 12 as as ADP iajs-2560 12 13 a a DET iajs-2560 12 14 generalization generalization NOUN iajs-2560 12 15 of of ADP iajs-2560 12 16 a a DET iajs-2560 12 17 prime prime ADJ iajs-2560 12 18 submodule submodule NOUN iajs-2560 12 19 , , PUNCT iajs-2560 12 20 where where SCONJ iajs-2560 12 21 a a DET iajs-2560 12 22 proper proper ADJ iajs-2560 12 23 submodule submodule NOUN iajs-2560 12 24 𝐻 𝐻 PROPN iajs-2560 12 25 of of ADP iajs-2560 12 26 an an DET iajs-2560 12 27 𝑅-module 𝑅-module PROPN iajs-2560 12 28 ∁ ∁ PROPN iajs-2560 12 29 is be AUX iajs-2560 12 30 said say VERB iajs-2560 12 31 to to PART iajs-2560 12 32 be be AUX iajs-2560 12 33 a a DET iajs-2560 12 34 primary primary ADJ iajs-2560 12 35 submodule submodule NOUN iajs-2560 12 36 if if SCONJ iajs-2560 12 37 whenever whenever SCONJ iajs-2560 12 38 π‘Žπ‘₯ π‘Žπ‘₯ NOUN iajs-2560 12 39 ∈ ∈ PROPN iajs-2560 12 40 𝐻 𝐻 PROPN iajs-2560 12 41 , , PUNCT iajs-2560 12 42 for for ADP iajs-2560 12 43 π‘Ž π‘Ž PROPN iajs-2560 12 44 ∈ ∈ PROPN iajs-2560 12 45 𝑅 𝑅 PROPN iajs-2560 12 46 , , PUNCT iajs-2560 12 47 π‘₯ π‘₯ PRON iajs-2560 12 48 ∈ ∈ PROPN iajs-2560 12 49 ∁ ∁ PROPN iajs-2560 12 50 implies imply VERB iajs-2560 12 51 that that SCONJ iajs-2560 12 52 either either CCONJ iajs-2560 12 53 π‘₯ π‘₯ DET iajs-2560 12 54 ∈ ∈ PROPN iajs-2560 12 55 𝐻 𝐻 PROPN iajs-2560 12 56 or or CCONJ iajs-2560 12 57 π‘Žπ‘›βˆ π‘Žπ‘›βˆ VERB iajs-2560 12 58 βŠ† βŠ† NUM iajs-2560 12 59 𝐻 𝐻 PROPN iajs-2560 12 60 for for ADP iajs-2560 12 61 some some DET iajs-2560 12 62 𝑛 𝑛 PRON iajs-2560 12 63 ∈ ∈ PROPN iajs-2560 12 64 𝑍+ 𝑍+ NOUN iajs-2560 12 65 . . PUNCT iajs-2560 13 1 the the DET iajs-2560 13 2 well well ADV iajs-2560 13 3 - - PUNCT iajs-2560 13 4 known know VERB iajs-2560 13 5 generalization generalization NOUN iajs-2560 13 6 of of ADP iajs-2560 13 7 prime prime ADJ iajs-2560 13 8 submodules submodule NOUN iajs-2560 13 9 is be AUX iajs-2560 13 10 the the DET iajs-2560 13 11 concept concept NOUN iajs-2560 13 12 of of ADP iajs-2560 13 13 2 2 NUM iajs-2560 13 14 - - PUNCT iajs-2560 13 15 absorbing absorb VERB iajs-2560 13 16 submodule submodule NOUN iajs-2560 13 17 , , PUNCT iajs-2560 13 18 where where SCONJ iajs-2560 13 19 a a DET iajs-2560 13 20 proper proper ADJ iajs-2560 13 21 ibn ibn PROPN iajs-2560 13 22 al al PROPN iajs-2560 13 23 haitham haitham PROPN iajs-2560 13 24 journal journal PROPN iajs-2560 13 25 for for ADP iajs-2560 13 26 pure pure ADJ iajs-2560 13 27 and and CCONJ iajs-2560 13 28 applied apply VERB iajs-2560 13 29 science science NOUN iajs-2560 13 30 journal journal PROPN iajs-2560 13 31 homepage homepage NOUN iajs-2560 13 32 : : PUNCT iajs-2560 13 33 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2560 13 34 doi doi NOUN iajs-2560 13 35 : : PUNCT iajs-2560 13 36 10.30526/34.1.2560 10.30526/34.1.2560 PROPN iajs-2560 13 37 article article NOUN iajs-2560 13 38 history history NOUN iajs-2560 13 39 : : PUNCT iajs-2560 13 40 received receive VERB iajs-2560 13 41 4 4 NUM iajs-2560 13 42 , , PUNCT iajs-2560 13 43 februaryber,2020 februaryber,2020 NOUN iajs-2560 13 44 , , PUNCT iajs-2560 13 45 accepted accept VERB iajs-2560 13 46 , , PUNCT iajs-2560 13 47 13 13 NUM iajs-2560 13 48 , , PUNCT iajs-2560 13 49 february february PROPN iajs-2560 13 50 2020 2020 NUM iajs-2560 13 51 , , PUNCT iajs-2560 13 52 published publish VERB iajs-2560 13 53 in in ADP iajs-2560 13 54 january january PROPN iajs-2560 13 55 2021 2021 NUM iajs-2560 13 56 directorate directorate ADJ iajs-2560 13 57 general general NOUN iajs-2560 13 58 of of ADP iajs-2560 13 59 education education NOUN iajs-2560 13 60 salahaddin salahaddin PROPN iajs-2560 13 61 , , PUNCT iajs-2560 13 62 the the DET iajs-2560 13 63 ministry ministry PROPN iajs-2560 13 64 of of ADP iajs-2560 13 65 education education PROPN iajs-2560 13 66 , , PUNCT iajs-2560 13 67 tikrit tikrit NOUN iajs-2560 13 68 , , PUNCT iajs-2560 13 69 iraq iraq PROPN iajs-2560 13 70 . . PUNCT iajs-2560 14 1 directorate directorate ADJ iajs-2560 14 2 general general NOUN iajs-2560 14 3 of of ADP iajs-2560 14 4 education education NOUN iajs-2560 14 5 salahaddin salahaddin PROPN iajs-2560 14 6 , , PUNCT iajs-2560 14 7 the the DET iajs-2560 14 8 ministry ministry PROPN iajs-2560 14 9 of of ADP iajs-2560 14 10 education education PROPN iajs-2560 14 11 , , PUNCT iajs-2560 14 12 tikrit tikrit NOUN iajs-2560 14 13 , , PUNCT iajs-2560 14 14 iraq iraq PROPN iajs-2560 14 15 . . PUNCT iajs-2560 15 1 department department PROPN iajs-2560 15 2 of of ADP iajs-2560 15 3 mathematics mathematics PROPN iajs-2560 15 4 , , PUNCT iajs-2560 15 5 college college NOUN iajs-2560 15 6 of of ADP iajs-2560 15 7 computer computer NOUN iajs-2560 15 8 sciences sciences PROPN iajs-2560 15 9 and and CCONJ iajs-2560 15 10 mathematics mathematic NOUN iajs-2560 15 11 , , PUNCT iajs-2560 15 12 tikrit tikrit NOUN iajs-2560 15 13 university university NOUN iajs-2560 15 14 , , PUNCT iajs-2560 15 15 tikrit tikrit NOUN iajs-2560 15 16 , , PUNCT iajs-2560 15 17 iraq iraq PROPN iajs-2560 15 18 . . PUNCT iajs-2560 16 1 mailto:omar.aldoori87@gmail.com mailto:omar.aldoori87@gmail.com X iajs-2560 16 2 mailto:ali.shebl@st.tu.edu.iq mailto:ali.shebl@st.tu.edu.iq PROPN iajs-2560 16 3 mailto:h.mohammadali@tu.edu.iq mailto:h.mohammadali@tu.edu.iq VERB iajs-2560 16 4 117 117 NUM iajs-2560 16 5 ibn ibn PROPN iajs-2560 16 6 al al PROPN iajs-2560 16 7 - - PUNCT iajs-2560 16 8 haitham haitham PROPN iajs-2560 16 9 jour jour X iajs-2560 16 10 . . PROPN iajs-2560 17 1 for for ADP iajs-2560 17 2 pure pure ADJ iajs-2560 17 3 & & CCONJ iajs-2560 17 4 appl appl PROPN iajs-2560 17 5 . . PUNCT iajs-2560 18 1 sci sci PROPN iajs-2560 18 2 . . PROPN iajs-2560 19 1 34 34 NUM iajs-2560 19 2 ( ( PUNCT iajs-2560 19 3 1 1 NUM iajs-2560 19 4 ) ) PUNCT iajs-2560 19 5 2021 2021 NUM iajs-2560 19 6 submodule submodule NOUN iajs-2560 19 7 𝐻 𝐻 PROPN iajs-2560 19 8 of of ADP iajs-2560 19 9 an an DET iajs-2560 19 10 𝑅-module 𝑅-module PROPN iajs-2560 19 11 ∁ ∁ PROPN iajs-2560 19 12 is be AUX iajs-2560 19 13 called call VERB iajs-2560 19 14 2 2 NUM iajs-2560 19 15 - - PUNCT iajs-2560 19 16 absorbing absorb VERB iajs-2560 19 17 submodule submodule NOUN iajs-2560 19 18 if if SCONJ iajs-2560 19 19 whenever whenever SCONJ iajs-2560 19 20 π‘Žπ‘π‘₯ π‘Žπ‘π‘₯ PROPN iajs-2560 19 21 ∈ ∈ PROPN iajs-2560 19 22 𝐻 𝐻 PROPN iajs-2560 19 23 , , PUNCT iajs-2560 19 24 for for ADP iajs-2560 19 25 π‘Ž,𝑏 π‘Ž,𝑏 NOUN iajs-2560 19 26 ∈ ∈ PROPN iajs-2560 19 27 𝑅 𝑅 PROPN iajs-2560 19 28 , , PUNCT iajs-2560 19 29 π‘₯ π‘₯ PRON iajs-2560 19 30 ∈ ∈ PROPN iajs-2560 19 31 ∁ ∁ PROPN iajs-2560 19 32 implies imply VERB iajs-2560 19 33 that that SCONJ iajs-2560 19 34 either either CCONJ iajs-2560 19 35 π‘Žπ‘₯ π‘Žπ‘₯ NOUN iajs-2560 19 36 ∈ ∈ PROPN iajs-2560 19 37 𝐻 𝐻 PROPN iajs-2560 19 38 or or CCONJ iajs-2560 19 39 𝑏π‘₯ 𝑏π‘₯ NOUN iajs-2560 19 40 ∈ ∈ PROPN iajs-2560 19 41 𝐻 𝐻 PROPN iajs-2560 19 42 or or CCONJ iajs-2560 19 43 π‘Žπ‘βˆ π‘Žπ‘βˆ ADP iajs-2560 19 44 βŠ† βŠ† NUM iajs-2560 19 45 𝐻[3 𝐻[3 PROPN iajs-2560 19 46 ] ] PUNCT iajs-2560 19 47 is be AUX iajs-2560 19 48 studied study VERB iajs-2560 19 49 extensively extensively ADV iajs-2560 19 50 . . PUNCT iajs-2560 20 1 there there PRON iajs-2560 20 2 are be VERB iajs-2560 20 3 many many ADJ iajs-2560 20 4 generalizations generalization NOUN iajs-2560 20 5 of of ADP iajs-2560 20 6 the the DET iajs-2560 20 7 concept concept NOUN iajs-2560 20 8 of of ADP iajs-2560 20 9 2 2 NUM iajs-2560 20 10 - - PUNCT iajs-2560 20 11 absorbing absorbing ADJ iajs-2560 20 12 , , PUNCT iajs-2560 20 13 for for ADP iajs-2560 20 14 example example NOUN iajs-2560 20 15 see see VERB iajs-2560 20 16 [ [ PUNCT iajs-2560 20 17 4 4 NUM iajs-2560 20 18 … … SYM iajs-2560 20 19 6 6 NUM iajs-2560 20 20 ] ] PUNCT iajs-2560 20 21 . . PUNCT iajs-2560 21 1 dubey dubey PROPN iajs-2560 21 2 in in ADP iajs-2560 21 3 [ [ X iajs-2560 21 4 7 7 NUM iajs-2560 21 5 ] ] PUNCT iajs-2560 21 6 introduced introduce VERB iajs-2560 21 7 the the DET iajs-2560 21 8 concept concept NOUN iajs-2560 21 9 of of ADP iajs-2560 21 10 2 2 NUM iajs-2560 21 11 - - PUNCT iajs-2560 21 12 absorbing absorbing ADJ iajs-2560 21 13 primary primary ADJ iajs-2560 21 14 submodule submodule NOUN iajs-2560 21 15 , , PUNCT iajs-2560 21 16 where where SCONJ iajs-2560 21 17 a a DET iajs-2560 21 18 proper proper ADJ iajs-2560 21 19 submodule submodule NOUN iajs-2560 21 20 𝐻 𝐻 PROPN iajs-2560 21 21 of of ADP iajs-2560 21 22 an an DET iajs-2560 21 23 𝑅-module 𝑅-module PROPN iajs-2560 21 24 ∁ ∁ PROPN iajs-2560 21 25 is be AUX iajs-2560 21 26 called call VERB iajs-2560 21 27 2 2 NUM iajs-2560 21 28 - - PUNCT iajs-2560 21 29 absorbing absorbing ADJ iajs-2560 21 30 primary primary ADJ iajs-2560 21 31 submodule submodule NOUN iajs-2560 21 32 if if SCONJ iajs-2560 21 33 whenever whenever SCONJ iajs-2560 21 34 π‘Žπ‘π‘₯ π‘Žπ‘π‘₯ PROPN iajs-2560 21 35 ∈ ∈ PROPN iajs-2560 21 36 𝐻 𝐻 PROPN iajs-2560 21 37 , , PUNCT iajs-2560 21 38 for for ADP iajs-2560 21 39 π‘Ž,𝑏 π‘Ž,𝑏 NOUN iajs-2560 21 40 ∈ ∈ PROPN iajs-2560 21 41 𝑅 𝑅 PROPN iajs-2560 21 42 , , PUNCT iajs-2560 21 43 π‘₯ π‘₯ PRON iajs-2560 21 44 ∈ ∈ PROPN iajs-2560 21 45 ∁ ∁ PROPN iajs-2560 21 46 implies imply VERB iajs-2560 21 47 that that SCONJ iajs-2560 21 48 either either CCONJ iajs-2560 21 49 π‘Žπ‘₯ π‘Žπ‘₯ PROPN iajs-2560 21 50 ∈ ∈ PROPN iajs-2560 21 51 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 X iajs-2560 21 52 ) ) PUNCT iajs-2560 21 53 or or CCONJ iajs-2560 21 54 𝑏π‘₯ 𝑏π‘₯ PROPN iajs-2560 21 55 ∈ ∈ PROPN iajs-2560 21 56 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 NOUN iajs-2560 21 57 ) ) PUNCT iajs-2560 21 58 or or CCONJ iajs-2560 21 59 π‘Žπ‘ π‘Žπ‘ ADP iajs-2560 21 60 ∈ ∈ PROPN iajs-2560 21 61 [ [ X iajs-2560 21 62 𝐻:𝑅 𝐻:𝑅 PROPN iajs-2560 21 63 ∁ ∁ PROPN iajs-2560 21 64 ] ] X iajs-2560 21 65 , , PUNCT iajs-2560 21 66 where where SCONJ iajs-2560 21 67 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 X iajs-2560 21 68 ) ) PUNCT iajs-2560 21 69 is be AUX iajs-2560 21 70 the the DET iajs-2560 21 71 intersection intersection NOUN iajs-2560 21 72 of of ADP iajs-2560 21 73 all all DET iajs-2560 21 74 prime prime ADJ iajs-2560 21 75 submodule submodule NOUN iajs-2560 21 76 of of ADP iajs-2560 21 77 ∁ ∁ PROPN iajs-2560 21 78 containing contain VERB iajs-2560 21 79 𝐻. 𝐻. PROPN iajs-2560 21 80 recall recall NOUN iajs-2560 21 81 that that SCONJ iajs-2560 21 82 an an DET iajs-2560 21 83 𝑅-module 𝑅-module PROPN iajs-2560 21 84 ∁ ∁ PROPN iajs-2560 21 85 is be AUX iajs-2560 21 86 multiplication multiplication NOUN iajs-2560 21 87 if if SCONJ iajs-2560 21 88 every every DET iajs-2560 21 89 submodule submodule NOUN iajs-2560 21 90 𝐻 𝐻 PROPN iajs-2560 21 91 of of ADP iajs-2560 21 92 ∁ ∁ PROPN iajs-2560 21 93 is be AUX iajs-2560 21 94 of of ADP iajs-2560 21 95 the the DET iajs-2560 21 96 form form NOUN iajs-2560 21 97 𝐻 𝐻 NOUN iajs-2560 21 98 = = PUNCT iajs-2560 21 99 𝐽∁ 𝐽∁ X iajs-2560 21 100 for for ADP iajs-2560 21 101 some some DET iajs-2560 21 102 ideal ideal ADJ iajs-2560 21 103 𝐽 𝐽 PROPN iajs-2560 21 104 of of ADP iajs-2560 21 105 𝑅 𝑅 PROPN iajs-2560 22 1 [ [ NOUN iajs-2560 22 2 8 8 NUM iajs-2560 22 3 ] ] PUNCT iajs-2560 22 4 . . PUNCT iajs-2560 23 1 again again ADV iajs-2560 23 2 recall recall VERB iajs-2560 23 3 that that SCONJ iajs-2560 23 4 an an DET iajs-2560 23 5 𝑅-module 𝑅-module PROPN iajs-2560 23 6 ∁ ∁ PROPN iajs-2560 23 7 is be AUX iajs-2560 23 8 said say VERB iajs-2560 23 9 to to PART iajs-2560 23 10 be be AUX iajs-2560 23 11 faithful faithful ADJ iajs-2560 23 12 if if SCONJ iajs-2560 23 13 π‘Žπ‘›π‘›π‘…(∁ π‘Žπ‘›π‘›π‘…(∁ PROPN iajs-2560 23 14 ) ) PUNCT iajs-2560 23 15 = = PUNCT iajs-2560 23 16 ( ( PUNCT iajs-2560 23 17 0 0 NUM iajs-2560 23 18 ) ) PUNCT iajs-2560 23 19 . . PUNCT iajs-2560 24 1 and and CCONJ iajs-2560 24 2 a a DET iajs-2560 24 3 ring ring NOUN iajs-2560 24 4 𝑅 𝑅 PROPN iajs-2560 24 5 is be AUX iajs-2560 24 6 said say VERB iajs-2560 24 7 to to PART iajs-2560 24 8 be be AUX iajs-2560 24 9 artinian artinian ADJ iajs-2560 24 10 if if SCONJ iajs-2560 24 11 𝑅 𝑅 PROPN iajs-2560 24 12 satisfies satisfie NOUN iajs-2560 24 13 d.cc d.cc VERB iajs-2560 24 14 on on ADP iajs-2560 24 15 ideals ideal NOUN iajs-2560 24 16 of of ADP iajs-2560 24 17 𝑅 𝑅 PROPN iajs-2560 24 18 , , PUNCT iajs-2560 24 19 that that PRON iajs-2560 24 20 is be AUX iajs-2560 24 21 𝐼1 𝐼1 ADJ iajs-2560 24 22 βŠ‡ βŠ‡ PROPN iajs-2560 24 23 𝐼2 𝐼2 PROPN iajs-2560 24 24 βŠ‡ βŠ‡ PROPN iajs-2560 24 25 β‹― β‹― PROPN iajs-2560 24 26 βŠ‡ βŠ‡ PROPN iajs-2560 24 27 β‹― β‹― PROPN iajs-2560 24 28 , , PUNCT iajs-2560 24 29 then then ADV iajs-2560 24 30 there there PRON iajs-2560 24 31 exists exist VERB iajs-2560 24 32 𝑛 𝑛 PRON iajs-2560 24 33 ∈ ∈ NOUN iajs-2560 24 34 𝑍+ 𝑍+ NOUN iajs-2560 25 1 such such ADJ iajs-2560 25 2 that that SCONJ iajs-2560 25 3 𝐼𝑛 𝐼𝑛 PROPN iajs-2560 25 4 = = SYM iajs-2560 25 5 πΌπ‘š πΌπ‘š PROPN iajs-2560 25 6 for for ADP iajs-2560 25 7 some some DET iajs-2560 25 8 𝑛 𝑛 PART iajs-2560 25 9 > > X iajs-2560 25 10 π‘š π‘š X iajs-2560 26 1 [ [ X iajs-2560 26 2 11 11 NUM iajs-2560 26 3 ] ] PUNCT iajs-2560 26 4 . . PUNCT iajs-2560 27 1 finally finally ADV iajs-2560 27 2 a a DET iajs-2560 27 3 ring ring NOUN iajs-2560 27 4 𝑅 𝑅 PROPN iajs-2560 27 5 is be AUX iajs-2560 27 6 called call VERB iajs-2560 27 7 a a DET iajs-2560 27 8 good good ADJ iajs-2560 27 9 ring ring NOUN iajs-2560 27 10 , , PUNCT iajs-2560 27 11 if if SCONJ iajs-2560 27 12 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 27 13 ) ) PUNCT iajs-2560 27 14 . . PUNCT iajs-2560 28 1 ∁= ∁= PROPN iajs-2560 28 2 𝐽(∁ 𝐽(∁ PROPN iajs-2560 28 3 ) ) PUNCT iajs-2560 28 4 where where SCONJ iajs-2560 28 5 ∁ ∁ PROPN iajs-2560 28 6 is be AUX iajs-2560 28 7 an an DET iajs-2560 28 8 𝑅-module 𝑅-module PROPN iajs-2560 29 1 [ [ X iajs-2560 29 2 12 12 NUM iajs-2560 29 3 ] ] PUNCT iajs-2560 29 4 . . PUNCT iajs-2560 30 1 2 2 X iajs-2560 30 2 . . X iajs-2560 30 3 nearly nearly ADV iajs-2560 30 4 primary-2 primary-2 NOUN iajs-2560 30 5 - - PUNCT iajs-2560 30 6 absorbing absorbing ADJ iajs-2560 30 7 submodules submodule NOUN iajs-2560 30 8 in in ADP iajs-2560 30 9 this this DET iajs-2560 30 10 paper paper NOUN iajs-2560 30 11 , , PUNCT iajs-2560 30 12 we we PRON iajs-2560 30 13 will will AUX iajs-2560 30 14 introduce introduce VERB iajs-2560 30 15 the the DET iajs-2560 30 16 concept concept NOUN iajs-2560 30 17 of of ADP iajs-2560 30 18 nearly nearly ADV iajs-2560 30 19 primary-2 primary-2 NOUN iajs-2560 30 20 - - PUNCT iajs-2560 30 21 absorbing absorb VERB iajs-2560 30 22 submodule submodule NOUN iajs-2560 30 23 and and CCONJ iajs-2560 30 24 give give VERB iajs-2560 30 25 some some DET iajs-2560 30 26 basic basic ADJ iajs-2560 30 27 results result NOUN iajs-2560 30 28 of of ADP iajs-2560 30 29 these these DET iajs-2560 30 30 classes class NOUN iajs-2560 30 31 of of ADP iajs-2560 30 32 submodules submodule NOUN iajs-2560 30 33 . . PUNCT iajs-2560 31 1 and and CCONJ iajs-2560 31 2 discuss discuss VERB iajs-2560 31 3 on on ADP iajs-2560 31 4 the the DET iajs-2560 31 5 relationships relationship NOUN iajs-2560 31 6 with with ADP iajs-2560 31 7 class class NOUN iajs-2560 31 8 of of ADP iajs-2560 31 9 2 2 NUM iajs-2560 31 10 - - PUNCT iajs-2560 31 11 absorbing absorb VERB iajs-2560 31 12 submodules submodule NOUN iajs-2560 31 13 and and CCONJ iajs-2560 31 14 nearly nearly ADV iajs-2560 31 15 primary-2 primary-2 NOUN iajs-2560 31 16 - - PUNCT iajs-2560 31 17 absorbing absorbing ADJ iajs-2560 31 18 submodules submodule NOUN iajs-2560 31 19 . . PUNCT iajs-2560 32 1 definition definition NOUN iajs-2560 32 2 1 1 NUM iajs-2560 32 3 a a DET iajs-2560 32 4 proper proper ADJ iajs-2560 32 5 submodule submodule NOUN iajs-2560 32 6 𝐻 𝐻 PROPN iajs-2560 32 7 of of ADP iajs-2560 32 8 an an DET iajs-2560 32 9 𝑅-module 𝑅-module PROPN iajs-2560 32 10 ∁ ∁ PROPN iajs-2560 32 11 is be AUX iajs-2560 32 12 said say VERB iajs-2560 32 13 to to PART iajs-2560 32 14 be be AUX iajs-2560 32 15 nearly nearly ADV iajs-2560 32 16 primary-2 primary-2 NOUN iajs-2560 32 17 - - PUNCT iajs-2560 32 18 absorbing absorb VERB iajs-2560 32 19 submodule submodule NOUN iajs-2560 32 20 of of ADP iajs-2560 32 21 ∁ ∁ PROPN iajs-2560 32 22 , , PUNCT iajs-2560 32 23 if if SCONJ iajs-2560 32 24 whenever whenever SCONJ iajs-2560 32 25 π‘Žπ‘π‘₯ π‘Žπ‘π‘₯ PROPN iajs-2560 32 26 ∈ ∈ PROPN iajs-2560 32 27 𝐻 𝐻 PROPN iajs-2560 32 28 , , PUNCT iajs-2560 32 29 for for ADP iajs-2560 32 30 π‘Ž π‘Ž NOUN iajs-2560 32 31 , , PUNCT iajs-2560 32 32 𝑏 𝑏 PROPN iajs-2560 32 33 ∈ ∈ PROPN iajs-2560 32 34 𝑅 𝑅 PROPN iajs-2560 32 35 , , PUNCT iajs-2560 32 36 π‘₯ π‘₯ PRON iajs-2560 32 37 ∈ ∈ PROPN iajs-2560 32 38 ∁ ∁ PROPN iajs-2560 32 39 implies imply VERB iajs-2560 32 40 that that SCONJ iajs-2560 32 41 either either CCONJ iajs-2560 32 42 π‘Žπ‘₯ π‘Žπ‘₯ PROPN iajs-2560 32 43 ∈ ∈ PROPN iajs-2560 32 44 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 X iajs-2560 32 45 ) ) PUNCT iajs-2560 32 46 + + CCONJ iajs-2560 32 47 𝐽(∁ 𝐽(∁ ADJ iajs-2560 32 48 ) ) PUNCT iajs-2560 32 49 or or CCONJ iajs-2560 32 50 𝑏π‘₯ 𝑏π‘₯ PROPN iajs-2560 32 51 ∈ ∈ PROPN iajs-2560 32 52 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 X iajs-2560 32 53 ) ) PUNCT iajs-2560 32 54 + + CCONJ iajs-2560 32 55 𝐽(∁ 𝐽(∁ ADJ iajs-2560 32 56 ) ) PUNCT iajs-2560 32 57 or or CCONJ iajs-2560 32 58 π‘Žπ‘ π‘Žπ‘ ADP iajs-2560 32 59 ∈ ∈ PROPN iajs-2560 33 1 [ [ X iajs-2560 33 2 𝐻 𝐻 PROPN iajs-2560 33 3 + + CCONJ iajs-2560 33 4 𝐽(∁):𝑅 𝐽(∁):𝑅 ADJ iajs-2560 33 5 ∁ ∁ NOUN iajs-2560 33 6 ] ] PUNCT iajs-2560 33 7 . . PUNCT iajs-2560 34 1 and and CCONJ iajs-2560 34 2 a a DET iajs-2560 34 3 proper proper ADJ iajs-2560 34 4 ideal ideal ADJ iajs-2560 34 5 𝐽 𝐽 PROPN iajs-2560 34 6 of of ADP iajs-2560 34 7 a a DET iajs-2560 34 8 ring ring NOUN iajs-2560 34 9 𝑅 𝑅 PROPN iajs-2560 34 10 is be AUX iajs-2560 34 11 called call VERB iajs-2560 34 12 nearly nearly ADV iajs-2560 34 13 primary-2 primary-2 NOUN iajs-2560 34 14 - - PUNCT iajs-2560 34 15 absorbing absorbing ADJ iajs-2560 34 16 ideal ideal NOUN iajs-2560 34 17 of of ADP iajs-2560 34 18 𝑅 𝑅 PROPN iajs-2560 34 19 , , PUNCT iajs-2560 34 20 if if SCONJ iajs-2560 34 21 𝐽 𝐽 PRON iajs-2560 34 22 is be AUX iajs-2560 34 23 nearly nearly ADV iajs-2560 34 24 primary-2 primary-2 NOUN iajs-2560 34 25 - - PUNCT iajs-2560 34 26 absorbing absorbing ADJ iajs-2560 34 27 submodules submodule NOUN iajs-2560 34 28 of of ADP iajs-2560 34 29 an an DET iajs-2560 34 30 𝑅module 𝑅module PROPN iajs-2560 34 31 𝑅. 𝑅. NOUN iajs-2560 34 32 remarks remark NOUN iajs-2560 34 33 and and CCONJ iajs-2560 34 34 examples example NOUN iajs-2560 34 35 2 2 NUM iajs-2560 34 36 1 1 NUM iajs-2560 34 37 . . PUNCT iajs-2560 35 1 it it PRON iajs-2560 35 2 is be AUX iajs-2560 35 3 clear clear ADJ iajs-2560 35 4 that that SCONJ iajs-2560 35 5 every every DET iajs-2560 35 6 2 2 NUM iajs-2560 35 7 - - PUNCT iajs-2560 35 8 absorbing absorb VERB iajs-2560 35 9 submodule submodule NOUN iajs-2560 35 10 of of ADP iajs-2560 35 11 an an DET iajs-2560 35 12 𝑅-module 𝑅-module PROPN iajs-2560 35 13 ∁ ∁ PROPN iajs-2560 35 14 is be AUX iajs-2560 35 15 a a DET iajs-2560 35 16 nearly nearly ADV iajs-2560 35 17 primary-2absorbing primary-2absorbe VERB iajs-2560 35 18 submodule submodule NOUN iajs-2560 35 19 , , PUNCT iajs-2560 35 20 while while SCONJ iajs-2560 35 21 the the DET iajs-2560 35 22 reverse reverse NOUN iajs-2560 35 23 does do AUX iajs-2560 35 24 not not PART iajs-2560 35 25 hold hold VERB iajs-2560 35 26 in in ADP iajs-2560 35 27 general general ADJ iajs-2560 35 28 , , PUNCT iajs-2560 35 29 the the DET iajs-2560 35 30 following follow VERB iajs-2560 35 31 example example NOUN iajs-2560 35 32 show show VERB iajs-2560 35 33 that that SCONJ iajs-2560 35 34 : : PUNCT iajs-2560 35 35 let let VERB iajs-2560 35 36 ∁= ∁= PROPN iajs-2560 35 37 𝑍16 𝑍16 PROPN iajs-2560 35 38 , , PUNCT iajs-2560 35 39 𝑅 𝑅 PROPN iajs-2560 35 40 = = PUNCT iajs-2560 35 41 𝑍 𝑍 PROPN iajs-2560 35 42 and and CCONJ iajs-2560 35 43 𝐾 𝐾 NOUN iajs-2560 35 44 = = PUNCT iajs-2560 35 45 〈 〈 PROPN iajs-2560 35 46 8Μ… 8Μ… NUM iajs-2560 35 47 βŒͺ βŒͺ NOUN iajs-2560 35 48 . . PUNCT iajs-2560 36 1 𝐾 𝐾 NOUN iajs-2560 36 2 is be AUX iajs-2560 36 3 not not PART iajs-2560 36 4 2 2 NUM iajs-2560 36 5 - - PUNCT iajs-2560 36 6 absorbing absorb VERB iajs-2560 36 7 submodule submodule NOUN iajs-2560 36 8 since since SCONJ iajs-2560 36 9 2.2 2.2 NUM iajs-2560 36 10 . . PUNCT iajs-2560 37 1 2Μ… 2Μ… NUM iajs-2560 37 2 ∈ ∈ PROPN iajs-2560 37 3 𝐾 𝐾 NOUN iajs-2560 37 4 where where SCONJ iajs-2560 37 5 2 2 NUM iajs-2560 37 6 ∈ ∈ PROPN iajs-2560 37 7 𝑅 𝑅 PROPN iajs-2560 37 8 = = PUNCT iajs-2560 37 9 𝑍 𝑍 PROPN iajs-2560 37 10 , , PUNCT iajs-2560 37 11 2Μ… 2Μ… PROPN iajs-2560 37 12 ∈ ∈ NOUN iajs-2560 37 13 𝑍16 𝑍16 PROPN iajs-2560 37 14 , , PUNCT iajs-2560 37 15 then then ADV iajs-2560 37 16 2 2 X iajs-2560 37 17 . . X iajs-2560 38 1 2Μ… 2Μ… NOUN iajs-2560 38 2 = = SYM iajs-2560 38 3 4Μ… 4Μ… ADJ iajs-2560 38 4 βˆ‰ βˆ‰ PROPN iajs-2560 38 5 𝐾 𝐾 PROPN iajs-2560 38 6 and and CCONJ iajs-2560 38 7 2.2 2.2 NUM iajs-2560 38 8 = = SYM iajs-2560 38 9 4 4 NUM iajs-2560 38 10 βˆ‰ βˆ‰ PROPN iajs-2560 39 1 [ [ X iajs-2560 39 2 𝐾:𝑅 𝐾:𝑅 PROPN iajs-2560 39 3 𝑍16 𝑍16 PROPN iajs-2560 39 4 ] ] X iajs-2560 39 5 = = SYM iajs-2560 39 6 8𝑍. 8𝑍. NUM iajs-2560 39 7 but but CCONJ iajs-2560 39 8 𝐾 𝐾 PROPN iajs-2560 39 9 is be AUX iajs-2560 39 10 a a DET iajs-2560 39 11 nearly nearly ADV iajs-2560 39 12 primary-2 primary-2 NOUN iajs-2560 39 13 - - PUNCT iajs-2560 39 14 absorbing absorb VERB iajs-2560 39 15 submodule submodule NOUN iajs-2560 39 16 of of ADP iajs-2560 39 17 𝑍16 𝑍16 PROPN iajs-2560 39 18 , , PUNCT iajs-2560 39 19 since since SCONJ iajs-2560 39 20 𝐽(𝑍16 𝐽(𝑍16 PROPN iajs-2560 39 21 ) ) PUNCT iajs-2560 39 22 = = PUNCT iajs-2560 40 1 〈 〈 PROPN iajs-2560 40 2 2Μ… 2Μ… NUM iajs-2560 40 3 βŒͺ βŒͺ NOUN iajs-2560 40 4 and and CCONJ iajs-2560 40 5 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 40 6 ) ) PUNCT iajs-2560 40 7 = = PUNCT iajs-2560 41 1 〈 〈 PROPN iajs-2560 41 2 2Μ… 2Μ… NOUN iajs-2560 41 3 βŒͺ βŒͺ NOUN iajs-2560 41 4 for for ADP iajs-2560 41 5 all all DET iajs-2560 41 6 π‘Ž π‘Ž NOUN iajs-2560 41 7 , , PUNCT iajs-2560 41 8 𝑏 𝑏 PROPN iajs-2560 41 9 ∈ ∈ PROPN iajs-2560 41 10 𝑍 𝑍 NOUN iajs-2560 41 11 , , PUNCT iajs-2560 41 12 π‘₯ π‘₯ PRON iajs-2560 41 13 ∈ ∈ PROPN iajs-2560 41 14 𝑍16 𝑍16 PROPN iajs-2560 41 15 with with ADP iajs-2560 41 16 π‘Žπ‘π‘₯ π‘Žπ‘π‘₯ NOUN iajs-2560 41 17 ∈ ∈ PROPN iajs-2560 41 18 𝐾 𝐾 PROPN iajs-2560 41 19 = = PUNCT iajs-2560 41 20 〈 〈 PROPN iajs-2560 41 21 8Μ… 8Μ… NUM iajs-2560 41 22 βŒͺ βŒͺ NOUN iajs-2560 41 23 implies imply VERB iajs-2560 41 24 that that SCONJ iajs-2560 41 25 either either CCONJ iajs-2560 41 26 π‘Žπ‘₯ π‘Žπ‘₯ X iajs-2560 41 27 ∈ ∈ PROPN iajs-2560 41 28 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 41 29 ) ) PUNCT iajs-2560 41 30 + + CCONJ iajs-2560 41 31 𝐽(𝑍16 𝐽(𝑍16 PROPN iajs-2560 41 32 ) ) PUNCT iajs-2560 41 33 = = PUNCT iajs-2560 41 34 〈 〈 PROPN iajs-2560 41 35 2Μ… 2Μ… NUM iajs-2560 41 36 βŒͺ βŒͺ NOUN iajs-2560 41 37 + + CCONJ iajs-2560 41 38 〈 〈 NOUN iajs-2560 41 39 2Μ… 2Μ… NOUN iajs-2560 41 40 βŒͺ βŒͺ NOUN iajs-2560 41 41 = = PUNCT iajs-2560 41 42 〈 〈 PROPN iajs-2560 41 43 2Μ… 2Μ… NOUN iajs-2560 41 44 βŒͺ βŒͺ NOUN iajs-2560 41 45 or or CCONJ iajs-2560 41 46 𝑏π‘₯ 𝑏π‘₯ ADP iajs-2560 41 47 ∈ ∈ PROPN iajs-2560 41 48 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 41 49 ) ) PUNCT iajs-2560 41 50 + + CCONJ iajs-2560 41 51 𝐽(𝑍16 𝐽(𝑍16 PROPN iajs-2560 41 52 ) ) PUNCT iajs-2560 41 53 = = PUNCT iajs-2560 41 54 〈 〈 PROPN iajs-2560 41 55 2Μ… 2Μ… NOUN iajs-2560 41 56 βŒͺ βŒͺ NOUN iajs-2560 41 57 or or CCONJ iajs-2560 41 58 π‘Žπ‘ π‘Žπ‘ PRON iajs-2560 41 59 ∈ ∈ PROPN iajs-2560 41 60 [ [ X iajs-2560 41 61 〈 〈 NOUN iajs-2560 41 62 8Μ… 8Μ… NUM iajs-2560 41 63 βŒͺ βŒͺ NOUN iajs-2560 41 64 + + CCONJ iajs-2560 41 65 𝐽(𝑍16):𝑍 𝐽(𝑍16):𝑍 X iajs-2560 41 66 𝑍16 𝑍16 PROPN iajs-2560 41 67 ] ] X iajs-2560 41 68 = = PUNCT iajs-2560 42 1 [ [ X iajs-2560 42 2 〈 〈 X iajs-2560 42 3 2Μ… 2Μ… NUM iajs-2560 42 4 βŒͺ βŒͺ NOUN iajs-2560 42 5 : : PUNCT iajs-2560 42 6 𝑍16 𝑍16 X iajs-2560 42 7 ] ] PUNCT iajs-2560 42 8 = = PUNCT iajs-2560 42 9 2𝑍. 2𝑍. NUM iajs-2560 42 10 that that PRON iajs-2560 42 11 is be AUX iajs-2560 42 12 2.2 2.2 NUM iajs-2560 42 13 . . PUNCT iajs-2560 43 1 2Μ… 2Μ… NUM iajs-2560 43 2 ∈ ∈ PROPN iajs-2560 43 3 𝐾 𝐾 PROPN iajs-2560 43 4 , , PUNCT iajs-2560 43 5 implies imply VERB iajs-2560 43 6 that that SCONJ iajs-2560 43 7 2 2 X iajs-2560 43 8 . . X iajs-2560 44 1 2Μ… 2Μ… NOUN iajs-2560 44 2 = = SYM iajs-2560 45 1 4Μ… 4Μ… PROPN iajs-2560 45 2 ∈ ∈ PROPN iajs-2560 45 3 〈 〈 NOUN iajs-2560 45 4 2Μ… 2Μ… NOUN iajs-2560 45 5 βŒͺ βŒͺ NOUN iajs-2560 45 6 + + CCONJ iajs-2560 45 7 〈 〈 NOUN iajs-2560 45 8 2Μ… 2Μ… NOUN iajs-2560 45 9 βŒͺ βŒͺ NOUN iajs-2560 45 10 = = PUNCT iajs-2560 45 11 〈 〈 PROPN iajs-2560 45 12 2Μ… 2Μ… NOUN iajs-2560 45 13 βŒͺ βŒͺ NOUN iajs-2560 45 14 or or CCONJ iajs-2560 45 15 2.2 2.2 NUM iajs-2560 45 16 = = SYM iajs-2560 45 17 4 4 NUM iajs-2560 45 18 ∈ ∈ NOUN iajs-2560 45 19 [ [ X iajs-2560 45 20 〈 〈 NOUN iajs-2560 45 21 8Μ… 8Μ… NUM iajs-2560 45 22 βŒͺ βŒͺ NOUN iajs-2560 45 23 + + CCONJ iajs-2560 45 24 〈 〈 PROPN iajs-2560 45 25 2Μ…βŒͺ:𝑍 2Μ…βŒͺ:𝑍 NUM iajs-2560 45 26 𝑍16 𝑍16 PROPN iajs-2560 45 27 ] ] X iajs-2560 45 28 = = SYM iajs-2560 45 29 2𝑍. 2𝑍. NUM iajs-2560 45 30 2 2 NUM iajs-2560 45 31 . . PUNCT iajs-2560 46 1 it it PRON iajs-2560 46 2 is be AUX iajs-2560 46 3 clear clear ADJ iajs-2560 46 4 that that SCONJ iajs-2560 46 5 every every DET iajs-2560 46 6 prime prime ADJ iajs-2560 46 7 submodule submodule NOUN iajs-2560 46 8 of of ADP iajs-2560 46 9 an an DET iajs-2560 46 10 𝑅-module 𝑅-module PROPN iajs-2560 46 11 ∁ ∁ PROPN iajs-2560 46 12 is be AUX iajs-2560 46 13 a a DET iajs-2560 46 14 nearly nearly ADV iajs-2560 46 15 primary-2 primary-2 NOUN iajs-2560 46 16 - - PUNCT iajs-2560 46 17 absorbing absorb VERB iajs-2560 46 18 submodule submodule NOUN iajs-2560 46 19 , , PUNCT iajs-2560 46 20 while while SCONJ iajs-2560 46 21 the the DET iajs-2560 46 22 reverse reverse NOUN iajs-2560 46 23 does do AUX iajs-2560 46 24 not not PART iajs-2560 46 25 hold hold VERB iajs-2560 46 26 in in ADP iajs-2560 46 27 general general ADJ iajs-2560 46 28 , , PUNCT iajs-2560 46 29 the the DET iajs-2560 46 30 following follow VERB iajs-2560 46 31 example example NOUN iajs-2560 46 32 show show VERB iajs-2560 46 33 that that SCONJ iajs-2560 46 34 : : PUNCT iajs-2560 46 35 let let VERB iajs-2560 46 36 ∁= ∁= PROPN iajs-2560 46 37 𝑍8 𝑍8 PROPN iajs-2560 46 38 , , PUNCT iajs-2560 46 39 𝑅 𝑅 PROPN iajs-2560 46 40 = = PUNCT iajs-2560 46 41 𝑍 𝑍 PROPN iajs-2560 46 42 and and CCONJ iajs-2560 46 43 𝐾 𝐾 PROPN iajs-2560 46 44 = = PUNCT iajs-2560 46 45 〈 〈 PROPN iajs-2560 46 46 4Μ… 4Μ… ADJ iajs-2560 46 47 βŒͺ βŒͺ NOUN iajs-2560 46 48 . . PUNCT iajs-2560 47 1 𝐾 𝐾 NOUN iajs-2560 47 2 is be AUX iajs-2560 47 3 not not PART iajs-2560 47 4 prime prime ADJ iajs-2560 47 5 submodule submodule NOUN iajs-2560 47 6 of of ADP iajs-2560 47 7 𝑍8 𝑍8 PROPN iajs-2560 47 8 , , PUNCT iajs-2560 47 9 since since SCONJ iajs-2560 47 10 2 2 NUM iajs-2560 47 11 . . PUNCT iajs-2560 47 12 2Μ… 2Μ… NUM iajs-2560 47 13 ∈ ∈ PROPN iajs-2560 47 14 𝐾 𝐾 PROPN iajs-2560 47 15 , , PUNCT iajs-2560 47 16 where where SCONJ iajs-2560 47 17 2 2 NUM iajs-2560 47 18 ∈ ∈ NOUN iajs-2560 47 19 𝑍 𝑍 NOUN iajs-2560 47 20 , , PUNCT iajs-2560 47 21 2Μ… 2Μ… PROPN iajs-2560 47 22 ∈ ∈ NOUN iajs-2560 47 23 𝑍8 𝑍8 NOUN iajs-2560 47 24 implies imply VERB iajs-2560 47 25 that that SCONJ iajs-2560 47 26 2Μ… 2Μ… PROPN iajs-2560 47 27 βˆ‰ βˆ‰ PROPN iajs-2560 47 28 𝐾 𝐾 PROPN iajs-2560 47 29 and and CCONJ iajs-2560 47 30 2 2 NUM iajs-2560 47 31 βˆ‰ βˆ‰ NOUN iajs-2560 47 32 [ [ X iajs-2560 47 33 𝐾:𝑍 𝐾:𝑍 PROPN iajs-2560 47 34 ∁ ∁ PROPN iajs-2560 47 35 ] ] X iajs-2560 47 36 = = SYM iajs-2560 47 37 4𝑍. 4𝑍. NUM iajs-2560 47 38 but but CCONJ iajs-2560 47 39 𝐾 𝐾 PROPN iajs-2560 47 40 is be AUX iajs-2560 47 41 a a DET iajs-2560 47 42 nearly nearly ADV iajs-2560 47 43 primary-2absorbing primary-2absorbe VERB iajs-2560 47 44 submodule submodule NOUN iajs-2560 47 45 of of ADP iajs-2560 47 46 ∁ ∁ PROPN iajs-2560 47 47 , , PUNCT iajs-2560 47 48 since since SCONJ iajs-2560 47 49 2.2 2.2 NUM iajs-2560 47 50 . . PUNCT iajs-2560 48 1 1Μ… 1Μ… NUM iajs-2560 48 2 ∈ ∈ PROPN iajs-2560 48 3 𝐾 𝐾 PROPN iajs-2560 48 4 , , PUNCT iajs-2560 48 5 where where SCONJ iajs-2560 48 6 2 2 NUM iajs-2560 48 7 ∈ ∈ PROPN iajs-2560 48 8 𝑍 𝑍 NOUN iajs-2560 48 9 , , PUNCT iajs-2560 48 10 1Μ… 1Μ… PROPN iajs-2560 48 11 ∈ ∈ PROPN iajs-2560 48 12 𝑍8 𝑍8 NOUN iajs-2560 48 13 2 2 NUM iajs-2560 48 14 implies imply VERB iajs-2560 48 15 that that SCONJ iajs-2560 48 16 2 2 X iajs-2560 48 17 . . X iajs-2560 48 18 1Μ… 1Μ… NUM iajs-2560 48 19 ∈ ∈ PROPN iajs-2560 48 20 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 48 21 ) ) PUNCT iajs-2560 48 22 + + CCONJ iajs-2560 48 23 𝐽(𝑍8 𝐽(𝑍8 NOUN iajs-2560 48 24 ) ) PUNCT iajs-2560 48 25 = = PUNCT iajs-2560 48 26 〈 〈 PROPN iajs-2560 48 27 2 2 NUM iajs-2560 48 28 βŒͺ βŒͺ NOUN iajs-2560 48 29 or or CCONJ iajs-2560 48 30 2.2 2.2 NUM iajs-2560 48 31 = = SYM iajs-2560 48 32 4 4 NUM iajs-2560 48 33 ∈ ∈ NOUN iajs-2560 48 34 [ [ X iajs-2560 48 35 𝐾 𝐾 X iajs-2560 48 36 + + NUM iajs-2560 48 37 𝐽(∁):𝑍 𝐽(∁):𝑍 ADV iajs-2560 48 38 ∁ ∁ NOUN iajs-2560 48 39 ] ] X iajs-2560 48 40 = = SYM iajs-2560 48 41 2𝑍. 2𝑍. NUM iajs-2560 48 42 3 3 NUM iajs-2560 48 43 . . PUNCT iajs-2560 49 1 it it PRON iajs-2560 49 2 is be AUX iajs-2560 49 3 clear clear ADJ iajs-2560 49 4 that that SCONJ iajs-2560 49 5 every every DET iajs-2560 49 6 primary primary ADJ iajs-2560 49 7 submodule submodule NOUN iajs-2560 49 8 of of ADP iajs-2560 49 9 an an DET iajs-2560 49 10 𝑅-module 𝑅-module PROPN iajs-2560 49 11 ∁ ∁ PROPN iajs-2560 49 12 is be AUX iajs-2560 49 13 a a DET iajs-2560 49 14 nearly nearly ADV iajs-2560 49 15 primary-2 primary-2 NOUN iajs-2560 49 16 - - PUNCT iajs-2560 49 17 absorbing absorb VERB iajs-2560 49 18 submodule submodule NOUN iajs-2560 49 19 , , PUNCT iajs-2560 49 20 while while SCONJ iajs-2560 49 21 the the DET iajs-2560 49 22 reverse reverse NOUN iajs-2560 49 23 does do AUX iajs-2560 49 24 not not PART iajs-2560 49 25 hold hold VERB iajs-2560 49 26 in in ADP iajs-2560 49 27 general general ADJ iajs-2560 49 28 . . PUNCT iajs-2560 50 1 the the DET iajs-2560 50 2 following follow VERB iajs-2560 50 3 example example NOUN iajs-2560 50 4 show show VERB iajs-2560 50 5 that that SCONJ iajs-2560 50 6 : : PUNCT iajs-2560 50 7 let let VERB iajs-2560 50 8 ∁= ∁= PROPN iajs-2560 50 9 𝑍6 𝑍6 PROPN iajs-2560 50 10 , , PUNCT iajs-2560 50 11 𝑅 𝑅 PROPN iajs-2560 50 12 = = PUNCT iajs-2560 50 13 𝑍 𝑍 PROPN iajs-2560 50 14 and and CCONJ iajs-2560 50 15 𝐾 𝐾 PROPN iajs-2560 50 16 = = PUNCT iajs-2560 50 17 〈 〈 PROPN iajs-2560 50 18 0Μ… 0Μ… PROPN iajs-2560 50 19 βŒͺ βŒͺ NOUN iajs-2560 50 20 is be AUX iajs-2560 50 21 a a DET iajs-2560 50 22 submodule submodule NOUN iajs-2560 50 23 of of ADP iajs-2560 50 24 ∁. ∁. NOUN iajs-2560 50 25 𝐾 𝐾 PROPN iajs-2560 50 26 is be AUX iajs-2560 50 27 a a DET iajs-2560 50 28 nearly nearly ADV iajs-2560 50 29 primary-2 primary-2 NOUN iajs-2560 50 30 - - PUNCT iajs-2560 50 31 absorbing absorb VERB iajs-2560 50 32 submodule submodule NOUN iajs-2560 50 33 of of ADP iajs-2560 50 34 ∁ ∁ PROPN iajs-2560 50 35 but but CCONJ iajs-2560 50 36 not not PART iajs-2560 50 37 primary primary ADJ iajs-2560 50 38 submodule submodule NOUN iajs-2560 50 39 , , PUNCT iajs-2560 50 40 since since SCONJ iajs-2560 50 41 3 3 NUM iajs-2560 50 42 ∈ ∈ PROPN iajs-2560 50 43 𝑍 𝑍 NOUN iajs-2560 50 44 , , PUNCT iajs-2560 50 45 2Μ… 2Μ… PROPN iajs-2560 50 46 ∈ ∈ PROPN iajs-2560 50 47 𝑍6 𝑍6 NOUN iajs-2560 50 48 such such ADJ iajs-2560 50 49 that that SCONJ iajs-2560 50 50 3 3 X iajs-2560 50 51 . . X iajs-2560 50 52 2Μ… 2Μ… NUM iajs-2560 50 53 ∈ ∈ PROPN iajs-2560 50 54 𝐾 𝐾 PROPN iajs-2560 50 55 , , PUNCT iajs-2560 50 56 but but CCONJ iajs-2560 50 57 118 118 NUM iajs-2560 50 58 ibn ibn PROPN iajs-2560 50 59 al al PROPN iajs-2560 50 60 - - PUNCT iajs-2560 50 61 haitham haitham PROPN iajs-2560 50 62 jour jour X iajs-2560 50 63 . . PROPN iajs-2560 51 1 for for ADP iajs-2560 51 2 pure pure ADJ iajs-2560 51 3 & & CCONJ iajs-2560 51 4 appl appl PROPN iajs-2560 51 5 . . PUNCT iajs-2560 52 1 sci sci PROPN iajs-2560 52 2 . . PROPN iajs-2560 53 1 34 34 NUM iajs-2560 53 2 ( ( PUNCT iajs-2560 53 3 1 1 NUM iajs-2560 53 4 ) ) PUNCT iajs-2560 53 5 2021 2021 NUM iajs-2560 53 6 2Μ… 2Μ… NUM iajs-2560 53 7 βˆ‰ βˆ‰ PROPN iajs-2560 53 8 𝐾 𝐾 PROPN iajs-2560 53 9 = = PUNCT iajs-2560 53 10 〈 〈 PROPN iajs-2560 53 11 0Μ… 0Μ… PROPN iajs-2560 53 12 βŒͺ βŒͺ NOUN iajs-2560 53 13 and and CCONJ iajs-2560 53 14 3 3 NUM iajs-2560 53 15 βˆ‰ βˆ‰ PROPN iajs-2560 53 16 √[〈0Μ… √[〈0Μ… PROPN iajs-2560 53 17 βŒͺ βŒͺ PROPN iajs-2560 53 18 : : PUNCT iajs-2560 53 19 𝑍6 𝑍6 PROPN iajs-2560 53 20 ] ] X iajs-2560 53 21 = = PUNCT iajs-2560 53 22 √6𝑍 √6𝑍 X iajs-2560 53 23 = = PUNCT iajs-2560 53 24 6𝑍. 6𝑍. NUM iajs-2560 54 1 but but CCONJ iajs-2560 54 2 𝐾 𝐾 PROPN iajs-2560 54 3 is be AUX iajs-2560 54 4 a a DET iajs-2560 54 5 nearly nearly ADV iajs-2560 54 6 primary-2 primary-2 NOUN iajs-2560 54 7 - - PUNCT iajs-2560 54 8 absorbing absorb VERB iajs-2560 54 9 submodule submodule NOUN iajs-2560 54 10 of of ADP iajs-2560 54 11 ∁= ∁= PROPN iajs-2560 54 12 𝑍6 𝑍6 PROPN iajs-2560 54 13 . . PUNCT iajs-2560 55 1 for for ADP iajs-2560 55 2 all all DET iajs-2560 55 3 π‘Ž,𝑏 π‘Ž,𝑏 NOUN iajs-2560 55 4 ∈ ∈ PROPN iajs-2560 55 5 𝑅,π‘₯ 𝑅,π‘₯ PUNCT iajs-2560 55 6 ∈ ∈ PROPN iajs-2560 55 7 ∁ ∁ PROPN iajs-2560 55 8 , , PUNCT iajs-2560 55 9 with with ADP iajs-2560 55 10 π‘Žπ‘π‘₯ π‘Žπ‘π‘₯ ADJ iajs-2560 55 11 ∈ ∈ PROPN iajs-2560 55 12 𝐾 𝐾 PROPN iajs-2560 55 13 implies imply VERB iajs-2560 55 14 that that SCONJ iajs-2560 55 15 either either CCONJ iajs-2560 55 16 π‘Žπ‘₯ π‘Žπ‘₯ X iajs-2560 55 17 ∈ ∈ PROPN iajs-2560 55 18 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 55 19 ) ) PUNCT iajs-2560 55 20 + + PUNCT iajs-2560 55 21 𝐽(∁ 𝐽(∁ ADJ iajs-2560 55 22 ) ) PUNCT iajs-2560 55 23 = = PUNCT iajs-2560 55 24 〈 〈 PROPN iajs-2560 55 25 0Μ… 0Μ… NUM iajs-2560 55 26 βŒͺ βŒͺ NOUN iajs-2560 55 27 + + CCONJ iajs-2560 55 28 〈 〈 PROPN iajs-2560 55 29 0 0 NUM iajs-2560 55 30 βŒͺ βŒͺ NOUN iajs-2560 55 31 = = SYM iajs-2560 55 32 〈 〈 PROPN iajs-2560 55 33 0Μ… 0Μ… NUM iajs-2560 55 34 βŒͺ βŒͺ NOUN iajs-2560 55 35 or or CCONJ iajs-2560 55 36 𝑏π‘₯ 𝑏π‘₯ ADP iajs-2560 55 37 ∈ ∈ PROPN iajs-2560 55 38 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 55 39 ) ) PUNCT iajs-2560 55 40 + + PUNCT iajs-2560 55 41 𝐽(∁ 𝐽(∁ ADJ iajs-2560 55 42 ) ) PUNCT iajs-2560 55 43 = = PUNCT iajs-2560 56 1 〈 〈 PROPN iajs-2560 56 2 0Μ… 0Μ… NUM iajs-2560 56 3 βŒͺ βŒͺ NOUN iajs-2560 56 4 + + CCONJ iajs-2560 56 5 〈 〈 PROPN iajs-2560 56 6 0 0 NUM iajs-2560 56 7 βŒͺ βŒͺ NOUN iajs-2560 56 8 = = SYM iajs-2560 56 9 〈 〈 PROPN iajs-2560 56 10 0Μ… 0Μ… NUM iajs-2560 56 11 βŒͺ βŒͺ NOUN iajs-2560 56 12 or or CCONJ iajs-2560 56 13 π‘Žπ‘ π‘Žπ‘ PRON iajs-2560 56 14 ∈ ∈ PROPN iajs-2560 56 15 [ [ X iajs-2560 56 16 〈 〈 NOUN iajs-2560 56 17 0Μ… 0Μ… NUM iajs-2560 56 18 βŒͺ βŒͺ NOUN iajs-2560 56 19 + + CCONJ iajs-2560 56 20 〈 〈 PROPN iajs-2560 56 21 0Μ…βŒͺ:𝑍 0Μ…βŒͺ:𝑍 NOUN iajs-2560 56 22 𝑍6 𝑍6 PROPN iajs-2560 56 23 ] ] X iajs-2560 57 1 = = PUNCT iajs-2560 57 2 6𝑍. 6𝑍. NOUN iajs-2560 57 3 that that PRON iajs-2560 57 4 is be AUX iajs-2560 57 5 2.3 2.3 NUM iajs-2560 57 6 . . PUNCT iajs-2560 58 1 1Μ… 1Μ… NUM iajs-2560 58 2 ∈ ∈ PROPN iajs-2560 58 3 𝐾,where 𝐾,where PUNCT iajs-2560 58 4 2,3 2,3 NUM iajs-2560 58 5 ∈ ∈ PROPN iajs-2560 58 6 𝑅 𝑅 PROPN iajs-2560 58 7 , , PUNCT iajs-2560 58 8 1Μ… 1Μ… PROPN iajs-2560 58 9 ∈ ∈ PROPN iajs-2560 58 10 𝑍6 𝑍6 PROPN iajs-2560 58 11 implies imply VERB iajs-2560 58 12 that that SCONJ iajs-2560 58 13 2 2 X iajs-2560 58 14 . . X iajs-2560 58 15 1Μ… 1Μ… NUM iajs-2560 58 16 = = SYM iajs-2560 58 17 2Μ… 2Μ… NUM iajs-2560 58 18 βˆ‰ βˆ‰ PROPN iajs-2560 58 19 〈 〈 PROPN iajs-2560 58 20 0Μ… 0Μ… PROPN iajs-2560 58 21 βŒͺ βŒͺ NOUN iajs-2560 58 22 but but CCONJ iajs-2560 58 23 2.3 2.3 NUM iajs-2560 58 24 = = SYM iajs-2560 58 25 6 6 NUM iajs-2560 58 26 ∈ ∈ NOUN iajs-2560 58 27 [ [ X iajs-2560 58 28 〈 〈 NOUN iajs-2560 58 29 0Μ… 0Μ… NUM iajs-2560 58 30 βŒͺ βŒͺ NOUN iajs-2560 58 31 + + CCONJ iajs-2560 58 32 〈 〈 PROPN iajs-2560 58 33 0Μ…βŒͺ:𝑍 0Μ…βŒͺ:𝑍 NOUN iajs-2560 58 34 𝑍6 𝑍6 PROPN iajs-2560 58 35 ] ] X iajs-2560 59 1 = = PUNCT iajs-2560 59 2 6𝑍. 6𝑍. NUM iajs-2560 59 3 the the DET iajs-2560 59 4 following following ADJ iajs-2560 59 5 results result NOUN iajs-2560 59 6 are be AUX iajs-2560 59 7 characterizations characterization NOUN iajs-2560 59 8 of of ADP iajs-2560 59 9 a a DET iajs-2560 59 10 nearly nearly ADV iajs-2560 59 11 primary-2 primary-2 NOUN iajs-2560 59 12 - - PUNCT iajs-2560 59 13 absorbing absorbing ADJ iajs-2560 59 14 submodules submodule NOUN iajs-2560 59 15 . . PUNCT iajs-2560 60 1 proposition proposition NOUN iajs-2560 60 2 3 3 NUM iajs-2560 60 3 let let VERB iajs-2560 60 4 ∁ ∁ NOUN iajs-2560 60 5 be be AUX iajs-2560 60 6 an an DET iajs-2560 60 7 𝑅-module 𝑅-module PROPN iajs-2560 60 8 and and CCONJ iajs-2560 60 9 𝐾 𝐾 PROPN iajs-2560 60 10 is be AUX iajs-2560 60 11 a a DET iajs-2560 60 12 proper proper ADJ iajs-2560 60 13 submodule submodule NOUN iajs-2560 60 14 of of ADP iajs-2560 60 15 ∁. ∁. NOUN iajs-2560 60 16 then then ADV iajs-2560 60 17 𝐾 𝐾 PROPN iajs-2560 60 18 is be AUX iajs-2560 60 19 a a DET iajs-2560 60 20 nearly nearly ADV iajs-2560 60 21 primary-2absorbing primary-2absorbe VERB iajs-2560 60 22 submodule submodule NOUN iajs-2560 60 23 of of ADP iajs-2560 60 24 ∁ ∁ PROPN iajs-2560 60 25 if if SCONJ iajs-2560 60 26 and and CCONJ iajs-2560 60 27 only only ADV iajs-2560 60 28 if if SCONJ iajs-2560 60 29 for for ADP iajs-2560 60 30 each each DET iajs-2560 60 31 π‘Ÿ π‘Ÿ NOUN iajs-2560 60 32 , , PUNCT iajs-2560 60 33 𝑠 𝑠 PROPN iajs-2560 60 34 ∈ ∈ PROPN iajs-2560 60 35 𝑅 𝑅 PROPN iajs-2560 60 36 with with ADP iajs-2560 60 37 π‘Ÿπ‘  π‘Ÿπ‘  PROPN iajs-2560 60 38 βˆ‰ βˆ‰ PROPN iajs-2560 61 1 [ [ X iajs-2560 61 2 𝐾 𝐾 PROPN iajs-2560 61 3 + + CCONJ iajs-2560 61 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 61 5 ∁ ∁ NOUN iajs-2560 61 6 ] ] PUNCT iajs-2560 61 7 , , PUNCT iajs-2560 61 8 [ [ X iajs-2560 61 9 𝐾:∁ 𝐾:∁ PROPN iajs-2560 61 10 π‘Ÿπ‘  π‘Ÿπ‘  NUM iajs-2560 61 11 ] ] X iajs-2560 61 12 βŠ† βŠ† NUM iajs-2560 62 1 [ [ X iajs-2560 62 2 π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ PROPN iajs-2560 62 3 π‘Ÿ π‘Ÿ NOUN iajs-2560 62 4 ] ] X iajs-2560 62 5 βˆͺ βˆͺ ADP iajs-2560 62 6 [ [ X iajs-2560 62 7 π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ PROPN iajs-2560 62 8 𝑠 𝑠 PROPN iajs-2560 62 9 ] ] PUNCT iajs-2560 62 10 . . PUNCT iajs-2560 63 1 proof proof NOUN iajs-2560 63 2 : : PUNCT iajs-2560 63 3 ( ( PUNCT iajs-2560 63 4 β‡’ β‡’ NOUN iajs-2560 63 5 ) ) PUNCT iajs-2560 63 6 let let VERB iajs-2560 63 7 π‘₯ π‘₯ PRON iajs-2560 63 8 ∈ ∈ PROPN iajs-2560 63 9 [ [ X iajs-2560 63 10 𝐾:∁ 𝐾:∁ PROPN iajs-2560 63 11 π‘Ÿπ‘  π‘Ÿπ‘  PROPN iajs-2560 63 12 ] ] PUNCT iajs-2560 63 13 , , PUNCT iajs-2560 63 14 where where SCONJ iajs-2560 63 15 π‘Ÿ π‘Ÿ X iajs-2560 63 16 , , PUNCT iajs-2560 63 17 𝑠 𝑠 PROPN iajs-2560 63 18 ∈ ∈ PROPN iajs-2560 63 19 𝑅 𝑅 PROPN iajs-2560 63 20 and and CCONJ iajs-2560 63 21 π‘Ÿπ‘  π‘Ÿπ‘  PROPN iajs-2560 63 22 βˆ‰ βˆ‰ PROPN iajs-2560 64 1 [ [ X iajs-2560 64 2 𝐾 𝐾 PROPN iajs-2560 64 3 + + CCONJ iajs-2560 64 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 64 5 ∁ ∁ NOUN iajs-2560 64 6 ] ] PUNCT iajs-2560 64 7 , , PUNCT iajs-2560 64 8 implies imply VERB iajs-2560 64 9 that that SCONJ iajs-2560 64 10 π‘Ÿπ‘ π‘₯ π‘Ÿπ‘ π‘₯ NUM iajs-2560 64 11 ∈ ∈ PROPN iajs-2560 64 12 𝐾. 𝐾. PROPN iajs-2560 64 13 but but CCONJ iajs-2560 64 14 𝐾 𝐾 PROPN iajs-2560 64 15 is be AUX iajs-2560 64 16 a a DET iajs-2560 64 17 nearly nearly ADV iajs-2560 64 18 primary-2 primary-2 NOUN iajs-2560 64 19 - - PUNCT iajs-2560 64 20 absorbing absorb VERB iajs-2560 64 21 submodule submodule NOUN iajs-2560 64 22 of of ADP iajs-2560 64 23 ∁ ∁ PROPN iajs-2560 64 24 , , PUNCT iajs-2560 64 25 and and CCONJ iajs-2560 64 26 π‘Ÿπ‘  π‘Ÿπ‘  PROPN iajs-2560 64 27 βˆ‰ βˆ‰ PROPN iajs-2560 65 1 [ [ X iajs-2560 65 2 𝐾 𝐾 PROPN iajs-2560 65 3 + + CCONJ iajs-2560 65 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 65 5 ∁ ∁ NOUN iajs-2560 65 6 ] ] NUM iajs-2560 65 7 , , PUNCT iajs-2560 65 8 then then ADV iajs-2560 65 9 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PROPN iajs-2560 65 10 ∈ ∈ PROPN iajs-2560 65 11 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 65 12 ) ) PUNCT iajs-2560 65 13 + + PUNCT iajs-2560 65 14 𝐽(∁ 𝐽(∁ ADJ iajs-2560 65 15 ) ) PUNCT iajs-2560 65 16 or or CCONJ iajs-2560 65 17 𝑠π‘₯ 𝑠π‘₯ ADP iajs-2560 65 18 ∈ ∈ PROPN iajs-2560 65 19 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 65 20 ) ) PUNCT iajs-2560 65 21 + + CCONJ iajs-2560 65 22 𝐽(∁ 𝐽(∁ ADJ iajs-2560 65 23 ) ) PUNCT iajs-2560 65 24 . . PUNCT iajs-2560 66 1 that that PRON iajs-2560 66 2 is be AUX iajs-2560 66 3 either either CCONJ iajs-2560 66 4 π‘₯ π‘₯ DET iajs-2560 66 5 ∈ ∈ PROPN iajs-2560 66 6 [ [ X iajs-2560 66 7 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 66 8 ) ) PUNCT iajs-2560 67 1 + + CCONJ iajs-2560 67 2 𝐽(∁):∁ 𝐽(∁):∁ PRON iajs-2560 67 3 π‘Ÿ π‘Ÿ NOUN iajs-2560 67 4 ] ] PUNCT iajs-2560 67 5 or or CCONJ iajs-2560 67 6 π‘₯ π‘₯ PRON iajs-2560 67 7 ∈ ∈ PROPN iajs-2560 68 1 [ [ X iajs-2560 68 2 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 68 3 ) ) PUNCT iajs-2560 68 4 + + CCONJ iajs-2560 68 5 𝐽(∁):∁ 𝐽(∁):∁ PRON iajs-2560 68 6 𝑠 𝑠 X iajs-2560 68 7 ] ] PUNCT iajs-2560 68 8 , , PUNCT iajs-2560 68 9 thus thus ADV iajs-2560 68 10 π‘₯ π‘₯ DET iajs-2560 68 11 ∈ ∈ NOUN iajs-2560 68 12 [ [ X iajs-2560 68 13 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 68 14 ) ) PUNCT iajs-2560 69 1 + + CCONJ iajs-2560 69 2 𝐽(∁):∁ 𝐽(∁):∁ PRON iajs-2560 69 3 π‘Ÿ π‘Ÿ NOUN iajs-2560 69 4 ] ] X iajs-2560 69 5 βˆͺ βˆͺ ADP iajs-2560 69 6 [ [ X iajs-2560 69 7 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 69 8 ) ) PUNCT iajs-2560 70 1 + + CCONJ iajs-2560 70 2 𝑠𝐽(∁):∁ 𝑠𝐽(∁):∁ PROPN iajs-2560 70 3 𝑠 𝑠 NOUN iajs-2560 70 4 ] ] PUNCT iajs-2560 70 5 . . PUNCT iajs-2560 71 1 hence hence ADV iajs-2560 71 2 , , PUNCT iajs-2560 72 1 [ [ X iajs-2560 72 2 𝐾:∁ 𝐾:∁ PROPN iajs-2560 72 3 π‘Ÿπ‘  π‘Ÿπ‘  NUM iajs-2560 72 4 ] ] X iajs-2560 72 5 βŠ† βŠ† NUM iajs-2560 72 6 [ [ X iajs-2560 72 7 π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ PROPN iajs-2560 72 8 π‘Ÿ π‘Ÿ NOUN iajs-2560 72 9 ] ] X iajs-2560 72 10 βˆͺ βˆͺ ADP iajs-2560 72 11 [ [ X iajs-2560 72 12 π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ PROPN iajs-2560 72 13 𝑠 𝑠 PROPN iajs-2560 72 14 ] ] X iajs-2560 72 15 . . PUNCT iajs-2560 73 1 ( ( PUNCT iajs-2560 73 2 ⇐ ⇐ ADJ iajs-2560 73 3 ) ) PUNCT iajs-2560 73 4 let let VERB iajs-2560 73 5 π‘Ÿπ‘ π‘₯ π‘Ÿπ‘ π‘₯ NUM iajs-2560 73 6 ∈ ∈ PROPN iajs-2560 73 7 𝐾 𝐾 PROPN iajs-2560 73 8 , , PUNCT iajs-2560 73 9 where where SCONJ iajs-2560 73 10 π‘₯ π‘₯ DET iajs-2560 73 11 ∈ ∈ PROPN iajs-2560 73 12 ∁ ∁ PROPN iajs-2560 73 13 and and CCONJ iajs-2560 73 14 π‘Ÿ π‘Ÿ NOUN iajs-2560 73 15 , , PUNCT iajs-2560 73 16 𝑠 𝑠 PROPN iajs-2560 73 17 ∈ ∈ PROPN iajs-2560 73 18 𝑅 𝑅 PROPN iajs-2560 73 19 with with ADP iajs-2560 73 20 π‘Ÿπ‘  π‘Ÿπ‘  PROPN iajs-2560 73 21 βˆ‰ βˆ‰ PROPN iajs-2560 74 1 [ [ X iajs-2560 74 2 𝐾 𝐾 PROPN iajs-2560 74 3 + + CCONJ iajs-2560 74 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 74 5 ∁ ∁ NOUN iajs-2560 74 6 ] ] PUNCT iajs-2560 74 7 . . PUNCT iajs-2560 75 1 it it PRON iajs-2560 75 2 follows follow VERB iajs-2560 75 3 that that SCONJ iajs-2560 75 4 π‘₯ π‘₯ PRON iajs-2560 75 5 ∈ ∈ PROPN iajs-2560 76 1 [ [ X iajs-2560 76 2 𝐾:∁ 𝐾:∁ PROPN iajs-2560 76 3 π‘Ÿπ‘  π‘Ÿπ‘  PROPN iajs-2560 76 4 ] ] PUNCT iajs-2560 76 5 , , PUNCT iajs-2560 76 6 by by ADP iajs-2560 76 7 hypothesis hypothesis NOUN iajs-2560 76 8 π‘₯ π‘₯ X iajs-2560 76 9 ∈ ∈ PROPN iajs-2560 77 1 [ [ X iajs-2560 77 2 π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ PROPN iajs-2560 77 3 π‘Ÿ π‘Ÿ NOUN iajs-2560 77 4 ] ] X iajs-2560 77 5 βˆͺ βˆͺ ADP iajs-2560 77 6 [ [ X iajs-2560 77 7 π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ PROPN iajs-2560 77 8 𝑠 𝑠 PROPN iajs-2560 77 9 ] ] PUNCT iajs-2560 77 10 . . PUNCT iajs-2560 78 1 hence hence ADV iajs-2560 78 2 π‘₯ π‘₯ PRON iajs-2560 78 3 ∈ ∈ PROPN iajs-2560 79 1 [ [ X iajs-2560 79 2 π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ PROPN iajs-2560 79 3 π‘Ÿ π‘Ÿ NOUN iajs-2560 79 4 ] ] X iajs-2560 79 5 or or CCONJ iajs-2560 79 6 π‘₯ π‘₯ DET iajs-2560 79 7 ∈ ∈ PROPN iajs-2560 79 8 [ [ X iajs-2560 79 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐾):∁ PROPN iajs-2560 79 10 𝑠 𝑠 PROPN iajs-2560 79 11 ] ] X iajs-2560 79 12 . . PUNCT iajs-2560 80 1 therefore therefore ADV iajs-2560 80 2 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PROPN iajs-2560 80 3 ∈ ∈ PROPN iajs-2560 80 4 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 PART iajs-2560 80 5 ) ) PUNCT iajs-2560 80 6 + + PUNCT iajs-2560 80 7 𝐽(∁ 𝐽(∁ ADJ iajs-2560 80 8 ) ) PUNCT iajs-2560 80 9 or or CCONJ iajs-2560 80 10 𝑠π‘₯ 𝑠π‘₯ ADP iajs-2560 80 11 ∈ ∈ PROPN iajs-2560 80 12 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 80 13 ) ) PUNCT iajs-2560 80 14 + + PUNCT iajs-2560 80 15 𝐽(∁ 𝐽(∁ ADJ iajs-2560 80 16 ) ) PUNCT iajs-2560 80 17 , , PUNCT iajs-2560 80 18 that that PRON iajs-2560 80 19 is is ADV iajs-2560 80 20 𝐾 𝐾 PROPN iajs-2560 80 21 is be AUX iajs-2560 80 22 a a DET iajs-2560 80 23 nearly nearly ADV iajs-2560 80 24 primary-2 primary-2 NOUN iajs-2560 80 25 - - PUNCT iajs-2560 80 26 absorbing absorbing ADJ iajs-2560 80 27 submodule submodule NOUN iajs-2560 80 28 of of ADP iajs-2560 80 29 ∁. ∁. NOUN iajs-2560 80 30 proposition proposition NOUN iajs-2560 80 31 4 4 NUM iajs-2560 80 32 let let VERB iajs-2560 80 33 ∁ ∁ NOUN iajs-2560 80 34 be be AUX iajs-2560 80 35 an an DET iajs-2560 80 36 𝑅-module 𝑅-module PROPN iajs-2560 80 37 and and CCONJ iajs-2560 80 38 𝐿 𝐿 PROPN iajs-2560 80 39 be be AUX iajs-2560 80 40 a a DET iajs-2560 80 41 proper proper ADJ iajs-2560 80 42 submodule submodule NOUN iajs-2560 80 43 of of ADP iajs-2560 80 44 ∁ ∁ PROPN iajs-2560 80 45 . . PUNCT iajs-2560 81 1 then then ADV iajs-2560 81 2 𝐿 𝐿 PROPN iajs-2560 81 3 is be AUX iajs-2560 81 4 a a DET iajs-2560 81 5 nearly nearly ADV iajs-2560 81 6 primary-2absorbing primary-2absorbe VERB iajs-2560 81 7 submodule submodule NOUN iajs-2560 81 8 of of ADP iajs-2560 81 9 ∁ ∁ PROPN iajs-2560 81 10 if if SCONJ iajs-2560 81 11 and and CCONJ iajs-2560 81 12 only only ADV iajs-2560 81 13 if if SCONJ iajs-2560 81 14 π‘Žπ‘πΎ π‘Žπ‘πΎ PROPN iajs-2560 81 15 βŠ† βŠ† NUM iajs-2560 81 16 𝐿 𝐿 PROPN iajs-2560 81 17 for for ADP iajs-2560 81 18 π‘Ž π‘Ž NOUN iajs-2560 81 19 , , PUNCT iajs-2560 81 20 𝑏 𝑏 PROPN iajs-2560 81 21 ∈ ∈ PROPN iajs-2560 81 22 𝑅 𝑅 PROPN iajs-2560 81 23 and and CCONJ iajs-2560 81 24 𝐾 𝐾 PROPN iajs-2560 81 25 is be AUX iajs-2560 81 26 a a DET iajs-2560 81 27 submodule submodule NOUN iajs-2560 81 28 of of ADP iajs-2560 81 29 ∁ ∁ PROPN iajs-2560 81 30 , , PUNCT iajs-2560 81 31 with with ADP iajs-2560 81 32 π‘Žπ‘ π‘Žπ‘ PROPN iajs-2560 81 33 βˆ‰ βˆ‰ PROPN iajs-2560 82 1 [ [ X iajs-2560 82 2 𝐿 𝐿 PROPN iajs-2560 82 3 + + PROPN iajs-2560 82 4 𝐽 𝐽 PROPN iajs-2560 82 5 ( ( PUNCT iajs-2560 82 6 ∁ ∁ PROPN iajs-2560 82 7 ): ): PUNCT iajs-2560 82 8 𝑅 𝑅 PROPN iajs-2560 82 9 ∁ ∁ PROPN iajs-2560 82 10 ] ] PUNCT iajs-2560 82 11 , , PUNCT iajs-2560 82 12 implies imply VERB iajs-2560 82 13 that that SCONJ iajs-2560 82 14 π‘ŽπΎ π‘ŽπΎ NOUN iajs-2560 82 15 βŠ† βŠ† NUM iajs-2560 82 16 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 82 17 ) ) PUNCT iajs-2560 83 1 + + CCONJ iajs-2560 83 2 𝐽 𝐽 PROPN iajs-2560 83 3 ( ( PUNCT iajs-2560 83 4 ∁ ∁ PROPN iajs-2560 83 5 ) ) PUNCT iajs-2560 83 6 or or CCONJ iajs-2560 83 7 𝑏𝐾 𝑏𝐾 PROPN iajs-2560 83 8 βŠ† βŠ† NUM iajs-2560 83 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 83 10 ) ) PUNCT iajs-2560 84 1 + + CCONJ iajs-2560 84 2 𝐽 𝐽 PROPN iajs-2560 84 3 ( ( PUNCT iajs-2560 84 4 ∁ ∁ PROPN iajs-2560 84 5 ) ) PUNCT iajs-2560 84 6 . . PUNCT iajs-2560 85 1 proof proof NOUN iajs-2560 85 2 : : PUNCT iajs-2560 85 3 ( ( PUNCT iajs-2560 85 4 β‡’ β‡’ NOUN iajs-2560 85 5 ) ) PUNCT iajs-2560 85 6 let let VERB iajs-2560 85 7 𝐿 𝐿 PROPN iajs-2560 85 8 be be AUX iajs-2560 85 9 a a DET iajs-2560 85 10 nearly nearly ADV iajs-2560 85 11 primary-2 primary-2 NOUN iajs-2560 85 12 - - PUNCT iajs-2560 85 13 absorbing absorb VERB iajs-2560 85 14 submodule submodule NOUN iajs-2560 85 15 of of ADP iajs-2560 85 16 ∁ ∁ PROPN iajs-2560 85 17 , , PUNCT iajs-2560 85 18 π‘Žπ‘πΎ π‘Žπ‘πΎ PROPN iajs-2560 85 19 βŠ† βŠ† NUM iajs-2560 85 20 𝐿 𝐿 PROPN iajs-2560 85 21 with with ADP iajs-2560 85 22 π‘Ÿ π‘Ÿ NOUN iajs-2560 85 23 , , PUNCT iajs-2560 85 24 𝑠 𝑠 PROPN iajs-2560 85 25 ∈ ∈ PROPN iajs-2560 85 26 𝑅 𝑅 PROPN iajs-2560 85 27 and and CCONJ iajs-2560 85 28 𝐾 𝐾 PROPN iajs-2560 85 29 is be AUX iajs-2560 85 30 a a DET iajs-2560 85 31 submodule submodule NOUN iajs-2560 85 32 of of ADP iajs-2560 85 33 ∁ ∁ PROPN iajs-2560 85 34 with with ADP iajs-2560 85 35 π‘Žπ‘ π‘Žπ‘ PROPN iajs-2560 85 36 βˆ‰ βˆ‰ PROPN iajs-2560 86 1 [ [ X iajs-2560 86 2 𝐿 𝐿 PROPN iajs-2560 86 3 + + PROPN iajs-2560 86 4 𝐽 𝐽 PROPN iajs-2560 86 5 ( ( PUNCT iajs-2560 86 6 ∁ ∁ PROPN iajs-2560 86 7 ): ): PUNCT iajs-2560 86 8 𝑅 𝑅 PROPN iajs-2560 86 9 ∁ ∁ PROPN iajs-2560 86 10 ] ] PUNCT iajs-2560 86 11 . . PUNCT iajs-2560 87 1 assume assume VERB iajs-2560 87 2 that that SCONJ iajs-2560 87 3 π‘ŽπΎ π‘ŽπΎ NOUN iajs-2560 87 4 ⊈ ⊈ PROPN iajs-2560 87 5 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 87 6 ) ) PUNCT iajs-2560 88 1 + + CCONJ iajs-2560 88 2 𝐽 𝐽 PROPN iajs-2560 88 3 ( ( PUNCT iajs-2560 88 4 ∁ ∁ PROPN iajs-2560 88 5 ) ) PUNCT iajs-2560 88 6 and and CCONJ iajs-2560 88 7 𝑏𝐾 𝑏𝐾 PROPN iajs-2560 88 8 ⊈ ⊈ PROPN iajs-2560 88 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 88 10 ) ) PUNCT iajs-2560 89 1 + + CCONJ iajs-2560 89 2 𝐽 𝐽 PROPN iajs-2560 89 3 ( ( PUNCT iajs-2560 89 4 ∁ ∁ PROPN iajs-2560 89 5 ) ) PUNCT iajs-2560 89 6 , , PUNCT iajs-2560 89 7 then then ADV iajs-2560 89 8 π‘Žπ‘˜1 π‘Žπ‘˜1 PROPN iajs-2560 89 9 βˆ‰ βˆ‰ PROPN iajs-2560 89 10 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 89 11 ) ) PUNCT iajs-2560 90 1 + + CCONJ iajs-2560 90 2 𝐽 𝐽 PROPN iajs-2560 90 3 ( ( PUNCT iajs-2560 90 4 ∁ ∁ PROPN iajs-2560 90 5 ) ) PUNCT iajs-2560 90 6 and and CCONJ iajs-2560 90 7 π‘π‘˜2 π‘π‘˜2 PROPN iajs-2560 90 8 βˆ‰ βˆ‰ PROPN iajs-2560 90 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 PROPN iajs-2560 90 10 ) ) PUNCT iajs-2560 91 1 + + CCONJ iajs-2560 91 2 𝐽 𝐽 PROPN iajs-2560 91 3 ( ( PUNCT iajs-2560 91 4 ∁ ∁ PROPN iajs-2560 91 5 ) ) PUNCT iajs-2560 91 6 for for ADP iajs-2560 91 7 some some DET iajs-2560 91 8 π‘˜1 π‘˜1 NOUN iajs-2560 91 9 , , PUNCT iajs-2560 91 10 π‘˜2 π‘˜2 PROPN iajs-2560 91 11 ∈ ∈ PROPN iajs-2560 91 12 𝐾. 𝐾. PROPN iajs-2560 91 13 now now ADV iajs-2560 91 14 we we PRON iajs-2560 91 15 have have VERB iajs-2560 91 16 π‘Žπ‘π‘˜1 π‘Žπ‘π‘˜1 PROPN iajs-2560 91 17 ∈ ∈ PROPN iajs-2560 91 18 𝐿 𝐿 PROPN iajs-2560 91 19 but but CCONJ iajs-2560 91 20 𝐿 𝐿 PROPN iajs-2560 91 21 is be AUX iajs-2560 91 22 a a DET iajs-2560 91 23 nearly nearly ADV iajs-2560 91 24 primary-2 primary-2 NOUN iajs-2560 91 25 - - PUNCT iajs-2560 91 26 absorbing absorb VERB iajs-2560 91 27 submodule submodule NOUN iajs-2560 91 28 of of ADP iajs-2560 91 29 ∁ ∁ PROPN iajs-2560 91 30 and and CCONJ iajs-2560 91 31 π‘Žπ‘ π‘Žπ‘ PROPN iajs-2560 91 32 βˆ‰ βˆ‰ PROPN iajs-2560 92 1 [ [ X iajs-2560 92 2 𝐿 𝐿 PROPN iajs-2560 92 3 + + PROPN iajs-2560 92 4 𝐽 𝐽 PROPN iajs-2560 92 5 ( ( PUNCT iajs-2560 92 6 ∁ ∁ PROPN iajs-2560 92 7 ): ): PUNCT iajs-2560 92 8 𝑅 𝑅 PROPN iajs-2560 92 9 ∁ ∁ PROPN iajs-2560 92 10 ] ] PUNCT iajs-2560 92 11 and and CCONJ iajs-2560 92 12 π‘Žπ‘˜1 π‘Žπ‘˜1 PROPN iajs-2560 92 13 βˆ‰ βˆ‰ PROPN iajs-2560 92 14 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 92 15 ) ) PUNCT iajs-2560 92 16 + + CCONJ iajs-2560 92 17 𝐽 𝐽 PROPN iajs-2560 92 18 ( ( PUNCT iajs-2560 92 19 ∁ ∁ PROPN iajs-2560 92 20 ) ) PUNCT iajs-2560 92 21 , , PUNCT iajs-2560 92 22 then then ADV iajs-2560 92 23 π‘π‘˜1 π‘π‘˜1 PROPN iajs-2560 92 24 ∈ ∈ PROPN iajs-2560 92 25 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 92 26 ) ) PUNCT iajs-2560 93 1 + + CCONJ iajs-2560 93 2 𝐽 𝐽 PROPN iajs-2560 93 3 ( ( PUNCT iajs-2560 93 4 ∁ ∁ PROPN iajs-2560 93 5 ) ) PUNCT iajs-2560 93 6 . . PUNCT iajs-2560 94 1 also also ADV iajs-2560 94 2 , , PUNCT iajs-2560 94 3 since since SCONJ iajs-2560 94 4 π‘Žπ‘π‘˜2 π‘Žπ‘π‘˜2 PROPN iajs-2560 94 5 ∈ ∈ PROPN iajs-2560 94 6 𝐿 𝐿 PROPN iajs-2560 94 7 and and CCONJ iajs-2560 94 8 π‘Žπ‘ π‘Žπ‘ PROPN iajs-2560 94 9 βˆ‰ βˆ‰ PROPN iajs-2560 95 1 [ [ X iajs-2560 95 2 𝐿 𝐿 PROPN iajs-2560 95 3 + + PROPN iajs-2560 95 4 𝐽 𝐽 PROPN iajs-2560 95 5 ( ( PUNCT iajs-2560 95 6 ∁ ∁ PROPN iajs-2560 95 7 ): ): PUNCT iajs-2560 95 8 𝑅 𝑅 PROPN iajs-2560 95 9 ∁ ∁ PROPN iajs-2560 95 10 ] ] PUNCT iajs-2560 95 11 and and CCONJ iajs-2560 95 12 π‘π‘˜2 π‘π‘˜2 PROPN iajs-2560 95 13 βˆ‰ βˆ‰ PROPN iajs-2560 95 14 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 PROPN iajs-2560 95 15 ) ) PUNCT iajs-2560 95 16 + + CCONJ iajs-2560 95 17 𝐽 𝐽 PROPN iajs-2560 95 18 ( ( PUNCT iajs-2560 95 19 ∁ ∁ PROPN iajs-2560 95 20 ) ) PUNCT iajs-2560 95 21 , , PUNCT iajs-2560 95 22 then then ADV iajs-2560 95 23 π‘Žπ‘˜2 π‘Žπ‘˜2 PROPN iajs-2560 95 24 ∈ ∈ PROPN iajs-2560 95 25 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 95 26 ) ) PUNCT iajs-2560 96 1 + + CCONJ iajs-2560 96 2 𝐽 𝐽 PROPN iajs-2560 96 3 ( ( PUNCT iajs-2560 96 4 ∁ ∁ PROPN iajs-2560 96 5 ) ) PUNCT iajs-2560 96 6 . . PUNCT iajs-2560 97 1 again again ADV iajs-2560 97 2 since since SCONJ iajs-2560 97 3 π‘Žπ‘(π‘˜1 π‘Žπ‘(π‘˜1 NOUN iajs-2560 97 4 + + CCONJ iajs-2560 97 5 π‘˜2 π‘˜2 PROPN iajs-2560 97 6 ) ) PUNCT iajs-2560 97 7 ∈ ∈ PROPN iajs-2560 97 8 𝐿 𝐿 PROPN iajs-2560 97 9 and and CCONJ iajs-2560 97 10 π‘Žπ‘ π‘Žπ‘ PROPN iajs-2560 97 11 βˆ‰ βˆ‰ PROPN iajs-2560 97 12 [ [ X iajs-2560 97 13 𝐿 𝐿 PROPN iajs-2560 97 14 + + PROPN iajs-2560 97 15 𝐽 𝐽 PROPN iajs-2560 97 16 ( ( PUNCT iajs-2560 97 17 ∁ ∁ PROPN iajs-2560 97 18 ): ): PUNCT iajs-2560 97 19 𝑅 𝑅 PROPN iajs-2560 97 20 ∁ ∁ PROPN iajs-2560 97 21 ] ] X iajs-2560 98 1 we we PRON iajs-2560 98 2 have have VERB iajs-2560 98 3 π‘Ž(π‘˜1 π‘Ž(π‘˜1 NOUN iajs-2560 98 4 + + CCONJ iajs-2560 98 5 π‘˜2 π‘˜2 ADJ iajs-2560 98 6 ) ) PUNCT iajs-2560 98 7 ∈ ∈ PROPN iajs-2560 98 8 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 98 9 ) ) PUNCT iajs-2560 99 1 + + CCONJ iajs-2560 99 2 𝐽 𝐽 PROPN iajs-2560 99 3 ( ( PUNCT iajs-2560 99 4 ∁ ∁ PROPN iajs-2560 99 5 ) ) PUNCT iajs-2560 99 6 or or CCONJ iajs-2560 99 7 𝑏(π‘˜1 𝑏(π‘˜1 NOUN iajs-2560 99 8 + + CCONJ iajs-2560 99 9 π‘˜2 π‘˜2 ADJ iajs-2560 99 10 ) ) PUNCT iajs-2560 99 11 ∈ ∈ PROPN iajs-2560 99 12 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 99 13 ) ) PUNCT iajs-2560 100 1 + + CCONJ iajs-2560 100 2 𝐽 𝐽 PROPN iajs-2560 100 3 ( ( PUNCT iajs-2560 100 4 ∁ ∁ PROPN iajs-2560 100 5 ) ) PUNCT iajs-2560 100 6 . . PUNCT iajs-2560 101 1 suppose suppose VERB iajs-2560 101 2 that that SCONJ iajs-2560 101 3 π‘Ž(π‘˜1 π‘Ž(π‘˜1 NOUN iajs-2560 101 4 + + X iajs-2560 101 5 π‘˜2 π‘˜2 PROPN iajs-2560 101 6 ) ) PUNCT iajs-2560 102 1 = = VERB iajs-2560 102 2 π‘Žπ‘˜1 π‘Žπ‘˜1 PROPN iajs-2560 102 3 + + CCONJ iajs-2560 102 4 π‘Žπ‘˜2 π‘Žπ‘˜2 PROPN iajs-2560 102 5 ∈ ∈ PROPN iajs-2560 102 6 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 102 7 ) ) PUNCT iajs-2560 103 1 + + CCONJ iajs-2560 103 2 𝐽 𝐽 PROPN iajs-2560 103 3 ( ( PUNCT iajs-2560 103 4 ∁ ∁ PROPN iajs-2560 103 5 ) ) PUNCT iajs-2560 103 6 , , PUNCT iajs-2560 103 7 but but CCONJ iajs-2560 103 8 π‘Žπ‘˜2 π‘Žπ‘˜2 PROPN iajs-2560 103 9 ∈ ∈ PROPN iajs-2560 103 10 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 103 11 ) ) PUNCT iajs-2560 104 1 + + CCONJ iajs-2560 104 2 𝐽 𝐽 PROPN iajs-2560 104 3 ( ( PUNCT iajs-2560 104 4 ∁ ∁ PROPN iajs-2560 104 5 ) ) PUNCT iajs-2560 104 6 , , PUNCT iajs-2560 104 7 it it PRON iajs-2560 104 8 follows follow VERB iajs-2560 104 9 that that SCONJ iajs-2560 104 10 π‘Žπ‘˜1 π‘Žπ‘˜1 PROPN iajs-2560 104 11 ∈ ∈ PROPN iajs-2560 104 12 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 104 13 ) ) PUNCT iajs-2560 105 1 + + CCONJ iajs-2560 105 2 𝐽 𝐽 PROPN iajs-2560 105 3 ( ( PUNCT iajs-2560 105 4 ∁ ∁ PROPN iajs-2560 105 5 ) ) PUNCT iajs-2560 105 6 a a DET iajs-2560 105 7 contradiction contradiction NOUN iajs-2560 105 8 . . PUNCT iajs-2560 106 1 suppose suppose VERB iajs-2560 106 2 that that SCONJ iajs-2560 106 3 𝑏(π‘˜1 𝑏(π‘˜1 PROPN iajs-2560 106 4 + + SYM iajs-2560 106 5 π‘˜2 π‘˜2 NOUN iajs-2560 106 6 ) ) PUNCT iajs-2560 106 7 = = SYM iajs-2560 107 1 π‘π‘˜1 π‘π‘˜1 NOUN iajs-2560 107 2 + + CCONJ iajs-2560 107 3 π‘π‘˜2 π‘π‘˜2 PROPN iajs-2560 107 4 ∈ ∈ PROPN iajs-2560 107 5 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 107 6 ) ) PUNCT iajs-2560 108 1 + + CCONJ iajs-2560 108 2 𝐽 𝐽 PROPN iajs-2560 108 3 ( ( PUNCT iajs-2560 108 4 ∁ ∁ PROPN iajs-2560 108 5 ) ) PUNCT iajs-2560 108 6 , , PUNCT iajs-2560 108 7 but but CCONJ iajs-2560 108 8 π‘π‘˜1 π‘π‘˜1 PROPN iajs-2560 108 9 ∈ ∈ PROPN iajs-2560 108 10 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 108 11 ) ) PUNCT iajs-2560 109 1 + + CCONJ iajs-2560 109 2 𝐽 𝐽 PROPN iajs-2560 109 3 ( ( PUNCT iajs-2560 109 4 ∁ ∁ PROPN iajs-2560 109 5 ) ) PUNCT iajs-2560 109 6 , , PUNCT iajs-2560 109 7 we we PRON iajs-2560 109 8 have have VERB iajs-2560 109 9 π‘π‘˜2 π‘π‘˜2 PROPN iajs-2560 109 10 ∈ ∈ NOUN iajs-2560 109 11 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 109 12 ) ) PUNCT iajs-2560 110 1 + + CCONJ iajs-2560 110 2 𝐽 𝐽 PROPN iajs-2560 110 3 ( ( PUNCT iajs-2560 110 4 ∁ ∁ PROPN iajs-2560 110 5 ) ) PUNCT iajs-2560 110 6 a a DET iajs-2560 110 7 contradiction contradiction NOUN iajs-2560 110 8 . . PUNCT iajs-2560 111 1 hence hence ADV iajs-2560 111 2 π‘ŽπΎ π‘ŽπΎ VERB iajs-2560 111 3 βŠ† βŠ† NUM iajs-2560 111 4 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 111 5 ) ) PUNCT iajs-2560 112 1 + + CCONJ iajs-2560 112 2 𝐽 𝐽 PROPN iajs-2560 112 3 ( ( PUNCT iajs-2560 112 4 ∁ ∁ PROPN iajs-2560 112 5 ) ) PUNCT iajs-2560 112 6 or or CCONJ iajs-2560 112 7 𝑏𝐾 𝑏𝐾 PROPN iajs-2560 112 8 βŠ† βŠ† NUM iajs-2560 112 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 112 10 ) ) PUNCT iajs-2560 113 1 + + CCONJ iajs-2560 113 2 𝐽 𝐽 PROPN iajs-2560 113 3 ( ( PUNCT iajs-2560 113 4 ∁ ∁ PROPN iajs-2560 113 5 ) ) PUNCT iajs-2560 113 6 . . PUNCT iajs-2560 114 1 ( ( PUNCT iajs-2560 114 2 ⇐ ⇐ NOUN iajs-2560 114 3 ) ) PUNCT iajs-2560 114 4 let let VERB iajs-2560 114 5 π‘Žπ‘π‘₯ π‘Žπ‘π‘₯ NOUN iajs-2560 114 6 ∈ ∈ PROPN iajs-2560 114 7 𝐿 𝐿 PROPN iajs-2560 114 8 , , PUNCT iajs-2560 114 9 where where SCONJ iajs-2560 114 10 π‘₯ π‘₯ DET iajs-2560 114 11 ∈ ∈ PROPN iajs-2560 114 12 ∁ ∁ PROPN iajs-2560 114 13 and and CCONJ iajs-2560 114 14 π‘Ž π‘Ž NOUN iajs-2560 114 15 , , PUNCT iajs-2560 114 16 𝑏 𝑏 PROPN iajs-2560 114 17 ∈ ∈ PROPN iajs-2560 114 18 𝑅 𝑅 PROPN iajs-2560 114 19 with with ADP iajs-2560 114 20 π‘Žπ‘ π‘Žπ‘ PROPN iajs-2560 114 21 βˆ‰ βˆ‰ PROPN iajs-2560 114 22 [ [ X iajs-2560 114 23 𝐿 𝐿 PROPN iajs-2560 114 24 + + PROPN iajs-2560 114 25 𝐽 𝐽 PROPN iajs-2560 114 26 ( ( PUNCT iajs-2560 114 27 ∁ ∁ PROPN iajs-2560 114 28 ): ): PUNCT iajs-2560 114 29 𝑅 𝑅 PROPN iajs-2560 114 30 ∁ ∁ PROPN iajs-2560 114 31 ] ] PUNCT iajs-2560 114 32 . . PUNCT iajs-2560 115 1 so so ADV iajs-2560 115 2 that that PRON iajs-2560 115 3 π‘Žπ‘(π‘₯ π‘Žπ‘(π‘₯ NOUN iajs-2560 115 4 ) ) PUNCT iajs-2560 115 5 βŠ† βŠ† X iajs-2560 115 6 𝐿 𝐿 PROPN iajs-2560 115 7 , , PUNCT iajs-2560 115 8 it it PRON iajs-2560 115 9 follows follow VERB iajs-2560 115 10 by by ADP iajs-2560 115 11 hypothesis hypothesis NOUN iajs-2560 115 12 π‘Ž(π‘₯ π‘Ž(π‘₯ PROPN iajs-2560 115 13 ) ) PUNCT iajs-2560 115 14 βŠ† βŠ† NUM iajs-2560 115 15 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 115 16 ) ) PUNCT iajs-2560 116 1 + + CCONJ iajs-2560 116 2 𝐽 𝐽 PROPN iajs-2560 116 3 ( ( PUNCT iajs-2560 116 4 ∁ ∁ PROPN iajs-2560 116 5 ) ) PUNCT iajs-2560 116 6 or or CCONJ iajs-2560 116 7 𝑏(π‘₯ 𝑏(π‘₯ NOUN iajs-2560 116 8 ) ) PUNCT iajs-2560 116 9 βŠ† βŠ† NUM iajs-2560 116 10 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 116 11 ) ) PUNCT iajs-2560 117 1 + + CCONJ iajs-2560 117 2 𝐽 𝐽 PROPN iajs-2560 117 3 ( ( PUNCT iajs-2560 117 4 ∁ ∁ PROPN iajs-2560 117 5 ) ) PUNCT iajs-2560 117 6 . . PUNCT iajs-2560 118 1 that that PRON iajs-2560 118 2 is be AUX iajs-2560 118 3 π‘Žπ‘₯ π‘Žπ‘₯ X iajs-2560 118 4 ∈ ∈ NOUN iajs-2560 118 5 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 118 6 ) ) PUNCT iajs-2560 119 1 + + CCONJ iajs-2560 119 2 𝐽 𝐽 PROPN iajs-2560 119 3 ( ( PUNCT iajs-2560 119 4 ∁ ∁ PROPN iajs-2560 119 5 ) ) PUNCT iajs-2560 119 6 or or CCONJ iajs-2560 119 7 𝑏π‘₯ 𝑏π‘₯ PROPN iajs-2560 119 8 ∈ ∈ PROPN iajs-2560 119 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 119 10 ) ) PUNCT iajs-2560 120 1 + + CCONJ iajs-2560 120 2 𝐽 𝐽 PROPN iajs-2560 120 3 ( ( PUNCT iajs-2560 120 4 ∁ ∁ PROPN iajs-2560 120 5 ) ) PUNCT iajs-2560 120 6 . . PUNCT iajs-2560 121 1 hence hence ADV iajs-2560 121 2 𝐿 𝐿 PROPN iajs-2560 121 3 is be AUX iajs-2560 121 4 a a DET iajs-2560 121 5 nearly nearly ADV iajs-2560 121 6 primary-2 primary-2 NOUN iajs-2560 121 7 - - PUNCT iajs-2560 121 8 absorbing absorb VERB iajs-2560 121 9 submodule submodule NOUN iajs-2560 121 10 of of ADP iajs-2560 121 11 ∁. ∁. PROPN iajs-2560 121 12 119 119 NUM iajs-2560 121 13 ibn ibn PROPN iajs-2560 121 14 al al PROPN iajs-2560 121 15 - - PUNCT iajs-2560 121 16 haitham haitham PROPN iajs-2560 121 17 jour jour X iajs-2560 121 18 . . PROPN iajs-2560 122 1 for for ADP iajs-2560 122 2 pure pure ADJ iajs-2560 122 3 & & CCONJ iajs-2560 122 4 appl appl PROPN iajs-2560 122 5 . . PUNCT iajs-2560 123 1 sci sci PROPN iajs-2560 123 2 . . PROPN iajs-2560 124 1 34 34 NUM iajs-2560 124 2 ( ( PUNCT iajs-2560 124 3 1 1 NUM iajs-2560 124 4 ) ) PUNCT iajs-2560 124 5 2021 2021 NUM iajs-2560 124 6 proposition proposition NOUN iajs-2560 124 7 5 5 NUM iajs-2560 124 8 let let VERB iajs-2560 124 9 ∁ ∁ NOUN iajs-2560 124 10 be be AUX iajs-2560 124 11 an an DET iajs-2560 124 12 𝑅-module 𝑅-module PROPN iajs-2560 124 13 and and CCONJ iajs-2560 124 14 𝐿 𝐿 PROPN iajs-2560 124 15 be be AUX iajs-2560 124 16 a a DET iajs-2560 124 17 proper proper ADJ iajs-2560 124 18 submodule submodule NOUN iajs-2560 124 19 of of ADP iajs-2560 124 20 ∁. ∁. NOUN iajs-2560 124 21 then then ADV iajs-2560 124 22 𝐿 𝐿 PROPN iajs-2560 124 23 is be AUX iajs-2560 124 24 a a DET iajs-2560 124 25 nearly nearly ADV iajs-2560 124 26 primary-2absorbing primary-2absorbe VERB iajs-2560 124 27 submodule submodule NOUN iajs-2560 124 28 of of ADP iajs-2560 124 29 ∁ ∁ PROPN iajs-2560 124 30 if if SCONJ iajs-2560 124 31 and and CCONJ iajs-2560 124 32 only only ADV iajs-2560 124 33 if if SCONJ iajs-2560 124 34 𝐼𝐽𝐾 𝐼𝐽𝐾 PROPN iajs-2560 124 35 βŠ† βŠ† NUM iajs-2560 124 36 𝐿 𝐿 PROPN iajs-2560 124 37 , , PUNCT iajs-2560 124 38 where where SCONJ iajs-2560 124 39 𝐼 𝐼 PROPN iajs-2560 124 40 , , PUNCT iajs-2560 124 41 𝐽 𝐽 PROPN iajs-2560 124 42 are be AUX iajs-2560 124 43 ideals ideal NOUN iajs-2560 124 44 of of ADP iajs-2560 124 45 𝑅 𝑅 PROPN iajs-2560 124 46 and and CCONJ iajs-2560 124 47 𝐾 𝐾 PROPN iajs-2560 124 48 is be AUX iajs-2560 124 49 a a DET iajs-2560 124 50 submodule submodule NOUN iajs-2560 124 51 of of ADP iajs-2560 124 52 ∁ ∁ PROPN iajs-2560 124 53 , , PUNCT iajs-2560 124 54 implies imply VERB iajs-2560 124 55 that that SCONJ iajs-2560 124 56 either either CCONJ iajs-2560 124 57 𝐼𝐽 𝐼𝐽 PROPN iajs-2560 124 58 βŠ† βŠ† NUM iajs-2560 124 59 [ [ X iajs-2560 124 60 𝐿 𝐿 PROPN iajs-2560 124 61 + + CCONJ iajs-2560 124 62 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 124 63 ∁ ∁ PROPN iajs-2560 124 64 ] ] PUNCT iajs-2560 124 65 or or CCONJ iajs-2560 124 66 𝐼𝐾 𝐼𝐾 PROPN iajs-2560 124 67 βŠ† βŠ† NUM iajs-2560 124 68 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 124 69 ) ) PUNCT iajs-2560 125 1 + + CCONJ iajs-2560 125 2 𝐽 𝐽 PROPN iajs-2560 125 3 ( ( PUNCT iajs-2560 125 4 ∁ ∁ PROPN iajs-2560 125 5 ) ) PUNCT iajs-2560 125 6 or or CCONJ iajs-2560 125 7 𝐽𝐾 𝐽𝐾 PROPN iajs-2560 125 8 βŠ† βŠ† NUM iajs-2560 125 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 125 10 ) ) PUNCT iajs-2560 126 1 + + CCONJ iajs-2560 126 2 𝐽 𝐽 PROPN iajs-2560 126 3 ( ( PUNCT iajs-2560 126 4 ∁ ∁ PROPN iajs-2560 126 5 ) ) PUNCT iajs-2560 126 6 . . PUNCT iajs-2560 127 1 proof proof NOUN iajs-2560 127 2 : : PUNCT iajs-2560 127 3 ( ( PUNCT iajs-2560 127 4 ⇐ ⇐ ADJ iajs-2560 127 5 ) ) PUNCT iajs-2560 127 6 clear clear ADJ iajs-2560 127 7 . . PUNCT iajs-2560 128 1 ( ( PUNCT iajs-2560 128 2 β‡’ β‡’ PROPN iajs-2560 128 3 ) ) PUNCT iajs-2560 128 4 assume assume VERB iajs-2560 128 5 that that SCONJ iajs-2560 128 6 𝐿 𝐿 PROPN iajs-2560 128 7 is be AUX iajs-2560 128 8 a a DET iajs-2560 128 9 nearly nearly ADV iajs-2560 128 10 primary-2 primary-2 NOUN iajs-2560 128 11 - - PUNCT iajs-2560 128 12 absorbing absorb VERB iajs-2560 128 13 submodule submodule NOUN iajs-2560 128 14 of of ADP iajs-2560 128 15 ∁ ∁ PROPN iajs-2560 128 16 , , PUNCT iajs-2560 128 17 and and CCONJ iajs-2560 128 18 𝐼𝐽𝐾 𝐼𝐽𝐾 PROPN iajs-2560 128 19 βŠ† βŠ† NUM iajs-2560 128 20 𝐿 𝐿 PROPN iajs-2560 128 21 , , PUNCT iajs-2560 128 22 where where SCONJ iajs-2560 128 23 𝐼 𝐼 PROPN iajs-2560 128 24 , , PUNCT iajs-2560 128 25 𝐽 𝐽 PROPN iajs-2560 128 26 are be AUX iajs-2560 128 27 ideals ideal NOUN iajs-2560 128 28 of of ADP iajs-2560 128 29 𝑅 𝑅 PROPN iajs-2560 128 30 and and CCONJ iajs-2560 128 31 𝐾 𝐾 PROPN iajs-2560 128 32 is be AUX iajs-2560 128 33 a a DET iajs-2560 128 34 submodule submodule NOUN iajs-2560 128 35 of of ADP iajs-2560 128 36 ∁ ∁ PROPN iajs-2560 128 37 and and CCONJ iajs-2560 128 38 𝐼𝐽 𝐼𝐽 NOUN iajs-2560 128 39 ⊈ ⊈ PUNCT iajs-2560 129 1 [ [ X iajs-2560 129 2 𝐿 𝐿 PROPN iajs-2560 129 3 + + CCONJ iajs-2560 129 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 129 5 ∁ ∁ NOUN iajs-2560 129 6 ] ] PUNCT iajs-2560 129 7 . . PUNCT iajs-2560 130 1 we we PRON iajs-2560 130 2 must must AUX iajs-2560 130 3 prove prove VERB iajs-2560 130 4 that that SCONJ iajs-2560 130 5 𝐼𝐾 𝐼𝐾 PROPN iajs-2560 130 6 βŠ† βŠ† NUM iajs-2560 130 7 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 130 8 ) ) PUNCT iajs-2560 131 1 + + CCONJ iajs-2560 131 2 𝐽 𝐽 PROPN iajs-2560 131 3 ( ( PUNCT iajs-2560 131 4 ∁ ∁ PROPN iajs-2560 131 5 ) ) PUNCT iajs-2560 131 6 or or CCONJ iajs-2560 131 7 𝐽𝐾 𝐽𝐾 PROPN iajs-2560 131 8 βŠ† βŠ† NUM iajs-2560 131 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 131 10 ) ) PUNCT iajs-2560 132 1 + + CCONJ iajs-2560 132 2 𝐽 𝐽 PROPN iajs-2560 132 3 ( ( PUNCT iajs-2560 132 4 ∁ ∁ PROPN iajs-2560 132 5 ) ) PUNCT iajs-2560 132 6 . . PUNCT iajs-2560 133 1 suppose suppose VERB iajs-2560 134 1 that that SCONJ iajs-2560 134 2 𝐼𝐾 𝐼𝐾 PROPN iajs-2560 134 3 ⊈ ⊈ X iajs-2560 134 4 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 X iajs-2560 134 5 ) ) PUNCT iajs-2560 134 6 + + CCONJ iajs-2560 134 7 𝐽 𝐽 PROPN iajs-2560 134 8 ( ( PUNCT iajs-2560 134 9 ∁ ∁ PROPN iajs-2560 134 10 ) ) PUNCT iajs-2560 134 11 and and CCONJ iajs-2560 134 12 𝐽𝐾 𝐽𝐾 PROPN iajs-2560 134 13 ⊈ ⊈ PROPN iajs-2560 134 14 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 134 15 ) ) PUNCT iajs-2560 135 1 + + CCONJ iajs-2560 135 2 𝐽 𝐽 PROPN iajs-2560 135 3 ( ( PUNCT iajs-2560 135 4 ∁ ∁ PROPN iajs-2560 135 5 ) ) PUNCT iajs-2560 135 6 . . PUNCT iajs-2560 136 1 it it PRON iajs-2560 136 2 follows follow VERB iajs-2560 136 3 that that SCONJ iajs-2560 136 4 there there PRON iajs-2560 136 5 exists exist VERB iajs-2560 136 6 π‘Ÿ1 π‘Ÿ1 PROPN iajs-2560 136 7 ∈ ∈ PROPN iajs-2560 136 8 𝐼 𝐼 PROPN iajs-2560 136 9 and and CCONJ iajs-2560 136 10 π‘Ÿ2 π‘Ÿ2 NOUN iajs-2560 136 11 ∈ ∈ PROPN iajs-2560 136 12 𝐽 𝐽 NOUN iajs-2560 136 13 such such ADJ iajs-2560 136 14 that that SCONJ iajs-2560 136 15 π‘Ÿ1𝐾 π‘Ÿ1𝐾 NOUN iajs-2560 136 16 ⊈ ⊈ PROPN iajs-2560 136 17 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 136 18 ) ) PUNCT iajs-2560 137 1 + + CCONJ iajs-2560 137 2 𝐽 𝐽 PROPN iajs-2560 137 3 ( ( PUNCT iajs-2560 137 4 ∁ ∁ PROPN iajs-2560 137 5 ) ) PUNCT iajs-2560 137 6 and and CCONJ iajs-2560 137 7 π‘Ÿ2𝐾 π‘Ÿ2𝐾 NUM iajs-2560 137 8 ⊈ ⊈ NUM iajs-2560 137 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 137 10 ) ) PUNCT iajs-2560 138 1 + + CCONJ iajs-2560 138 2 𝐽 𝐽 PROPN iajs-2560 138 3 ( ( PUNCT iajs-2560 138 4 ∁ ∁ PROPN iajs-2560 138 5 ) ) PUNCT iajs-2560 138 6 . . PUNCT iajs-2560 139 1 now now ADV iajs-2560 139 2 π‘Ÿ1π‘Ÿ2𝐾 π‘Ÿ1π‘Ÿ2𝐾 NOUN iajs-2560 139 3 βŠ† βŠ† NUM iajs-2560 139 4 𝐿 𝐿 NOUN iajs-2560 139 5 with with ADP iajs-2560 139 6 π‘Ÿ1𝐾 π‘Ÿ1𝐾 NOUN iajs-2560 139 7 ⊈ ⊈ X iajs-2560 139 8 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 139 9 ) ) PUNCT iajs-2560 140 1 + + CCONJ iajs-2560 140 2 𝐽 𝐽 PROPN iajs-2560 140 3 ( ( PUNCT iajs-2560 140 4 ∁ ∁ PROPN iajs-2560 140 5 ) ) PUNCT iajs-2560 140 6 and and CCONJ iajs-2560 140 7 π‘Ÿ2𝐾 π‘Ÿ2𝐾 NUM iajs-2560 140 8 ⊈ ⊈ NUM iajs-2560 140 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 140 10 ) ) PUNCT iajs-2560 141 1 + + CCONJ iajs-2560 141 2 𝐽 𝐽 PROPN iajs-2560 141 3 ( ( PUNCT iajs-2560 141 4 ∁ ∁ PROPN iajs-2560 141 5 ) ) PUNCT iajs-2560 141 6 and and CCONJ iajs-2560 141 7 𝐿 𝐿 PROPN iajs-2560 141 8 is be AUX iajs-2560 141 9 a a DET iajs-2560 141 10 nearly nearly ADV iajs-2560 141 11 primary-2 primary-2 NOUN iajs-2560 141 12 - - PUNCT iajs-2560 141 13 absorbing absorb VERB iajs-2560 141 14 submodule submodule NOUN iajs-2560 141 15 of of ADP iajs-2560 141 16 ∁ ∁ PROPN iajs-2560 141 17 , , PUNCT iajs-2560 141 18 impling imple VERB iajs-2560 141 19 that that SCONJ iajs-2560 141 20 by by ADP iajs-2560 141 21 proposition(4 proposition(4 PROPN iajs-2560 141 22 ) ) PUNCT iajs-2560 141 23 π‘Ÿ1π‘Ÿ2 π‘Ÿ1π‘Ÿ2 NUM iajs-2560 141 24 ∈ ∈ PROPN iajs-2560 141 25 [ [ X iajs-2560 141 26 𝐿 𝐿 PROPN iajs-2560 141 27 + + CCONJ iajs-2560 141 28 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 141 29 ∁ ∁ NUM iajs-2560 141 30 ] ] PUNCT iajs-2560 141 31 . . PUNCT iajs-2560 142 1 since since SCONJ iajs-2560 142 2 𝐼𝐽 𝐼𝐽 NOUN iajs-2560 142 3 ⊈ ⊈ PROPN iajs-2560 143 1 [ [ X iajs-2560 143 2 𝐿 𝐿 PROPN iajs-2560 143 3 + + CCONJ iajs-2560 143 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 143 5 ∁ ∁ NOUN iajs-2560 143 6 ] ] PUNCT iajs-2560 143 7 . . PUNCT iajs-2560 144 1 it it PRON iajs-2560 144 2 follows follow VERB iajs-2560 144 3 that that SCONJ iajs-2560 144 4 there there PRON iajs-2560 144 5 exists exist VERB iajs-2560 144 6 𝑠1 𝑠1 PROPN iajs-2560 144 7 ∈ ∈ PROPN iajs-2560 144 8 𝐼 𝐼 PROPN iajs-2560 144 9 , , PUNCT iajs-2560 144 10 𝑠2 𝑠2 PROPN iajs-2560 144 11 ∈ ∈ PROPN iajs-2560 144 12 𝐽 𝐽 PROPN iajs-2560 145 1 such such ADJ iajs-2560 145 2 that that SCONJ iajs-2560 145 3 𝑠1𝑠2 𝑠1𝑠2 PROPN iajs-2560 145 4 βˆ‰ βˆ‰ PROPN iajs-2560 145 5 [ [ X iajs-2560 145 6 𝐿 𝐿 PROPN iajs-2560 145 7 + + CCONJ iajs-2560 145 8 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 145 9 ∁ ∁ NUM iajs-2560 145 10 ] ] PUNCT iajs-2560 145 11 . . PUNCT iajs-2560 146 1 since since SCONJ iajs-2560 146 2 𝑠1𝑠2𝐾 𝑠1𝑠2𝐾 PROPN iajs-2560 146 3 βŠ† βŠ† NUM iajs-2560 146 4 𝐿 𝐿 PROPN iajs-2560 146 5 , , PUNCT iajs-2560 146 6 and and CCONJ iajs-2560 146 7 𝑠1𝑠2 𝑠1𝑠2 ADP iajs-2560 146 8 βˆ‰ βˆ‰ X iajs-2560 146 9 [ [ X iajs-2560 146 10 𝐿 𝐿 PROPN iajs-2560 146 11 + + CCONJ iajs-2560 146 12 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 146 13 ∁ ∁ NOUN iajs-2560 146 14 ] ] PUNCT iajs-2560 146 15 , , PUNCT iajs-2560 146 16 we we PRON iajs-2560 146 17 have have AUX iajs-2560 146 18 by by ADP iajs-2560 146 19 proposition(4 proposition(4 PROPN iajs-2560 146 20 ) ) PUNCT iajs-2560 146 21 either either CCONJ iajs-2560 146 22 𝑠1𝐾 𝑠1𝐾 ADJ iajs-2560 146 23 βŠ† βŠ† NUM iajs-2560 146 24 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 146 25 ) ) PUNCT iajs-2560 147 1 + + CCONJ iajs-2560 147 2 𝐽 𝐽 PROPN iajs-2560 147 3 ( ( PUNCT iajs-2560 147 4 ∁ ∁ PROPN iajs-2560 147 5 ) ) PUNCT iajs-2560 147 6 or or CCONJ iajs-2560 147 7 𝑠2𝐾 𝑠2𝐾 PRON iajs-2560 147 8 βŠ† βŠ† NUM iajs-2560 147 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 147 10 ) ) PUNCT iajs-2560 148 1 + + CCONJ iajs-2560 148 2 𝐽 𝐽 PROPN iajs-2560 148 3 ( ( PUNCT iajs-2560 148 4 ∁ ∁ PROPN iajs-2560 148 5 ) ) PUNCT iajs-2560 148 6 . . PUNCT iajs-2560 149 1 now now ADV iajs-2560 149 2 , , PUNCT iajs-2560 149 3 we we PRON iajs-2560 149 4 discuss discuss VERB iajs-2560 149 5 the the DET iajs-2560 149 6 following follow VERB iajs-2560 149 7 cases case NOUN iajs-2560 149 8 : : PUNCT iajs-2560 149 9 case case NOUN iajs-2560 149 10 one one NUM iajs-2560 149 11 : : PUNCT iajs-2560 149 12 suppose suppose VERB iajs-2560 149 13 that that SCONJ iajs-2560 149 14 𝑠1𝐾 𝑠1𝐾 VERB iajs-2560 149 15 βŠ† βŠ† NUM iajs-2560 149 16 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 149 17 ) ) PUNCT iajs-2560 150 1 + + CCONJ iajs-2560 150 2 𝐽 𝐽 PROPN iajs-2560 150 3 ( ( PUNCT iajs-2560 150 4 ∁ ∁ PROPN iajs-2560 150 5 ) ) PUNCT iajs-2560 150 6 but but CCONJ iajs-2560 150 7 𝑠2𝐾 𝑠2𝐾 DET iajs-2560 150 8 ⊈ ⊈ NUM iajs-2560 150 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 150 10 ) ) PUNCT iajs-2560 151 1 + + CCONJ iajs-2560 151 2 𝐽 𝐽 PROPN iajs-2560 151 3 ( ( PUNCT iajs-2560 151 4 ∁ ∁ PROPN iajs-2560 151 5 ) ) PUNCT iajs-2560 151 6 since since SCONJ iajs-2560 151 7 π‘Ÿ1𝑠2𝐾 π‘Ÿ1𝑠2𝐾 PROPN iajs-2560 151 8 βŠ† βŠ† NUM iajs-2560 151 9 𝐿 𝐿 PROPN iajs-2560 151 10 and and CCONJ iajs-2560 151 11 𝐿 𝐿 PROPN iajs-2560 151 12 is be AUX iajs-2560 151 13 a a DET iajs-2560 151 14 nearly nearly ADV iajs-2560 151 15 primary-2 primary-2 NOUN iajs-2560 151 16 - - PUNCT iajs-2560 151 17 absorbing absorb VERB iajs-2560 151 18 submodule submodule NOUN iajs-2560 151 19 of of ADP iajs-2560 151 20 ∁ ∁ PROPN iajs-2560 151 21 with with ADP iajs-2560 151 22 𝑠2𝐾 𝑠2𝐾 PRON iajs-2560 151 23 ⊈ ⊈ NUM iajs-2560 151 24 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 151 25 ) ) PUNCT iajs-2560 152 1 + + CCONJ iajs-2560 152 2 𝐽 𝐽 PROPN iajs-2560 152 3 ( ( PUNCT iajs-2560 152 4 ∁ ∁ PROPN iajs-2560 152 5 ) ) PUNCT iajs-2560 152 6 and and CCONJ iajs-2560 152 7 π‘Ÿ1𝐾 π‘Ÿ1𝐾 NOUN iajs-2560 152 8 ⊈ ⊈ NUM iajs-2560 152 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 152 10 ) ) PUNCT iajs-2560 153 1 + + CCONJ iajs-2560 153 2 𝐽 𝐽 PROPN iajs-2560 153 3 ( ( PUNCT iajs-2560 153 4 ∁ ∁ PROPN iajs-2560 153 5 ) ) PUNCT iajs-2560 153 6 , , PUNCT iajs-2560 153 7 implies imply VERB iajs-2560 153 8 that that SCONJ iajs-2560 153 9 π‘Ÿ1𝑠2 π‘Ÿ1𝑠2 PRON iajs-2560 154 1 ∈ ∈ PRON iajs-2560 154 2 [ [ X iajs-2560 154 3 𝐿 𝐿 PROPN iajs-2560 154 4 + + CCONJ iajs-2560 154 5 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 154 6 ∁ ∁ NOUN iajs-2560 154 7 ] ] PUNCT iajs-2560 154 8 by by ADP iajs-2560 154 9 proposition(4 proposition(4 PROPN iajs-2560 154 10 ) ) PUNCT iajs-2560 154 11 . . PUNCT iajs-2560 155 1 also also ADV iajs-2560 155 2 since since SCONJ iajs-2560 155 3 𝑠1𝐾 𝑠1𝐾 ADJ iajs-2560 155 4 βŠ† βŠ† NUM iajs-2560 155 5 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 155 6 ) ) PUNCT iajs-2560 156 1 + + CCONJ iajs-2560 156 2 𝐽 𝐽 PROPN iajs-2560 156 3 ( ( PUNCT iajs-2560 156 4 ∁ ∁ PROPN iajs-2560 156 5 ) ) PUNCT iajs-2560 156 6 but but CCONJ iajs-2560 156 7 π‘Ÿ1𝐾 π‘Ÿ1𝐾 NOUN iajs-2560 156 8 ⊈ ⊈ NUM iajs-2560 156 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 156 10 ) ) PUNCT iajs-2560 157 1 + + CCONJ iajs-2560 157 2 𝐽 𝐽 PROPN iajs-2560 157 3 ( ( PUNCT iajs-2560 157 4 ∁ ∁ PROPN iajs-2560 157 5 ) ) PUNCT iajs-2560 157 6 , , PUNCT iajs-2560 157 7 it it PRON iajs-2560 157 8 follows follow VERB iajs-2560 157 9 that that PRON iajs-2560 157 10 ( ( PUNCT iajs-2560 157 11 π‘Ÿ1 π‘Ÿ1 NOUN iajs-2560 157 12 + + CCONJ iajs-2560 157 13 𝑠1)𝐾 𝑠1)𝐾 NOUN iajs-2560 157 14 ⊈ ⊈ X iajs-2560 157 15 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 157 16 ) ) PUNCT iajs-2560 158 1 + + CCONJ iajs-2560 158 2 𝐽 𝐽 PROPN iajs-2560 158 3 ( ( PUNCT iajs-2560 158 4 ∁ ∁ PROPN iajs-2560 158 5 ) ) PUNCT iajs-2560 158 6 . . PUNCT iajs-2560 159 1 since since SCONJ iajs-2560 159 2 ( ( PUNCT iajs-2560 159 3 π‘Ÿ1 π‘Ÿ1 PROPN iajs-2560 159 4 + + CCONJ iajs-2560 159 5 𝑠1)𝑠2𝐾 𝑠1)𝑠2𝐾 VERB iajs-2560 159 6 βŠ† βŠ† NUM iajs-2560 159 7 𝐿 𝐿 PROPN iajs-2560 159 8 and and CCONJ iajs-2560 159 9 𝑠2𝐾 𝑠2𝐾 DET iajs-2560 159 10 ⊈ ⊈ NUM iajs-2560 159 11 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 159 12 ) ) PUNCT iajs-2560 159 13 + + CCONJ iajs-2560 159 14 𝐽 𝐽 PROPN iajs-2560 159 15 ( ( PUNCT iajs-2560 159 16 ∁ ∁ PROPN iajs-2560 159 17 ) ) PUNCT iajs-2560 159 18 and and CCONJ iajs-2560 159 19 ( ( PUNCT iajs-2560 159 20 π‘Ÿ1 π‘Ÿ1 NOUN iajs-2560 159 21 + + CCONJ iajs-2560 159 22 𝑠1)𝐾 𝑠1)𝐾 NOUN iajs-2560 159 23 ⊈ ⊈ X iajs-2560 159 24 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 159 25 ) ) PUNCT iajs-2560 159 26 + + CCONJ iajs-2560 159 27 𝐽 𝐽 PROPN iajs-2560 159 28 ( ( PUNCT iajs-2560 159 29 ∁ ∁ PROPN iajs-2560 159 30 ) ) PUNCT iajs-2560 159 31 implies imply VERB iajs-2560 159 32 that that SCONJ iajs-2560 159 33 by by ADP iajs-2560 159 34 proposition(4 proposition(4 PROPN iajs-2560 159 35 ) ) PUNCT iajs-2560 159 36 ( ( PUNCT iajs-2560 159 37 π‘Ÿ1 π‘Ÿ1 NOUN iajs-2560 159 38 + + CCONJ iajs-2560 159 39 𝑠1)𝑠2 𝑠1)𝑠2 PROPN iajs-2560 159 40 ∈ ∈ PROPN iajs-2560 159 41 [ [ X iajs-2560 159 42 𝐿 𝐿 PROPN iajs-2560 159 43 + + CCONJ iajs-2560 159 44 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 159 45 ∁ ∁ NOUN iajs-2560 159 46 ] ] PUNCT iajs-2560 159 47 . . PUNCT iajs-2560 160 1 that that PRON iajs-2560 160 2 is be AUX iajs-2560 160 3 ( ( PUNCT iajs-2560 160 4 π‘Ÿ1 π‘Ÿ1 NOUN iajs-2560 160 5 + + CCONJ iajs-2560 160 6 𝑠1)𝑠2 𝑠1)𝑠2 PROPN iajs-2560 160 7 = = SYM iajs-2560 160 8 π‘Ÿ1𝑠2 π‘Ÿ1𝑠2 PROPN iajs-2560 160 9 + + CCONJ iajs-2560 160 10 𝑠1𝑠2 𝑠1𝑠2 X iajs-2560 160 11 ∈ ∈ PROPN iajs-2560 161 1 [ [ X iajs-2560 161 2 𝐿 𝐿 PROPN iajs-2560 161 3 + + CCONJ iajs-2560 161 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 161 5 ∁ ∁ PROPN iajs-2560 161 6 ] ] PUNCT iajs-2560 161 7 and and CCONJ iajs-2560 161 8 π‘Ÿ1𝑠2 π‘Ÿ1𝑠2 NUM iajs-2560 161 9 ∈ ∈ PROPN iajs-2560 161 10 [ [ X iajs-2560 161 11 𝐿 𝐿 PROPN iajs-2560 161 12 + + CCONJ iajs-2560 161 13 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 161 14 ∁ ∁ NOUN iajs-2560 161 15 ] ] PUNCT iajs-2560 161 16 , , PUNCT iajs-2560 161 17 implies imply VERB iajs-2560 161 18 that that SCONJ iajs-2560 161 19 𝑠1𝑠2 𝑠1𝑠2 PROPN iajs-2560 161 20 ∈ ∈ PROPN iajs-2560 161 21 [ [ X iajs-2560 161 22 𝐿 𝐿 PROPN iajs-2560 161 23 + + CCONJ iajs-2560 161 24 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 161 25 ∁ ∁ NOUN iajs-2560 161 26 ] ] X iajs-2560 161 27 a a DET iajs-2560 161 28 contradiction contradiction NOUN iajs-2560 161 29 . . PUNCT iajs-2560 162 1 case case NOUN iajs-2560 162 2 two two NUM iajs-2560 162 3 : : PUNCT iajs-2560 162 4 let let VERB iajs-2560 162 5 it it PRON iajs-2560 162 6 be be AUX iajs-2560 162 7 𝑠2𝐾 𝑠2𝐾 PRON iajs-2560 162 8 βŠ† βŠ† NUM iajs-2560 162 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 162 10 ) ) PUNCT iajs-2560 163 1 + + CCONJ iajs-2560 163 2 𝐽 𝐽 PROPN iajs-2560 163 3 ( ( PUNCT iajs-2560 163 4 ∁ ∁ PROPN iajs-2560 163 5 ) ) PUNCT iajs-2560 163 6 but but CCONJ iajs-2560 163 7 𝑠1𝐾 𝑠1𝐾 VERB iajs-2560 163 8 ⊈ ⊈ NUM iajs-2560 163 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 163 10 ) ) PUNCT iajs-2560 164 1 + + CCONJ iajs-2560 164 2 𝐽 𝐽 PROPN iajs-2560 164 3 ( ( PUNCT iajs-2560 164 4 ∁ ∁ PROPN iajs-2560 164 5 ) ) PUNCT iajs-2560 164 6 in in ADP iajs-2560 164 7 similarly similarly ADV iajs-2560 164 8 steps step NOUN iajs-2560 164 9 of of ADP iajs-2560 164 10 case case NOUN iajs-2560 164 11 one one NUM iajs-2560 164 12 we we PRON iajs-2560 164 13 get get VERB iajs-2560 164 14 a a DET iajs-2560 164 15 contradiction contradiction NOUN iajs-2560 164 16 . . PUNCT iajs-2560 165 1 case case NOUN iajs-2560 165 2 three three NUM iajs-2560 165 3 : : PUNCT iajs-2560 165 4 assume assume VERB iajs-2560 165 5 that that SCONJ iajs-2560 165 6 𝑠1𝐾 𝑠1𝐾 VERB iajs-2560 165 7 βŠ† βŠ† NUM iajs-2560 165 8 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 165 9 ) ) PUNCT iajs-2560 166 1 + + CCONJ iajs-2560 166 2 𝐽 𝐽 PROPN iajs-2560 166 3 ( ( PUNCT iajs-2560 166 4 ∁ ∁ PROPN iajs-2560 166 5 ) ) PUNCT iajs-2560 166 6 but but CCONJ iajs-2560 166 7 𝑠2𝐾 𝑠2𝐾 PRON iajs-2560 166 8 βŠ† βŠ† NUM iajs-2560 166 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 166 10 ) ) PUNCT iajs-2560 167 1 + + CCONJ iajs-2560 167 2 𝐽 𝐽 PROPN iajs-2560 167 3 ( ( PUNCT iajs-2560 167 4 ∁ ∁ PROPN iajs-2560 167 5 ) ) PUNCT iajs-2560 167 6 . . PUNCT iajs-2560 168 1 now now ADV iajs-2560 168 2 since since SCONJ iajs-2560 168 3 𝑠2𝐾 𝑠2𝐾 NOUN iajs-2560 168 4 βŠ† βŠ† NUM iajs-2560 168 5 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 168 6 ) ) PUNCT iajs-2560 168 7 + + CCONJ iajs-2560 168 8 𝐽 𝐽 PROPN iajs-2560 168 9 ( ( PUNCT iajs-2560 168 10 ∁ ∁ PROPN iajs-2560 168 11 ) ) PUNCT iajs-2560 168 12 and and CCONJ iajs-2560 168 13 π‘Ÿ2𝐾 π‘Ÿ2𝐾 NUM iajs-2560 168 14 ⊈ ⊈ NUM iajs-2560 168 15 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 168 16 ) ) PUNCT iajs-2560 168 17 + + CCONJ iajs-2560 168 18 𝐽 𝐽 PROPN iajs-2560 168 19 ( ( PUNCT iajs-2560 168 20 ∁ ∁ PROPN iajs-2560 168 21 ) ) PUNCT iajs-2560 168 22 , , PUNCT iajs-2560 168 23 it it PRON iajs-2560 168 24 follows follow VERB iajs-2560 168 25 that that SCONJ iajs-2560 168 26 ( ( PUNCT iajs-2560 168 27 π‘Ÿ2 π‘Ÿ2 NOUN iajs-2560 168 28 + + CCONJ iajs-2560 168 29 𝑠2)𝐾 𝑠2)𝐾 X iajs-2560 168 30 ⊈ ⊈ X iajs-2560 168 31 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 168 32 ) ) PUNCT iajs-2560 169 1 + + CCONJ iajs-2560 169 2 𝐽 𝐽 PROPN iajs-2560 169 3 ( ( PUNCT iajs-2560 169 4 ∁ ∁ PROPN iajs-2560 169 5 ) ) PUNCT iajs-2560 169 6 . . PUNCT iajs-2560 170 1 we we PRON iajs-2560 170 2 have have VERB iajs-2560 170 3 π‘Ÿ1(π‘Ÿ2 π‘Ÿ1(π‘Ÿ2 NUM iajs-2560 170 4 + + CCONJ iajs-2560 170 5 𝑠2)𝐾 𝑠2)𝐾 NOUN iajs-2560 170 6 βŠ† βŠ† NUM iajs-2560 170 7 𝐿 𝐿 PROPN iajs-2560 170 8 and and CCONJ iajs-2560 170 9 π‘Ÿ1𝐾 π‘Ÿ1𝐾 NOUN iajs-2560 170 10 ⊈ ⊈ NUM iajs-2560 170 11 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 170 12 ) ) PUNCT iajs-2560 171 1 + + CCONJ iajs-2560 171 2 𝐽 𝐽 PROPN iajs-2560 171 3 ( ( PUNCT iajs-2560 171 4 ∁ ∁ PROPN iajs-2560 171 5 ) ) PUNCT iajs-2560 171 6 and and CCONJ iajs-2560 171 7 ( ( PUNCT iajs-2560 171 8 π‘Ÿ2 π‘Ÿ2 NOUN iajs-2560 171 9 + + CCONJ iajs-2560 171 10 𝑠2)𝐾 𝑠2)𝐾 X iajs-2560 171 11 ⊈ ⊈ X iajs-2560 171 12 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 171 13 ) ) PUNCT iajs-2560 172 1 + + CCONJ iajs-2560 172 2 𝐽 𝐽 PROPN iajs-2560 172 3 ( ( PUNCT iajs-2560 172 4 ∁ ∁ PROPN iajs-2560 172 5 ) ) PUNCT iajs-2560 172 6 , , PUNCT iajs-2560 172 7 by by ADP iajs-2560 172 8 proposition(4 proposition(4 PROPN iajs-2560 172 9 ) ) PUNCT iajs-2560 172 10 π‘Ÿ1(π‘Ÿ2 π‘Ÿ1(π‘Ÿ2 NUM iajs-2560 172 11 + + CCONJ iajs-2560 172 12 𝑠2 𝑠2 NOUN iajs-2560 172 13 ) ) PUNCT iajs-2560 172 14 = = SYM iajs-2560 172 15 π‘Ÿ1π‘Ÿ2 π‘Ÿ1π‘Ÿ2 PUNCT iajs-2560 172 16 + + SYM iajs-2560 172 17 π‘Ÿ1𝑠2 π‘Ÿ1𝑠2 X iajs-2560 172 18 ∈ ∈ PROPN iajs-2560 173 1 [ [ X iajs-2560 173 2 𝐿 𝐿 PROPN iajs-2560 173 3 + + CCONJ iajs-2560 173 4 𝐽(∁):𝑅 𝐽(∁):𝑅 ADJ iajs-2560 173 5 ∁].but ∁].but CCONJ iajs-2560 173 6 π‘Ÿ1π‘Ÿ2 π‘Ÿ1π‘Ÿ2 ADP iajs-2560 173 7 ∈ ∈ PROPN iajs-2560 173 8 [ [ X iajs-2560 173 9 𝐿 𝐿 PROPN iajs-2560 173 10 + + CCONJ iajs-2560 173 11 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 173 12 ∁ ∁ NUM iajs-2560 173 13 ] ] PUNCT iajs-2560 173 14 and and CCONJ iajs-2560 173 15 π‘Ÿ1π‘Ÿ2 π‘Ÿ1π‘Ÿ2 NUM iajs-2560 173 16 + + CCONJ iajs-2560 173 17 π‘Ÿ1𝑠2 π‘Ÿ1𝑠2 X iajs-2560 173 18 ∈ ∈ PROPN iajs-2560 173 19 [ [ X iajs-2560 173 20 𝐿 𝐿 PROPN iajs-2560 173 21 + + CCONJ iajs-2560 173 22 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 173 23 ∁ ∁ NOUN iajs-2560 173 24 ] ] PUNCT iajs-2560 173 25 . . PUNCT iajs-2560 174 1 it it PRON iajs-2560 174 2 follows follow VERB iajs-2560 174 3 that that SCONJ iajs-2560 174 4 π‘Ÿ1𝑠2 π‘Ÿ1𝑠2 PRON iajs-2560 175 1 ∈ ∈ PRON iajs-2560 175 2 [ [ X iajs-2560 175 3 𝐿 𝐿 PROPN iajs-2560 175 4 + + CCONJ iajs-2560 175 5 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 175 6 ∁ ∁ NOUN iajs-2560 175 7 ] ] PUNCT iajs-2560 175 8 . . PUNCT iajs-2560 176 1 now now ADV iajs-2560 176 2 , , PUNCT iajs-2560 176 3 since since SCONJ iajs-2560 176 4 𝑠1𝐾 𝑠1𝐾 ADJ iajs-2560 176 5 βŠ† βŠ† NUM iajs-2560 176 6 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 176 7 ) ) PUNCT iajs-2560 176 8 + + CCONJ iajs-2560 176 9 𝐽 𝐽 PROPN iajs-2560 176 10 ( ( PUNCT iajs-2560 176 11 ∁ ∁ PROPN iajs-2560 176 12 ) ) PUNCT iajs-2560 176 13 and and CCONJ iajs-2560 176 14 π‘Ÿ1𝐾 π‘Ÿ1𝐾 NOUN iajs-2560 176 15 ⊈ ⊈ NUM iajs-2560 176 16 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 176 17 ) ) PUNCT iajs-2560 177 1 + + CCONJ iajs-2560 177 2 𝐽 𝐽 PROPN iajs-2560 177 3 ( ( PUNCT iajs-2560 177 4 ∁ ∁ PROPN iajs-2560 177 5 ) ) PUNCT iajs-2560 177 6 , , PUNCT iajs-2560 177 7 implies imply VERB iajs-2560 177 8 that that SCONJ iajs-2560 177 9 ( ( PUNCT iajs-2560 177 10 π‘Ÿ1 π‘Ÿ1 NOUN iajs-2560 177 11 + + CCONJ iajs-2560 177 12 𝑠1)𝐾 𝑠1)𝐾 NOUN iajs-2560 177 13 ⊈ ⊈ X iajs-2560 177 14 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 177 15 ) ) PUNCT iajs-2560 178 1 + + CCONJ iajs-2560 178 2 𝐽 𝐽 PROPN iajs-2560 178 3 ( ( PUNCT iajs-2560 178 4 ∁ ∁ PROPN iajs-2560 178 5 ) ) PUNCT iajs-2560 178 6 since since SCONJ iajs-2560 178 7 ( ( PUNCT iajs-2560 178 8 π‘Ÿ1 π‘Ÿ1 PROPN iajs-2560 178 9 + + CCONJ iajs-2560 178 10 𝑠1)π‘Ÿ2𝐾 𝑠1)π‘Ÿ2𝐾 ADJ iajs-2560 178 11 βŠ† βŠ† NUM iajs-2560 178 12 𝐿 𝐿 PROPN iajs-2560 178 13 and and CCONJ iajs-2560 178 14 π‘Ÿ2𝐾 π‘Ÿ2𝐾 NUM iajs-2560 178 15 ⊈ ⊈ NUM iajs-2560 178 16 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 178 17 ) ) PUNCT iajs-2560 179 1 + + CCONJ iajs-2560 179 2 𝐽 𝐽 PROPN iajs-2560 179 3 ( ( PUNCT iajs-2560 179 4 ∁ ∁ PROPN iajs-2560 179 5 ) ) PUNCT iajs-2560 179 6 and and CCONJ iajs-2560 179 7 ( ( PUNCT iajs-2560 179 8 π‘Ÿ1 π‘Ÿ1 NOUN iajs-2560 179 9 + + CCONJ iajs-2560 179 10 𝑠1)𝐾 𝑠1)𝐾 NOUN iajs-2560 179 11 ⊈ ⊈ X iajs-2560 179 12 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 179 13 ) ) PUNCT iajs-2560 180 1 + + CCONJ iajs-2560 180 2 𝐽 𝐽 PROPN iajs-2560 180 3 ( ( PUNCT iajs-2560 180 4 ∁ ∁ PROPN iajs-2560 180 5 ) ) PUNCT iajs-2560 180 6 , , PUNCT iajs-2560 180 7 it it PRON iajs-2560 180 8 follows follow VERB iajs-2560 180 9 that that SCONJ iajs-2560 180 10 ( ( PUNCT iajs-2560 180 11 π‘Ÿ1 π‘Ÿ1 NOUN iajs-2560 180 12 + + CCONJ iajs-2560 180 13 𝑠1)π‘Ÿ2 𝑠1)π‘Ÿ2 X iajs-2560 180 14 = = SYM iajs-2560 181 1 π‘Ÿ1π‘Ÿ2 π‘Ÿ1π‘Ÿ2 NOUN iajs-2560 181 2 + + CCONJ iajs-2560 181 3 𝑠1π‘Ÿ2 𝑠1π‘Ÿ2 X iajs-2560 181 4 ∈ ∈ PROPN iajs-2560 181 5 [ [ X iajs-2560 181 6 𝐿 𝐿 PROPN iajs-2560 181 7 + + CCONJ iajs-2560 181 8 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 181 9 ∁ ∁ NOUN iajs-2560 181 10 ] ] PUNCT iajs-2560 181 11 by by ADP iajs-2560 181 12 proposition(4 proposition(4 PROPN iajs-2560 181 13 ) ) PUNCT iajs-2560 181 14 . . PUNCT iajs-2560 182 1 now now ADV iajs-2560 182 2 , , PUNCT iajs-2560 182 3 since since SCONJ iajs-2560 182 4 π‘Ÿ1π‘Ÿ2 π‘Ÿ1π‘Ÿ2 NUM iajs-2560 182 5 ∈ ∈ PROPN iajs-2560 183 1 [ [ X iajs-2560 183 2 𝐿 𝐿 PROPN iajs-2560 183 3 + + CCONJ iajs-2560 183 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 183 5 ∁ ∁ NUM iajs-2560 183 6 ] ] PUNCT iajs-2560 183 7 and and CCONJ iajs-2560 183 8 π‘Ÿ1π‘Ÿ2 π‘Ÿ1π‘Ÿ2 X iajs-2560 183 9 + + CCONJ iajs-2560 183 10 𝑠1π‘Ÿ2 𝑠1π‘Ÿ2 X iajs-2560 183 11 ∈ ∈ PROPN iajs-2560 183 12 [ [ X iajs-2560 183 13 𝐿 𝐿 PROPN iajs-2560 183 14 + + CCONJ iajs-2560 183 15 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 183 16 ∁ ∁ NOUN iajs-2560 183 17 ] ] PUNCT iajs-2560 183 18 , , PUNCT iajs-2560 183 19 implies imply VERB iajs-2560 183 20 that that SCONJ iajs-2560 183 21 𝑠1π‘Ÿ2 𝑠1π‘Ÿ2 PROPN iajs-2560 183 22 ∈ ∈ PROPN iajs-2560 183 23 [ [ X iajs-2560 183 24 𝐿 𝐿 PROPN iajs-2560 183 25 + + CCONJ iajs-2560 183 26 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 183 27 ∁ ∁ NOUN iajs-2560 183 28 ] ] PUNCT iajs-2560 183 29 . . PUNCT iajs-2560 184 1 also also ADV iajs-2560 184 2 , , PUNCT iajs-2560 184 3 since since SCONJ iajs-2560 184 4 ( ( PUNCT iajs-2560 184 5 π‘Ÿ1 π‘Ÿ1 NOUN iajs-2560 184 6 + + CCONJ iajs-2560 184 7 𝑠1)(π‘Ÿ2 𝑠1)(π‘Ÿ2 NOUN iajs-2560 184 8 + + CCONJ iajs-2560 184 9 𝑠2)𝐾 𝑠2)𝐾 NOUN iajs-2560 184 10 βŠ† βŠ† NUM iajs-2560 184 11 𝐿 𝐿 PROPN iajs-2560 184 12 and and CCONJ iajs-2560 184 13 ( ( PUNCT iajs-2560 184 14 π‘Ÿ1 π‘Ÿ1 PROPN iajs-2560 184 15 + + CCONJ iajs-2560 184 16 𝑠1)𝐾 𝑠1)𝐾 NOUN iajs-2560 184 17 ⊈ ⊈ X iajs-2560 184 18 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 184 19 ) ) PUNCT iajs-2560 185 1 + + CCONJ iajs-2560 185 2 𝐽 𝐽 PROPN iajs-2560 185 3 ( ( PUNCT iajs-2560 185 4 ∁ ∁ PROPN iajs-2560 185 5 ) ) PUNCT iajs-2560 185 6 and and CCONJ iajs-2560 185 7 ( ( PUNCT iajs-2560 185 8 π‘Ÿ2 π‘Ÿ2 NOUN iajs-2560 185 9 + + CCONJ iajs-2560 185 10 𝑠2)𝐾 𝑠2)𝐾 X iajs-2560 185 11 ⊈ ⊈ X iajs-2560 185 12 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 185 13 ) ) PUNCT iajs-2560 186 1 + + CCONJ iajs-2560 186 2 𝐽 𝐽 PROPN iajs-2560 186 3 ( ( PUNCT iajs-2560 186 4 ∁ ∁ PROPN iajs-2560 186 5 ) ) PUNCT iajs-2560 186 6 , , PUNCT iajs-2560 186 7 it it PRON iajs-2560 186 8 follows follow VERB iajs-2560 186 9 that that SCONJ iajs-2560 186 10 ( ( PUNCT iajs-2560 186 11 π‘Ÿ1 π‘Ÿ1 NOUN iajs-2560 186 12 + + CCONJ iajs-2560 186 13 𝑠1)(π‘Ÿ2 𝑠1)(π‘Ÿ2 NOUN iajs-2560 187 1 + + CCONJ iajs-2560 187 2 𝑠2 𝑠2 NOUN iajs-2560 187 3 ) ) PUNCT iajs-2560 187 4 = = SYM iajs-2560 187 5 π‘Ÿ1π‘Ÿ2 π‘Ÿ1π‘Ÿ2 PROPN iajs-2560 188 1 + + NOUN iajs-2560 188 2 π‘Ÿ1𝑠2 π‘Ÿ1𝑠2 PUNCT iajs-2560 188 3 + + X iajs-2560 188 4 𝑠1π‘Ÿ2 𝑠1π‘Ÿ2 X iajs-2560 188 5 + + X iajs-2560 188 6 𝑠1𝑠2 𝑠1𝑠2 PROPN iajs-2560 188 7 ∈ ∈ PROPN iajs-2560 188 8 [ [ X iajs-2560 188 9 𝐿 𝐿 PROPN iajs-2560 188 10 + + CCONJ iajs-2560 188 11 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 188 12 ∁ ∁ NOUN iajs-2560 188 13 ] ] PUNCT iajs-2560 188 14 by by ADP iajs-2560 188 15 proposition(4 proposition(4 PROPN iajs-2560 188 16 ) ) PUNCT iajs-2560 188 17 . . PUNCT iajs-2560 189 1 again again ADV iajs-2560 189 2 since since SCONJ iajs-2560 189 3 π‘Ÿ1π‘Ÿ2 π‘Ÿ1π‘Ÿ2 PROPN iajs-2560 189 4 , , PUNCT iajs-2560 189 5 π‘Ÿ1𝑠2 π‘Ÿ1𝑠2 AUX iajs-2560 189 6 , , PUNCT iajs-2560 189 7 𝑠1π‘Ÿ2 𝑠1π‘Ÿ2 PROPN iajs-2560 189 8 ∈ ∈ PROPN iajs-2560 189 9 [ [ X iajs-2560 189 10 𝐿 𝐿 PROPN iajs-2560 189 11 + + CCONJ iajs-2560 189 12 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 189 13 ∁ ∁ NOUN iajs-2560 189 14 ] ] PUNCT iajs-2560 189 15 , , PUNCT iajs-2560 189 16 we we PRON iajs-2560 189 17 get get VERB iajs-2560 189 18 that that DET iajs-2560 189 19 𝑠1𝑠2 𝑠1𝑠2 ADP iajs-2560 189 20 ∈ ∈ PROPN iajs-2560 189 21 [ [ X iajs-2560 189 22 𝐿 𝐿 PROPN iajs-2560 189 23 + + CCONJ iajs-2560 189 24 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 189 25 ∁ ∁ NOUN iajs-2560 189 26 ] ] X iajs-2560 189 27 a a DET iajs-2560 189 28 contradiction contradiction NOUN iajs-2560 189 29 . . PUNCT iajs-2560 190 1 thus thus ADV iajs-2560 190 2 , , PUNCT iajs-2560 190 3 we we PRON iajs-2560 190 4 have have VERB iajs-2560 190 5 either either CCONJ iajs-2560 190 6 𝐼𝐾 𝐼𝐾 PROPN iajs-2560 190 7 βŠ† βŠ† NUM iajs-2560 190 8 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 190 9 ) ) PUNCT iajs-2560 191 1 + + CCONJ iajs-2560 191 2 𝐽 𝐽 PROPN iajs-2560 191 3 ( ( PUNCT iajs-2560 191 4 ∁ ∁ PROPN iajs-2560 191 5 ) ) PUNCT iajs-2560 191 6 or or CCONJ iajs-2560 191 7 𝐽𝐾 𝐽𝐾 PROPN iajs-2560 191 8 βŠ† βŠ† NUM iajs-2560 191 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 191 10 ) ) PUNCT iajs-2560 192 1 + + CCONJ iajs-2560 192 2 𝐽 𝐽 PROPN iajs-2560 192 3 ( ( PUNCT iajs-2560 192 4 ∁ ∁ PROPN iajs-2560 192 5 ) ) PUNCT iajs-2560 192 6 . . PUNCT iajs-2560 193 1 proposition proposition NOUN iajs-2560 193 2 6 6 NUM iajs-2560 193 3 let let VERB iajs-2560 193 4 𝐻 𝐻 PRON iajs-2560 193 5 be be AUX iajs-2560 193 6 a a DET iajs-2560 193 7 proper proper ADJ iajs-2560 193 8 submodule submodule NOUN iajs-2560 193 9 of of ADP iajs-2560 193 10 an an DET iajs-2560 193 11 𝑅-module 𝑅-module PROPN iajs-2560 193 12 ∁ ∁ PROPN iajs-2560 193 13 , , PUNCT iajs-2560 193 14 with with ADP iajs-2560 193 15 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 NOUN iajs-2560 193 16 ) ) PUNCT iajs-2560 193 17 is be AUX iajs-2560 193 18 a a DET iajs-2560 193 19 prime prime ADJ iajs-2560 193 20 submodule submodule NOUN iajs-2560 193 21 of of ADP iajs-2560 193 22 ∁. ∁. NOUN iajs-2560 193 23 then then ADV iajs-2560 193 24 𝐻 𝐻 PROPN iajs-2560 193 25 is be AUX iajs-2560 193 26 a a DET iajs-2560 193 27 nearly nearly ADV iajs-2560 193 28 primary-2 primary-2 NOUN iajs-2560 193 29 - - PUNCT iajs-2560 193 30 absorbing absorb VERB iajs-2560 193 31 submodule submodule NOUN iajs-2560 193 32 of of ADP iajs-2560 193 33 ∁. ∁. PROPN iajs-2560 193 34 120 120 NUM iajs-2560 193 35 ibn ibn PROPN iajs-2560 193 36 al al PROPN iajs-2560 193 37 - - PUNCT iajs-2560 193 38 haitham haitham PROPN iajs-2560 193 39 jour jour X iajs-2560 193 40 . . PROPN iajs-2560 194 1 for for ADP iajs-2560 194 2 pure pure ADJ iajs-2560 194 3 & & CCONJ iajs-2560 194 4 appl appl PROPN iajs-2560 194 5 . . PUNCT iajs-2560 195 1 sci sci PROPN iajs-2560 195 2 . . PROPN iajs-2560 196 1 34 34 NUM iajs-2560 196 2 ( ( PUNCT iajs-2560 196 3 1 1 NUM iajs-2560 196 4 ) ) PUNCT iajs-2560 196 5 2021 2021 NUM iajs-2560 196 6 proof proof NOUN iajs-2560 196 7 : : PUNCT iajs-2560 196 8 suppose suppose VERB iajs-2560 196 9 that that SCONJ iajs-2560 196 10 π‘Žπ‘π‘₯ π‘Žπ‘π‘₯ PROPN iajs-2560 196 11 ∈ ∈ PROPN iajs-2560 196 12 𝐻 𝐻 PROPN iajs-2560 196 13 , , PUNCT iajs-2560 196 14 where where SCONJ iajs-2560 196 15 π‘Ž π‘Ž X iajs-2560 196 16 , , PUNCT iajs-2560 196 17 𝑏 𝑏 PROPN iajs-2560 196 18 ∈ ∈ PROPN iajs-2560 196 19 𝑅 𝑅 PROPN iajs-2560 196 20 , , PUNCT iajs-2560 196 21 π‘₯ π‘₯ PRON iajs-2560 196 22 ∈ ∈ ADJ iajs-2560 196 23 ∁ ∁ PROPN iajs-2560 196 24 and and CCONJ iajs-2560 196 25 𝑏π‘₯ 𝑏π‘₯ PROPN iajs-2560 196 26 βˆ‰ βˆ‰ PROPN iajs-2560 196 27 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 PROPN iajs-2560 196 28 ) ) PUNCT iajs-2560 196 29 + + CCONJ iajs-2560 196 30 𝐽 𝐽 PROPN iajs-2560 196 31 ( ( PUNCT iajs-2560 196 32 ∁ ∁ PROPN iajs-2560 196 33 ) ) PUNCT iajs-2560 196 34 . . PUNCT iajs-2560 197 1 since𝐻 since𝐻 PROPN iajs-2560 197 2 βŠ† βŠ† NUM iajs-2560 197 3 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 NOUN iajs-2560 197 4 ) ) PUNCT iajs-2560 197 5 , , PUNCT iajs-2560 197 6 then then ADV iajs-2560 197 7 π‘Ž(𝑏π‘₯ π‘Ž(𝑏π‘₯ X iajs-2560 197 8 ) ) PUNCT iajs-2560 197 9 ∈ ∈ PROPN iajs-2560 197 10 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 NOUN iajs-2560 197 11 ) ) PUNCT iajs-2560 197 12 , , PUNCT iajs-2560 197 13 but but CCONJ iajs-2560 197 14 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 NOUN iajs-2560 197 15 ) ) PUNCT iajs-2560 197 16 is be AUX iajs-2560 197 17 a a DET iajs-2560 197 18 prime prime ADJ iajs-2560 197 19 submodule submodule NOUN iajs-2560 197 20 of of ADP iajs-2560 197 21 ∁ ∁ PROPN iajs-2560 197 22 , , PUNCT iajs-2560 197 23 then then ADV iajs-2560 197 24 π‘ŽβˆβŠ† π‘ŽβˆβŠ† NOUN iajs-2560 197 25 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 X iajs-2560 197 26 ) ) PUNCT iajs-2560 197 27 βŠ† βŠ† NUM iajs-2560 197 28 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 NOUN iajs-2560 197 29 ) ) PUNCT iajs-2560 197 30 + + CCONJ iajs-2560 197 31 𝐽(∁ 𝐽(∁ ADJ iajs-2560 197 32 ) ) PUNCT iajs-2560 197 33 . . PUNCT iajs-2560 198 1 that that PRON iajs-2560 198 2 is be AUX iajs-2560 198 3 π‘Žπ‘₯ π‘Žπ‘₯ X iajs-2560 198 4 ∈ ∈ PROPN iajs-2560 198 5 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 π‘Ÿπ‘Žπ‘‘βˆ(𝐻 X iajs-2560 198 6 ) ) PUNCT iajs-2560 199 1 + + CCONJ iajs-2560 199 2 𝐽 𝐽 PROPN iajs-2560 199 3 ( ( PUNCT iajs-2560 199 4 ∁ ∁ PROPN iajs-2560 199 5 ) ) PUNCT iajs-2560 199 6 , , PUNCT iajs-2560 199 7 for for ADP iajs-2560 199 8 some some DET iajs-2560 199 9 π‘₯ π‘₯ PRON iajs-2560 199 10 ∈ ∈ NOUN iajs-2560 199 11 ∁. ∁. NOUN iajs-2560 199 12 thus thus ADV iajs-2560 199 13 𝐻 𝐻 PROPN iajs-2560 199 14 is be AUX iajs-2560 199 15 a a DET iajs-2560 199 16 nearly nearly ADV iajs-2560 199 17 primary-2 primary-2 NOUN iajs-2560 199 18 - - PUNCT iajs-2560 199 19 absorbing absorb VERB iajs-2560 199 20 submodule submodule NOUN iajs-2560 199 21 of of ADP iajs-2560 199 22 ∁. ∁. NOUN iajs-2560 199 23 before before SCONJ iajs-2560 199 24 we we PRON iajs-2560 199 25 introduce introduce VERB iajs-2560 199 26 the the DET iajs-2560 199 27 next next ADJ iajs-2560 199 28 result result NOUN iajs-2560 199 29 , , PUNCT iajs-2560 199 30 we we PRON iajs-2560 199 31 need need VERB iajs-2560 199 32 to to PART iajs-2560 199 33 recall recall VERB iajs-2560 199 34 the the DET iajs-2560 199 35 following follow VERB iajs-2560 199 36 lemma lemma PROPN iajs-2560 199 37 . . PUNCT iajs-2560 200 1 lemma lemma PROPN iajs-2560 200 2 7[10 7[10 NUM iajs-2560 200 3 ] ] X iajs-2560 200 4 if if SCONJ iajs-2560 200 5 𝑅 𝑅 PROPN iajs-2560 200 6 is be AUX iajs-2560 200 7 a a DET iajs-2560 200 8 good good ADJ iajs-2560 200 9 ring ring NOUN iajs-2560 200 10 , , PUNCT iajs-2560 200 11 ∁ ∁ PROPN iajs-2560 200 12 is be AUX iajs-2560 200 13 an an DET iajs-2560 200 14 𝑅-module 𝑅-module PROPN iajs-2560 200 15 and and CCONJ iajs-2560 200 16 𝑁 𝑁 PROPN iajs-2560 200 17 is be AUX iajs-2560 200 18 a a DET iajs-2560 200 19 submodule submodule NOUN iajs-2560 200 20 of of ADP iajs-2560 200 21 ∁ ∁ PROPN iajs-2560 200 22 , , PUNCT iajs-2560 200 23 then then ADV iajs-2560 200 24 𝐽(∁ 𝐽(∁ ADJ iajs-2560 200 25 ) ) PUNCT iajs-2560 200 26 ∩ ∩ NOUN iajs-2560 200 27 𝑁 𝑁 PROPN iajs-2560 200 28 = = SYM iajs-2560 200 29 𝐽(𝑁 𝐽(𝑁 PROPN iajs-2560 200 30 ) ) PUNCT iajs-2560 200 31 . . PUNCT iajs-2560 201 1 proposition proposition NOUN iajs-2560 201 2 8 8 NUM iajs-2560 201 3 let let VERB iajs-2560 201 4 𝑅 𝑅 PROPN iajs-2560 201 5 be be AUX iajs-2560 201 6 a a DET iajs-2560 201 7 good good ADJ iajs-2560 201 8 ring ring NOUN iajs-2560 201 9 , , PUNCT iajs-2560 201 10 𝐿 𝐿 PROPN iajs-2560 201 11 and and CCONJ iajs-2560 201 12 𝐾 𝐾 PROPN iajs-2560 201 13 are be AUX iajs-2560 201 14 proper proper ADJ iajs-2560 201 15 submodules submodule NOUN iajs-2560 201 16 of of ADP iajs-2560 201 17 an an DET iajs-2560 201 18 𝑅-module 𝑅-module PROPN iajs-2560 201 19 ∁ ∁ PROPN iajs-2560 201 20 with with ADP iajs-2560 201 21 𝐿 𝐿 PROPN iajs-2560 201 22 ⊊ ⊊ NOUN iajs-2560 201 23 𝐾 𝐾 PROPN iajs-2560 201 24 and and CCONJ iajs-2560 201 25 𝐽(∁ 𝐽(∁ PROPN iajs-2560 201 26 ) ) PUNCT iajs-2560 201 27 βŠ† βŠ† NUM iajs-2560 201 28 𝐾. 𝐾. NOUN iajs-2560 201 29 if if SCONJ iajs-2560 201 30 𝐿 𝐿 PROPN iajs-2560 201 31 is be AUX iajs-2560 201 32 a a DET iajs-2560 201 33 nearly nearly ADV iajs-2560 201 34 primary-2 primary-2 NOUN iajs-2560 201 35 - - PUNCT iajs-2560 201 36 absorbing absorb VERB iajs-2560 201 37 submodule submodule NOUN iajs-2560 201 38 of of ADP iajs-2560 201 39 ∁ ∁ PROPN iajs-2560 201 40 , , PUNCT iajs-2560 201 41 then then ADV iajs-2560 201 42 𝐿 𝐿 PROPN iajs-2560 201 43 is be AUX iajs-2560 201 44 a a DET iajs-2560 201 45 nearly nearly ADV iajs-2560 201 46 primary-2absorbing primary-2absorbe VERB iajs-2560 201 47 submodule submodule NOUN iajs-2560 201 48 of of ADP iajs-2560 201 49 𝐾. 𝐾. PROPN iajs-2560 201 50 proof proof NOUN iajs-2560 201 51 : : PUNCT iajs-2560 201 52 let let VERB iajs-2560 201 53 π‘Ÿπ‘ π‘₯ π‘Ÿπ‘ π‘₯ NUM iajs-2560 201 54 ∈ ∈ PROPN iajs-2560 201 55 𝐿 𝐿 PROPN iajs-2560 201 56 , , PUNCT iajs-2560 201 57 where where SCONJ iajs-2560 201 58 π‘Ÿ π‘Ÿ X iajs-2560 201 59 , , PUNCT iajs-2560 201 60 𝑠 𝑠 PROPN iajs-2560 201 61 ∈ ∈ PROPN iajs-2560 201 62 𝑅 𝑅 PROPN iajs-2560 201 63 , , PUNCT iajs-2560 201 64 π‘₯ π‘₯ PRON iajs-2560 201 65 ∈ ∈ PROPN iajs-2560 201 66 𝐾 𝐾 NOUN iajs-2560 201 67 βŠ† βŠ† NUM iajs-2560 201 68 ∁. ∁. NOUN iajs-2560 201 69 since since SCONJ iajs-2560 201 70 𝐿 𝐿 PROPN iajs-2560 201 71 is be AUX iajs-2560 201 72 a a DET iajs-2560 201 73 nearly nearly ADV iajs-2560 201 74 primary-2 primary-2 NOUN iajs-2560 201 75 - - PUNCT iajs-2560 201 76 absorbing absorb VERB iajs-2560 201 77 submodule submodule NOUN iajs-2560 201 78 of of ADP iajs-2560 201 79 ∁ ∁ PROPN iajs-2560 201 80 , , PUNCT iajs-2560 201 81 implies imply VERB iajs-2560 201 82 that that SCONJ iajs-2560 201 83 either either CCONJ iajs-2560 201 84 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PROPN iajs-2560 201 85 ∈ ∈ NOUN iajs-2560 201 86 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 201 87 ) ) PUNCT iajs-2560 201 88 + + SYM iajs-2560 201 89 𝐽(∁ 𝐽(∁ ADJ iajs-2560 201 90 ) ) PUNCT iajs-2560 201 91 or or CCONJ iajs-2560 201 92 𝑠π‘₯ 𝑠π‘₯ ADP iajs-2560 201 93 ∈ ∈ PROPN iajs-2560 201 94 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 201 95 ) ) PUNCT iajs-2560 201 96 + + SYM iajs-2560 201 97 𝐽(∁ 𝐽(∁ ADJ iajs-2560 201 98 ) ) PUNCT iajs-2560 201 99 or or CCONJ iajs-2560 201 100 π‘Ÿπ‘ βˆβŠ† π‘Ÿπ‘ βˆβŠ† NUM iajs-2560 201 101 𝐿 𝐿 PROPN iajs-2560 201 102 + + PROPN iajs-2560 201 103 𝐽(∁ 𝐽(∁ PROPN iajs-2560 201 104 ) ) PUNCT iajs-2560 201 105 . . PUNCT iajs-2560 202 1 that that PRON iajs-2560 202 2 is be AUX iajs-2560 202 3 either either CCONJ iajs-2560 202 4 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PRON iajs-2560 202 5 ∈ ∈ PROPN iajs-2560 202 6 ( ( PUNCT iajs-2560 202 7 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 X iajs-2560 202 8 ) ) PUNCT iajs-2560 202 9 + + SYM iajs-2560 202 10 𝐽(∁ 𝐽(∁ ADJ iajs-2560 202 11 ) ) PUNCT iajs-2560 202 12 ) ) PUNCT iajs-2560 202 13 ∩ ∩ PROPN iajs-2560 202 14 𝐾 𝐾 PROPN iajs-2560 202 15 or or CCONJ iajs-2560 202 16 𝑠π‘₯ 𝑠π‘₯ ADP iajs-2560 202 17 ∈ ∈ PROPN iajs-2560 202 18 ( ( PUNCT iajs-2560 202 19 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 X iajs-2560 202 20 ) ) PUNCT iajs-2560 202 21 + + SYM iajs-2560 202 22 𝐽(∁ 𝐽(∁ ADJ iajs-2560 202 23 ) ) PUNCT iajs-2560 202 24 ) ) PUNCT iajs-2560 202 25 ∩ ∩ PROPN iajs-2560 202 26 𝐾 𝐾 PROPN iajs-2560 202 27 or or CCONJ iajs-2560 202 28 π‘Ÿπ‘ βˆ π‘Ÿπ‘ βˆ ADJ iajs-2560 202 29 βŠ† βŠ† NUM iajs-2560 202 30 ( ( PUNCT iajs-2560 202 31 𝐿 𝐿 PROPN iajs-2560 202 32 + + PROPN iajs-2560 202 33 𝐽(∁ 𝐽(∁ PROPN iajs-2560 202 34 ) ) PUNCT iajs-2560 202 35 ) ) PUNCT iajs-2560 202 36 ∩ ∩ NOUN iajs-2560 203 1 𝐾. 𝐾. PROPN iajs-2560 203 2 but but CCONJ iajs-2560 203 3 by by ADP iajs-2560 203 4 modular modular ADJ iajs-2560 203 5 law law NOUN iajs-2560 203 6 we we PRON iajs-2560 203 7 have have VERB iajs-2560 203 8 ( ( PUNCT iajs-2560 203 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 203 10 ) ) PUNCT iajs-2560 203 11 + + SYM iajs-2560 203 12 𝐽(∁ 𝐽(∁ ADJ iajs-2560 203 13 ) ) PUNCT iajs-2560 203 14 ) ) PUNCT iajs-2560 203 15 ∩ ∩ NOUN iajs-2560 203 16 𝐾 𝐾 PROPN iajs-2560 203 17 = = SYM iajs-2560 203 18 ( ( PUNCT iajs-2560 203 19 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 203 20 ) ) PUNCT iajs-2560 203 21 ∩ ∩ ADJ iajs-2560 203 22 𝐾 𝐾 PROPN iajs-2560 203 23 ) ) PUNCT iajs-2560 203 24 + + CCONJ iajs-2560 203 25 ( ( PUNCT iajs-2560 203 26 𝐽(∁ 𝐽(∁ ADJ iajs-2560 203 27 ) ) PUNCT iajs-2560 203 28 ∩ ∩ ADJ iajs-2560 203 29 𝐾 𝐾 PROPN iajs-2560 203 30 ) ) PUNCT iajs-2560 203 31 = = SYM iajs-2560 203 32 ( ( PUNCT iajs-2560 203 33 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 203 34 ) ) PUNCT iajs-2560 203 35 ∩ ∩ ADJ iajs-2560 203 36 𝐾 𝐾 PROPN iajs-2560 203 37 ) ) PUNCT iajs-2560 204 1 + + NUM iajs-2560 204 2 𝐽(𝐾 𝐽(𝐾 NOUN iajs-2560 204 3 ) ) PUNCT iajs-2560 204 4 by by ADP iajs-2560 204 5 lemma(7 lemma(7 PROPN iajs-2560 204 6 ) ) PUNCT iajs-2560 204 7 . . PUNCT iajs-2560 205 1 thus thus ADV iajs-2560 205 2 we we PRON iajs-2560 205 3 have have VERB iajs-2560 205 4 either either CCONJ iajs-2560 205 5 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PRON iajs-2560 205 6 ∈ ∈ PROPN iajs-2560 205 7 ( ( PUNCT iajs-2560 205 8 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 205 9 ) ) PUNCT iajs-2560 205 10 ∩ ∩ ADJ iajs-2560 205 11 𝐾 𝐾 PROPN iajs-2560 205 12 ) ) PUNCT iajs-2560 205 13 + + NUM iajs-2560 205 14 𝐽(𝐾 𝐽(𝐾 NOUN iajs-2560 205 15 ) ) PUNCT iajs-2560 205 16 βŠ† βŠ† NUM iajs-2560 205 17 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 205 18 ) ) PUNCT iajs-2560 206 1 + + NUM iajs-2560 206 2 𝐽(𝐾 𝐽(𝐾 NOUN iajs-2560 206 3 ) ) PUNCT iajs-2560 206 4 or or CCONJ iajs-2560 206 5 𝑠π‘₯ 𝑠π‘₯ ADP iajs-2560 206 6 ∈∈ ∈∈ NOUN iajs-2560 206 7 ( ( PUNCT iajs-2560 206 8 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 206 9 ) ) PUNCT iajs-2560 206 10 ∩ ∩ ADJ iajs-2560 206 11 𝐾 𝐾 PROPN iajs-2560 206 12 ) ) PUNCT iajs-2560 206 13 + + NUM iajs-2560 206 14 𝐽(𝐾 𝐽(𝐾 NOUN iajs-2560 206 15 ) ) PUNCT iajs-2560 206 16 βŠ† βŠ† NUM iajs-2560 206 17 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 206 18 ) ) PUNCT iajs-2560 206 19 + + NUM iajs-2560 206 20 𝐽(𝐾 𝐽(𝐾 NOUN iajs-2560 206 21 ) ) PUNCT iajs-2560 206 22 or or CCONJ iajs-2560 206 23 π‘Ÿπ‘ βˆ π‘Ÿπ‘ βˆ ADJ iajs-2560 206 24 βŠ† βŠ† NUM iajs-2560 206 25 ( ( PUNCT iajs-2560 206 26 𝐿 𝐿 PROPN iajs-2560 206 27 ∩ ∩ ADJ iajs-2560 206 28 𝐾 𝐾 PROPN iajs-2560 206 29 ) ) PUNCT iajs-2560 206 30 + + CCONJ iajs-2560 206 31 ( ( PUNCT iajs-2560 206 32 𝐽(∁ 𝐽(∁ ADJ iajs-2560 206 33 ) ) PUNCT iajs-2560 206 34 ∩ ∩ ADJ iajs-2560 206 35 𝐾 𝐾 PROPN iajs-2560 206 36 ) ) PUNCT iajs-2560 206 37 = = SYM iajs-2560 206 38 ( ( PUNCT iajs-2560 206 39 𝐿 𝐿 PROPN iajs-2560 206 40 ∩ ∩ ADJ iajs-2560 206 41 𝐾 𝐾 PROPN iajs-2560 206 42 ) ) PUNCT iajs-2560 206 43 + + NUM iajs-2560 206 44 𝐽(𝐾 𝐽(𝐾 NOUN iajs-2560 206 45 ) ) PUNCT iajs-2560 206 46 βŠ† βŠ† NUM iajs-2560 206 47 𝐿 𝐿 PROPN iajs-2560 206 48 + + NOUN iajs-2560 206 49 𝐽(𝐾 𝐽(𝐾 NOUN iajs-2560 206 50 ) ) PUNCT iajs-2560 206 51 . . PUNCT iajs-2560 207 1 hence hence ADV iajs-2560 207 2 𝐿 𝐿 PROPN iajs-2560 207 3 is be AUX iajs-2560 207 4 a a DET iajs-2560 207 5 nearly nearly ADV iajs-2560 207 6 primary-2 primary-2 NOUN iajs-2560 207 7 - - PUNCT iajs-2560 207 8 absorbing absorb VERB iajs-2560 207 9 submodule submodule NOUN iajs-2560 207 10 of of ADP iajs-2560 207 11 𝐾. 𝐾. PROPN iajs-2560 207 12 recall recall VERB iajs-2560 207 13 that that PRON iajs-2560 207 14 for for ADP iajs-2560 207 15 any any DET iajs-2560 207 16 submodules submodule NOUN iajs-2560 207 17 𝐿 𝐿 PROPN iajs-2560 207 18 , , PUNCT iajs-2560 207 19 𝐾 𝐾 PROPN iajs-2560 207 20 of of ADP iajs-2560 207 21 a a DET iajs-2560 207 22 multiplication multiplication NOUN iajs-2560 207 23 𝑅-module 𝑅-module PROPN iajs-2560 207 24 ∁ ∁ PROPN iajs-2560 207 25 with with ADP iajs-2560 207 26 𝐿 𝐿 PROPN iajs-2560 207 27 = = SYM iajs-2560 207 28 𝐼∁ 𝐼∁ X iajs-2560 207 29 , , PUNCT iajs-2560 207 30 𝐾 𝐾 NOUN iajs-2560 207 31 = = PUNCT iajs-2560 207 32 𝐽∁ 𝐽∁ X iajs-2560 207 33 for for ADP iajs-2560 207 34 some some DET iajs-2560 207 35 ideals ideal NOUN iajs-2560 207 36 𝐼 𝐼 PROPN iajs-2560 207 37 and and CCONJ iajs-2560 207 38 𝐽 𝐽 PROPN iajs-2560 207 39 of of ADP iajs-2560 207 40 𝑅. 𝑅. NOUN iajs-2560 207 41 the the DET iajs-2560 207 42 product product NOUN iajs-2560 207 43 𝐿𝐾 𝐿𝐾 PROPN iajs-2560 207 44 = = SYM iajs-2560 207 45 𝐼∁. 𝐼∁. NOUN iajs-2560 207 46 𝐽∁= 𝐽∁= NUM iajs-2560 207 47 𝐼𝐽∁. 𝐼𝐽∁. SYM iajs-2560 207 48 that that PRON iajs-2560 207 49 is be AUX iajs-2560 207 50 𝐿𝐾 𝐿𝐾 PROPN iajs-2560 207 51 = = SYM iajs-2560 207 52 𝐼𝐾 𝐼𝐾 PROPN iajs-2560 207 53 , , PUNCT iajs-2560 207 54 in in ADP iajs-2560 207 55 particular particular ADJ iajs-2560 207 56 𝐿∁= 𝐿∁= X iajs-2560 207 57 𝐼∁∁= 𝐼∁∁= NUM iajs-2560 207 58 𝐼∁= 𝐼∁= NOUN iajs-2560 207 59 𝐿. 𝐿. VERB iajs-2560 207 60 also also ADV iajs-2560 207 61 for for ADP iajs-2560 207 62 any any DET iajs-2560 207 63 π‘₯ π‘₯ PROPN iajs-2560 207 64 ∈ ∈ ADJ iajs-2560 207 65 ∁ ∁ NOUN iajs-2560 207 66 we we PRON iajs-2560 207 67 have have VERB iajs-2560 207 68 𝐿π‘₯ 𝐿π‘₯ PROPN iajs-2560 207 69 = = SYM iajs-2560 207 70 𝐼π‘₯[13 𝐼π‘₯[13 PROPN iajs-2560 207 71 ] ] PUNCT iajs-2560 207 72 . . PUNCT iajs-2560 208 1 the the DET iajs-2560 208 2 following following ADJ iajs-2560 208 3 result result NOUN iajs-2560 208 4 gives give VERB iajs-2560 208 5 a a DET iajs-2560 208 6 characterization characterization NOUN iajs-2560 208 7 of of ADP iajs-2560 208 8 nearly nearly ADV iajs-2560 208 9 primary-2 primary-2 NOUN iajs-2560 208 10 - - PUNCT iajs-2560 208 11 absorbing absorbing ADJ iajs-2560 208 12 submodules submodule NOUN iajs-2560 208 13 in in ADP iajs-2560 208 14 class class NOUN iajs-2560 208 15 of of ADP iajs-2560 208 16 multiplication multiplication NOUN iajs-2560 208 17 modules module NOUN iajs-2560 208 18 . . PUNCT iajs-2560 209 1 proposition proposition NOUN iajs-2560 209 2 9 9 NUM iajs-2560 209 3 let let VERB iajs-2560 209 4 ∁ ∁ NOUN iajs-2560 209 5 be be AUX iajs-2560 209 6 a a DET iajs-2560 209 7 multiplication multiplication NOUN iajs-2560 209 8 𝑅-module 𝑅-module PROPN iajs-2560 209 9 and and CCONJ iajs-2560 209 10 𝐿 𝐿 PROPN iajs-2560 209 11 is be AUX iajs-2560 209 12 a a DET iajs-2560 209 13 proper proper ADJ iajs-2560 209 14 submodule submodule NOUN iajs-2560 209 15 of of ADP iajs-2560 209 16 ∁. ∁. NOUN iajs-2560 209 17 then then ADV iajs-2560 209 18 𝐿 𝐿 PROPN iajs-2560 209 19 is be AUX iajs-2560 209 20 a a DET iajs-2560 209 21 nearly nearly ADV iajs-2560 209 22 primary-2 primary-2 NOUN iajs-2560 209 23 - - PUNCT iajs-2560 209 24 absorbing absorb VERB iajs-2560 209 25 submodule submodule NOUN iajs-2560 209 26 of of ADP iajs-2560 209 27 ∁ ∁ PROPN iajs-2560 209 28 if if SCONJ iajs-2560 209 29 and and CCONJ iajs-2560 209 30 only only ADV iajs-2560 209 31 if if SCONJ iajs-2560 209 32 , , PUNCT iajs-2560 209 33 whenever whenever SCONJ iajs-2560 209 34 𝐿1𝐿2𝐿3 𝐿1𝐿2𝐿3 PROPN iajs-2560 209 35 βŠ† βŠ† NUM iajs-2560 209 36 𝐿 𝐿 PROPN iajs-2560 209 37 for for ADP iajs-2560 209 38 𝐿1 𝐿1 PROPN iajs-2560 209 39 , , PUNCT iajs-2560 209 40 𝐿2 𝐿2 PROPN iajs-2560 209 41 , , PUNCT iajs-2560 209 42 𝐿3 𝐿3 NOUN iajs-2560 209 43 are be AUX iajs-2560 209 44 submodules submodule NOUN iajs-2560 209 45 of of ADP iajs-2560 209 46 ∁ ∁ NUM iajs-2560 209 47 , , PUNCT iajs-2560 209 48 impling imple VERB iajs-2560 209 49 that that SCONJ iajs-2560 209 50 either either CCONJ iajs-2560 209 51 𝐿1𝐿3 𝐿1𝐿3 NOUN iajs-2560 209 52 βŠ† βŠ† NUM iajs-2560 209 53 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 209 54 ) ) PUNCT iajs-2560 209 55 + + SYM iajs-2560 209 56 𝐽(∁ 𝐽(∁ ADJ iajs-2560 209 57 ) ) PUNCT iajs-2560 209 58 or or CCONJ iajs-2560 209 59 𝐿2𝐿3 𝐿2𝐿3 PROPN iajs-2560 209 60 βŠ† βŠ† NUM iajs-2560 209 61 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 209 62 ) ) PUNCT iajs-2560 209 63 + + SYM iajs-2560 209 64 𝐽(∁ 𝐽(∁ ADJ iajs-2560 209 65 ) ) PUNCT iajs-2560 209 66 or or CCONJ iajs-2560 209 67 𝐿1𝐿2π‘Š 𝐿1𝐿2π‘Š NUM iajs-2560 209 68 βŠ† βŠ† NUM iajs-2560 209 69 𝐿 𝐿 PROPN iajs-2560 209 70 + + PROPN iajs-2560 209 71 𝐽(∁ 𝐽(∁ PROPN iajs-2560 209 72 ) ) PUNCT iajs-2560 209 73 . . PUNCT iajs-2560 210 1 proof proof NOUN iajs-2560 210 2 : : PUNCT iajs-2560 210 3 ( ( PUNCT iajs-2560 210 4 β‡’ β‡’ NOUN iajs-2560 210 5 ) ) PUNCT iajs-2560 210 6 let let VERB iajs-2560 210 7 𝐿 𝐿 PROPN iajs-2560 210 8 be be AUX iajs-2560 210 9 a a DET iajs-2560 210 10 nearly nearly ADV iajs-2560 210 11 primary-2 primary-2 NOUN iajs-2560 210 12 - - PUNCT iajs-2560 210 13 absorbing absorb VERB iajs-2560 210 14 submodule submodule NOUN iajs-2560 210 15 of of ADP iajs-2560 210 16 ∁ ∁ PROPN iajs-2560 210 17 and and CCONJ iajs-2560 210 18 𝐿1𝐿2𝐿3 𝐿1𝐿2𝐿3 PROPN iajs-2560 210 19 βŠ† βŠ† NUM iajs-2560 210 20 𝐿 𝐿 PROPN iajs-2560 210 21 for for ADP iajs-2560 210 22 𝐿1 𝐿1 PROPN iajs-2560 210 23 , , PUNCT iajs-2560 210 24 𝐿2 𝐿2 PROPN iajs-2560 210 25 , , PUNCT iajs-2560 210 26 𝐿3 𝐿3 NOUN iajs-2560 210 27 are be AUX iajs-2560 210 28 submodules submodule NOUN iajs-2560 210 29 of of ADP iajs-2560 210 30 ∁ ∁ NUM iajs-2560 210 31 , , PUNCT iajs-2560 210 32 with with ADP iajs-2560 210 33 𝐿1𝐿2 𝐿1𝐿2 ADJ iajs-2560 210 34 ∁ ∁ NOUN iajs-2560 210 35 ⊈ ⊈ PROPN iajs-2560 210 36 𝐿 𝐿 PROPN iajs-2560 210 37 + + PROPN iajs-2560 210 38 𝐽(∁ 𝐽(∁ PROPN iajs-2560 210 39 ) ) PUNCT iajs-2560 210 40 . . PUNCT iajs-2560 211 1 since since SCONJ iajs-2560 211 2 ∁ ∁ PROPN iajs-2560 211 3 is be AUX iajs-2560 211 4 a a DET iajs-2560 211 5 multiplication multiplication NOUN iajs-2560 211 6 , , PUNCT iajs-2560 211 7 then then ADV iajs-2560 211 8 𝐿1 𝐿1 VERB iajs-2560 211 9 = = PUNCT iajs-2560 211 10 𝐼1∁ 𝐼1∁ X iajs-2560 211 11 and and CCONJ iajs-2560 211 12 𝐿2 𝐿2 NOUN iajs-2560 212 1 = = SYM iajs-2560 212 2 𝐼2∁ 𝐼2∁ NOUN iajs-2560 212 3 for for ADP iajs-2560 212 4 some some DET iajs-2560 212 5 ideals ideal NOUN iajs-2560 212 6 𝐼1 𝐼1 NOUN iajs-2560 212 7 , , PUNCT iajs-2560 212 8 𝐼2 𝐼2 PROPN iajs-2560 212 9 , , PUNCT iajs-2560 212 10 𝐼3 𝐼3 NOUN iajs-2560 212 11 of of ADP iajs-2560 212 12 𝑅. 𝑅. NOUN iajs-2560 212 13 clearly clearly ADV iajs-2560 212 14 𝐼1𝐼2𝐿3 𝐼1𝐼2𝐿3 PROPN iajs-2560 212 15 βŠ† βŠ† NUM iajs-2560 212 16 𝐿 𝐿 PROPN iajs-2560 212 17 and and CCONJ iajs-2560 212 18 𝐼1𝐼2 𝐼1𝐼2 PRON iajs-2560 212 19 ⊈ ⊈ PUNCT iajs-2560 213 1 [ [ X iajs-2560 213 2 𝐿 𝐿 PROPN iajs-2560 213 3 + + CCONJ iajs-2560 213 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 213 5 ∁ ∁ NOUN iajs-2560 213 6 ] ] PUNCT iajs-2560 213 7 . . PUNCT iajs-2560 214 1 since since SCONJ iajs-2560 214 2 𝐿 𝐿 PROPN iajs-2560 214 3 is be AUX iajs-2560 214 4 a a DET iajs-2560 214 5 nearly nearly ADV iajs-2560 214 6 primary-2 primary-2 NOUN iajs-2560 214 7 - - PUNCT iajs-2560 214 8 absorbing absorb VERB iajs-2560 214 9 submodule submodule NOUN iajs-2560 214 10 of of ADP iajs-2560 214 11 ∁ ∁ PROPN iajs-2560 214 12 , , PUNCT iajs-2560 214 13 implies imply VERB iajs-2560 214 14 that that SCONJ iajs-2560 214 15 either either CCONJ iajs-2560 214 16 𝐼1𝐿3 𝐼1𝐿3 NUM iajs-2560 214 17 βŠ† βŠ† NUM iajs-2560 214 18 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 214 19 ) ) PUNCT iajs-2560 214 20 + + SYM iajs-2560 214 21 𝐽(∁ 𝐽(∁ ADJ iajs-2560 214 22 ) ) PUNCT iajs-2560 214 23 or or CCONJ iajs-2560 214 24 𝐼2𝐿3 𝐼2𝐿3 PROPN iajs-2560 214 25 βŠ† βŠ† NUM iajs-2560 214 26 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 214 27 ) ) PUNCT iajs-2560 214 28 + + SYM iajs-2560 214 29 𝐽(∁ 𝐽(∁ PROPN iajs-2560 214 30 ) ) PUNCT iajs-2560 214 31 , , PUNCT iajs-2560 214 32 it it PRON iajs-2560 214 33 follows follow VERB iajs-2560 214 34 that that SCONJ iajs-2560 214 35 either either CCONJ iajs-2560 214 36 𝐿1𝐿3 𝐿1𝐿3 NOUN iajs-2560 214 37 βŠ† βŠ† NUM iajs-2560 214 38 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 214 39 ) ) PUNCT iajs-2560 214 40 + + SYM iajs-2560 214 41 𝐽(∁ 𝐽(∁ ADJ iajs-2560 214 42 ) ) PUNCT iajs-2560 214 43 or or CCONJ iajs-2560 214 44 𝐿2𝐿3 𝐿2𝐿3 PROPN iajs-2560 214 45 βŠ† βŠ† NUM iajs-2560 214 46 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 214 47 ) ) PUNCT iajs-2560 214 48 + + SYM iajs-2560 214 49 𝐽(∁ 𝐽(∁ ADJ iajs-2560 214 50 ) ) PUNCT iajs-2560 214 51 ( ( PUNCT iajs-2560 214 52 ⇐ ⇐ NOUN iajs-2560 214 53 ) ) PUNCT iajs-2560 214 54 assume assume VERB iajs-2560 214 55 that that SCONJ iajs-2560 214 56 𝐼1𝐼2𝐾 𝐼1𝐼2𝐾 PROPN iajs-2560 214 57 βŠ† βŠ† NUM iajs-2560 214 58 𝐿 𝐿 PROPN iajs-2560 214 59 , , PUNCT iajs-2560 214 60 where where SCONJ iajs-2560 214 61 𝐼1 𝐼1 NOUN iajs-2560 214 62 , , PUNCT iajs-2560 214 63 𝐼2 𝐼2 NOUN iajs-2560 214 64 are be AUX iajs-2560 214 65 ideals ideal NOUN iajs-2560 214 66 of of ADP iajs-2560 214 67 𝑅 𝑅 NOUN iajs-2560 214 68 , , PUNCT iajs-2560 214 69 and and CCONJ iajs-2560 214 70 𝐾 𝐾 PROPN iajs-2560 214 71 is be AUX iajs-2560 214 72 a a DET iajs-2560 214 73 submodule submodule NOUN iajs-2560 214 74 of of ADP iajs-2560 214 75 ∁. ∁. NOUN iajs-2560 214 76 since since SCONJ iajs-2560 214 77 ∁ ∁ PROPN iajs-2560 214 78 is be AUX iajs-2560 214 79 a a DET iajs-2560 214 80 multiplication multiplication NOUN iajs-2560 214 81 , , PUNCT iajs-2560 214 82 then then ADV iajs-2560 214 83 𝐼1𝐼2𝐾 𝐼1𝐼2𝐾 NOUN iajs-2560 214 84 = = PUNCT iajs-2560 214 85 𝐿1𝐿2𝐾 𝐿1𝐿2𝐾 PROPN iajs-2560 215 1 βŠ† βŠ† X iajs-2560 215 2 𝐿 𝐿 PROPN iajs-2560 215 3 , , PUNCT iajs-2560 215 4 by by ADP iajs-2560 215 5 hypothesis hypothesis NOUN iajs-2560 215 6 either either CCONJ iajs-2560 215 7 𝐿1𝐾 𝐿1𝐾 NUM iajs-2560 215 8 βŠ† βŠ† NUM iajs-2560 215 9 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 215 10 ) ) PUNCT iajs-2560 215 11 + + SYM iajs-2560 215 12 𝐽(∁ 𝐽(∁ ADJ iajs-2560 215 13 ) ) PUNCT iajs-2560 215 14 or or CCONJ iajs-2560 215 15 𝐿2𝐾 𝐿2𝐾 PRON iajs-2560 215 16 βŠ† βŠ† NUM iajs-2560 215 17 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 215 18 ) ) PUNCT iajs-2560 215 19 + + SYM iajs-2560 215 20 𝐽(∁ 𝐽(∁ ADJ iajs-2560 215 21 ) ) PUNCT iajs-2560 215 22 or or CCONJ iajs-2560 215 23 𝐿1𝐿2 𝐿1𝐿2 VERB iajs-2560 215 24 βŠ† βŠ† NUM iajs-2560 215 25 [ [ X iajs-2560 215 26 𝐿 𝐿 PROPN iajs-2560 215 27 + + SYM iajs-2560 215 28 𝐽(∁ 𝐽(∁ PROPN iajs-2560 215 29 ) ) PUNCT iajs-2560 215 30 : : PUNCT iajs-2560 215 31 𝑅 𝑅 PROPN iajs-2560 215 32 ∁ ∁ PROPN iajs-2560 215 33 ] ] PUNCT iajs-2560 215 34 . . PUNCT iajs-2560 216 1 that that PRON iajs-2560 216 2 is be AUX iajs-2560 216 3 either either CCONJ iajs-2560 216 4 𝐼1𝐾 𝐼1𝐾 PROPN iajs-2560 216 5 βŠ† βŠ† NUM iajs-2560 216 6 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 216 7 ) ) PUNCT iajs-2560 217 1 + + CCONJ iajs-2560 217 2 𝐽(∁)or 𝐽(∁)or NUM iajs-2560 217 3 𝐼2𝐾 𝐼2𝐾 PRON iajs-2560 217 4 βŠ† βŠ† NUM iajs-2560 217 5 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 217 6 ) ) PUNCT iajs-2560 218 1 + + SYM iajs-2560 218 2 𝐽(∁ 𝐽(∁ ADJ iajs-2560 218 3 ) ) PUNCT iajs-2560 218 4 or or CCONJ iajs-2560 218 5 𝐼1𝐼2 𝐼1𝐼2 PRON iajs-2560 218 6 βŠ† βŠ† NUM iajs-2560 218 7 [ [ X iajs-2560 218 8 𝐿 𝐿 PROPN iajs-2560 218 9 + + SYM iajs-2560 218 10 𝐽(∁ 𝐽(∁ PROPN iajs-2560 218 11 ) ) PUNCT iajs-2560 218 12 : : PUNCT iajs-2560 219 1 𝑅 𝑅 PROPN iajs-2560 219 2 ∁ ∁ PROPN iajs-2560 219 3 ] ] PUNCT iajs-2560 219 4 . . PUNCT iajs-2560 220 1 then then ADV iajs-2560 220 2 by by ADP iajs-2560 220 3 proposition proposition NOUN iajs-2560 220 4 ( ( PUNCT iajs-2560 220 5 5 5 X iajs-2560 220 6 ) ) PUNCT iajs-2560 220 7 𝐿 𝐿 PROPN iajs-2560 220 8 is be AUX iajs-2560 220 9 a a DET iajs-2560 220 10 nearly nearly ADV iajs-2560 220 11 primary-2 primary-2 NOUN iajs-2560 220 12 - - PUNCT iajs-2560 220 13 absorbing absorb VERB iajs-2560 220 14 submodule submodule NOUN iajs-2560 220 15 of of ADP iajs-2560 220 16 ∁. ∁. PROPN iajs-2560 220 17 121 121 NUM iajs-2560 220 18 ibn ibn PROPN iajs-2560 220 19 al al PROPN iajs-2560 220 20 - - PUNCT iajs-2560 220 21 haitham haitham PROPN iajs-2560 220 22 jour jour X iajs-2560 220 23 . . PROPN iajs-2560 221 1 for for ADP iajs-2560 221 2 pure pure ADJ iajs-2560 221 3 & & CCONJ iajs-2560 221 4 appl appl PROPN iajs-2560 221 5 . . PUNCT iajs-2560 222 1 sci sci PROPN iajs-2560 222 2 . . PROPN iajs-2560 223 1 34 34 NUM iajs-2560 223 2 ( ( PUNCT iajs-2560 223 3 1 1 NUM iajs-2560 223 4 ) ) PUNCT iajs-2560 223 5 2021 2021 NUM iajs-2560 223 6 recall recall VERB iajs-2560 223 7 that that SCONJ iajs-2560 223 8 an an DET iajs-2560 223 9 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2560 223 10 𝑓 𝑓 X iajs-2560 223 11 : : PUNCT iajs-2560 223 12 ∁⟢ ∁⟢ PROPN iajs-2560 223 13 βˆΜ… βˆΜ… PROPN iajs-2560 223 14 is be AUX iajs-2560 223 15 called call VERB iajs-2560 223 16 a a DET iajs-2560 223 17 small small ADJ iajs-2560 223 18 epimorphism epimorphism NOUN iajs-2560 223 19 if if SCONJ iajs-2560 223 20 ker ker PROPN iajs-2560 223 21 ( ( PUNCT iajs-2560 223 22 𝑓 𝑓 X iajs-2560 223 23 ) ) PUNCT iajs-2560 223 24 is be AUX iajs-2560 223 25 a a DET iajs-2560 223 26 small small ADJ iajs-2560 223 27 submodule submodule NOUN iajs-2560 223 28 of of ADP iajs-2560 223 29 ∁ ∁ PROPN iajs-2560 223 30 [ [ X iajs-2560 223 31 10 10 NUM iajs-2560 223 32 ] ] PUNCT iajs-2560 223 33 . . PUNCT iajs-2560 224 1 lemma lemma PROPN iajs-2560 224 2 10 10 NUM iajs-2560 225 1 [ [ SYM iajs-2560 225 2 10 10 NUM iajs-2560 225 3 , , PUNCT iajs-2560 225 4 corollary corollary ADJ iajs-2560 225 5 ( ( PUNCT iajs-2560 225 6 9.1.5 9.1.5 NUM iajs-2560 225 7 ) ) PUNCT iajs-2560 225 8 ] ] PUNCT iajs-2560 225 9 let let VERB iajs-2560 225 10 ∁ ∁ PROPN iajs-2560 225 11 and and CCONJ iajs-2560 225 12 βˆΜ… βˆΜ… PROPN iajs-2560 225 13 be be AUX iajs-2560 225 14 an an DET iajs-2560 225 15 𝑅-module 𝑅-module PROPN iajs-2560 225 16 and and CCONJ iajs-2560 225 17 𝑁 𝑁 PROPN iajs-2560 225 18 be be VERB iajs-2560 225 19 a a DET iajs-2560 225 20 proper proper ADJ iajs-2560 225 21 submodule submodule NOUN iajs-2560 225 22 of of ADP iajs-2560 225 23 ∁. ∁. NOUN iajs-2560 225 24 if if SCONJ iajs-2560 225 25 𝑓 𝑓 X iajs-2560 225 26 : : PUNCT iajs-2560 225 27 ∁⟢ ∁⟢ PROPN iajs-2560 225 28 βˆΜ… βˆΜ… PROPN iajs-2560 225 29 is be AUX iajs-2560 225 30 an an DET iajs-2560 225 31 𝑅homomorphism 𝑅homomorphism PROPN iajs-2560 225 32 , , PUNCT iajs-2560 225 33 then then ADV iajs-2560 225 34 𝑓(𝐽(∁ 𝑓(𝐽(∁ PROPN iajs-2560 225 35 ) ) PUNCT iajs-2560 225 36 ) ) PUNCT iajs-2560 226 1 βŠ† βŠ† NUM iajs-2560 226 2 𝐽(βˆΜ… 𝐽(βˆΜ… NOUN iajs-2560 226 3 ) ) PUNCT iajs-2560 226 4 and and CCONJ iajs-2560 226 5 𝐽(𝑁 𝐽(𝑁 NOUN iajs-2560 226 6 ) ) PUNCT iajs-2560 226 7 βŠ† βŠ† NUM iajs-2560 226 8 𝐽(∁ 𝐽(∁ NOUN iajs-2560 226 9 ) ) PUNCT iajs-2560 226 10 . . PUNCT iajs-2560 227 1 if if SCONJ iajs-2560 227 2 𝑓 𝑓 PRON iajs-2560 227 3 is be AUX iajs-2560 227 4 a a DET iajs-2560 227 5 small small ADJ iajs-2560 227 6 epimorphism epimorphism NOUN iajs-2560 227 7 then then ADV iajs-2560 227 8 𝑓(𝐽(∁ 𝑓(𝐽(∁ PROPN iajs-2560 227 9 ) ) PUNCT iajs-2560 227 10 ) ) PUNCT iajs-2560 228 1 = = PUNCT iajs-2560 228 2 𝐽(βˆΜ… 𝐽(βˆΜ… X iajs-2560 228 3 ) ) PUNCT iajs-2560 228 4 = = SYM iajs-2560 228 5 𝐽(𝑓(∁ 𝐽(𝑓(∁ NUM iajs-2560 228 6 ) ) PUNCT iajs-2560 228 7 ) ) PUNCT iajs-2560 228 8 and and CCONJ iajs-2560 228 9 𝐽(∁ 𝐽(∁ ADP iajs-2560 228 10 ) ) PUNCT iajs-2560 228 11 = = SYM iajs-2560 228 12 π‘“βˆ’1(𝐽(βˆΜ… π‘“βˆ’1(𝐽(βˆΜ… NOUN iajs-2560 228 13 ) ) PUNCT iajs-2560 228 14 ) ) PUNCT iajs-2560 228 15 . . PUNCT iajs-2560 229 1 lemma lemma PROPN iajs-2560 229 2 11 11 NUM iajs-2560 230 1 [ [ X iajs-2560 230 2 2 2 NUM iajs-2560 230 3 ] ] PUNCT iajs-2560 230 4 let let VERB iajs-2560 230 5 𝑓 𝑓 X iajs-2560 230 6 : : PUNCT iajs-2560 230 7 ∁⟢ ∁⟢ PROPN iajs-2560 230 8 βˆΜ… βˆΜ… PROPN iajs-2560 230 9 be be AUX iajs-2560 230 10 an an DET iajs-2560 230 11 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2560 230 12 and and CCONJ iajs-2560 230 13 𝐿 𝐿 PROPN iajs-2560 230 14 is be AUX iajs-2560 230 15 a a DET iajs-2560 230 16 submodule submodule NOUN iajs-2560 230 17 of of ADP iajs-2560 230 18 βˆΜ… βˆΜ… PROPN iajs-2560 230 19 with with ADP iajs-2560 230 20 ker(𝑓 ker(𝑓 PROPN iajs-2560 230 21 ) ) PUNCT iajs-2560 230 22 βŠ† βŠ† NUM iajs-2560 230 23 𝐿 𝐿 PROPN iajs-2560 230 24 , , PUNCT iajs-2560 230 25 then then ADV iajs-2560 230 26 𝑓(π‘Ÿπ‘Žπ‘‘βˆ(𝐿 𝑓(π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 230 27 ) ) PUNCT iajs-2560 230 28 ) ) PUNCT iajs-2560 230 29 ) ) PUNCT iajs-2560 231 1 = = PUNCT iajs-2560 231 2 π‘Ÿπ‘Žπ‘‘βˆΜ…(𝑓(𝐿 π‘Ÿπ‘Žπ‘‘βˆΜ…(𝑓(𝐿 NUM iajs-2560 231 3 ) ) PUNCT iajs-2560 231 4 ) ) PUNCT iajs-2560 231 5 . . PUNCT iajs-2560 232 1 proposition proposition NOUN iajs-2560 232 2 12 12 NUM iajs-2560 232 3 let let VERB iajs-2560 232 4 𝑓 𝑓 X iajs-2560 232 5 : : PUNCT iajs-2560 232 6 ∁⟢ ∁⟢ PROPN iajs-2560 232 7 βˆΜ… βˆΜ… PROPN iajs-2560 232 8 be be AUX iajs-2560 232 9 a a DET iajs-2560 232 10 small small ADJ iajs-2560 232 11 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2560 232 12 and and CCONJ iajs-2560 232 13 οΏ½ οΏ½ NOUN iajs-2560 232 14 Μ… Μ… NOUN iajs-2560 232 15 οΏ½ οΏ½ NOUN iajs-2560 232 16 is be AUX iajs-2560 232 17 a a DET iajs-2560 232 18 nearly nearly ADV iajs-2560 232 19 primary-2 primary-2 NOUN iajs-2560 232 20 - - PUNCT iajs-2560 232 21 absorbing absorb VERB iajs-2560 232 22 submodule submodule NOUN iajs-2560 232 23 of of ADP iajs-2560 232 24 βˆΜ…. βˆΜ…. PROPN iajs-2560 232 25 then then ADV iajs-2560 232 26 π‘“βˆ’1( π‘“βˆ’1( PROPN iajs-2560 232 27 οΏ½ οΏ½ PROPN iajs-2560 232 28 Μ… Μ… NOUN iajs-2560 232 29 οΏ½ οΏ½ NOUN iajs-2560 232 30 ) ) PUNCT iajs-2560 232 31 be be VERB iajs-2560 232 32 a a DET iajs-2560 232 33 nearly nearly ADV iajs-2560 232 34 primary-2 primary-2 NOUN iajs-2560 232 35 - - PUNCT iajs-2560 232 36 absorbing absorb VERB iajs-2560 232 37 submodule submodule NOUN iajs-2560 232 38 of of ADP iajs-2560 232 39 ∁. ∁. NOUN iajs-2560 232 40 proof proof NOUN iajs-2560 232 41 : : PUNCT iajs-2560 232 42 let let VERB iajs-2560 232 43 π‘Ÿπ‘ π‘₯ π‘Ÿπ‘ π‘₯ NUM iajs-2560 232 44 ∈ ∈ PROPN iajs-2560 232 45 π‘“βˆ’1( π‘“βˆ’1( PROPN iajs-2560 232 46 οΏ½ οΏ½ PROPN iajs-2560 232 47 Μ… Μ… NOUN iajs-2560 232 48 οΏ½ οΏ½ NOUN iajs-2560 232 49 ) ) PUNCT iajs-2560 232 50 , , PUNCT iajs-2560 232 51 where where SCONJ iajs-2560 232 52 π‘Ÿ π‘Ÿ X iajs-2560 232 53 , , PUNCT iajs-2560 232 54 𝑠 𝑠 PROPN iajs-2560 232 55 ∈ ∈ PROPN iajs-2560 232 56 𝑅 𝑅 PROPN iajs-2560 232 57 , , PUNCT iajs-2560 232 58 π‘₯ π‘₯ PRON iajs-2560 232 59 ∈ ∈ ADJ iajs-2560 232 60 ∁ ∁ NUM iajs-2560 232 61 , , PUNCT iajs-2560 232 62 impling imple VERB iajs-2560 232 63 that that SCONJ iajs-2560 232 64 π‘Ÿπ‘ π‘“(π‘₯ π‘Ÿπ‘ π‘“(π‘₯ NOUN iajs-2560 232 65 ) ) PUNCT iajs-2560 232 66 ∈ ∈ PROPN iajs-2560 232 67 οΏ½ οΏ½ PROPN iajs-2560 232 68 Μ… Μ… NOUN iajs-2560 232 69 οΏ½ οΏ½ NOUN iajs-2560 232 70 . . PUNCT iajs-2560 233 1 since since SCONJ iajs-2560 233 2 οΏ½ οΏ½ PROPN iajs-2560 233 3 Μ… Μ… NOUN iajs-2560 233 4 οΏ½ οΏ½ NOUN iajs-2560 233 5 is be AUX iajs-2560 233 6 a a DET iajs-2560 233 7 nearly nearly ADV iajs-2560 233 8 primary-2 primary-2 NOUN iajs-2560 233 9 - - PUNCT iajs-2560 233 10 absorbing absorb VERB iajs-2560 233 11 submodule submodule NOUN iajs-2560 233 12 of of ADP iajs-2560 233 13 βˆΜ…. βˆΜ…. PRON iajs-2560 233 14 it it PRON iajs-2560 233 15 follows follow VERB iajs-2560 233 16 that that SCONJ iajs-2560 233 17 either either CCONJ iajs-2560 233 18 π‘Ÿπ‘“(π‘₯ π‘Ÿπ‘“(π‘₯ NOUN iajs-2560 233 19 ) ) PUNCT iajs-2560 233 20 ∈ ∈ PROPN iajs-2560 233 21 π‘Ÿπ‘Žπ‘‘βˆΜ…( π‘Ÿπ‘Žπ‘‘βˆΜ…( PROPN iajs-2560 233 22 οΏ½ οΏ½ PROPN iajs-2560 233 23 Μ… Μ… NOUN iajs-2560 233 24 οΏ½ οΏ½ NOUN iajs-2560 233 25 ) ) PUNCT iajs-2560 233 26 + + NUM iajs-2560 233 27 𝐽(βˆΜ… 𝐽(βˆΜ… NUM iajs-2560 233 28 ) ) PUNCT iajs-2560 233 29 or or CCONJ iajs-2560 233 30 𝑠𝑓(π‘₯ 𝑠𝑓(π‘₯ NOUN iajs-2560 233 31 ) ) PUNCT iajs-2560 233 32 ∈ ∈ PROPN iajs-2560 233 33 π‘Ÿπ‘Žπ‘‘βˆΜ…( π‘Ÿπ‘Žπ‘‘βˆΜ…( PROPN iajs-2560 233 34 οΏ½ οΏ½ PROPN iajs-2560 233 35 Μ… Μ… NOUN iajs-2560 233 36 οΏ½ οΏ½ NOUN iajs-2560 233 37 ) ) PUNCT iajs-2560 233 38 + + NUM iajs-2560 233 39 𝐽(βˆΜ… 𝐽(βˆΜ… NOUN iajs-2560 233 40 ) ) PUNCT iajs-2560 233 41 or or CCONJ iajs-2560 233 42 π‘ π‘ŸβˆΜ… π‘ π‘ŸβˆΜ… NOUN iajs-2560 233 43 βŠ† βŠ† NUM iajs-2560 233 44 οΏ½ οΏ½ NOUN iajs-2560 233 45 Μ… Μ… NOUN iajs-2560 233 46 οΏ½ οΏ½ NOUN iajs-2560 233 47 + + CCONJ iajs-2560 233 48 𝐽(βˆΜ… 𝐽(βˆΜ… NOUN iajs-2560 233 49 ) ) PUNCT iajs-2560 233 50 . . PUNCT iajs-2560 234 1 thus thus ADV iajs-2560 234 2 by by ADP iajs-2560 234 3 lemma lemma PROPN iajs-2560 234 4 ( ( PUNCT iajs-2560 234 5 10 10 NUM iajs-2560 234 6 ) ) PUNCT iajs-2560 234 7 and and CCONJ iajs-2560 234 8 by by ADP iajs-2560 234 9 lemma lemma PROPN iajs-2560 234 10 ( ( PUNCT iajs-2560 234 11 11 11 NUM iajs-2560 234 12 ) ) PUNCT iajs-2560 234 13 , , PUNCT iajs-2560 234 14 we we PRON iajs-2560 234 15 have have VERB iajs-2560 234 16 either either CCONJ iajs-2560 234 17 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PROPN iajs-2560 234 18 ∈ ∈ PROPN iajs-2560 234 19 π‘“βˆ’1(π‘Ÿπ‘Žπ‘‘βˆΜ…( π‘“βˆ’1(π‘Ÿπ‘Žπ‘‘βˆΜ…( PROPN iajs-2560 234 20 οΏ½ οΏ½ SYM iajs-2560 234 21 Μ… Μ… NOUN iajs-2560 234 22 οΏ½ οΏ½ NOUN iajs-2560 234 23 ) ) PUNCT iajs-2560 234 24 + + NUM iajs-2560 234 25 π‘“βˆ’1(𝐽(βˆΜ… π‘“βˆ’1(𝐽(βˆΜ… NOUN iajs-2560 234 26 ) ) PUNCT iajs-2560 234 27 ) ) PUNCT iajs-2560 235 1 = = SYM iajs-2560 235 2 π‘Ÿπ‘Žπ‘‘βˆ(π‘“βˆ’1( π‘Ÿπ‘Žπ‘‘βˆ(π‘“βˆ’1( NUM iajs-2560 235 3 οΏ½ οΏ½ NOUN iajs-2560 235 4 Μ… Μ… NOUN iajs-2560 235 5 οΏ½ οΏ½ NOUN iajs-2560 235 6 ) ) PUNCT iajs-2560 235 7 ) ) PUNCT iajs-2560 236 1 + + PUNCT iajs-2560 236 2 𝐽(∁ 𝐽(∁ ADV iajs-2560 236 3 ) ) PUNCT iajs-2560 236 4 or or CCONJ iajs-2560 236 5 𝑠π‘₯ 𝑠π‘₯ ADP iajs-2560 236 6 ∈ ∈ PROPN iajs-2560 236 7 π‘“βˆ’1(π‘Ÿπ‘Žπ‘‘βˆΜ…( π‘“βˆ’1(π‘Ÿπ‘Žπ‘‘βˆΜ…( PROPN iajs-2560 236 8 οΏ½ οΏ½ PROPN iajs-2560 236 9 Μ… Μ… NOUN iajs-2560 236 10 οΏ½ οΏ½ NOUN iajs-2560 236 11 ) ) PUNCT iajs-2560 236 12 + + NUM iajs-2560 236 13 π‘“βˆ’1(𝐽(βˆΜ… π‘“βˆ’1(𝐽(βˆΜ… NOUN iajs-2560 236 14 ) ) PUNCT iajs-2560 236 15 ) ) PUNCT iajs-2560 237 1 = = SYM iajs-2560 237 2 π‘Ÿπ‘Žπ‘‘βˆ(π‘“βˆ’1( π‘Ÿπ‘Žπ‘‘βˆ(π‘“βˆ’1( NUM iajs-2560 237 3 οΏ½ οΏ½ NOUN iajs-2560 237 4 Μ… Μ… NOUN iajs-2560 237 5 οΏ½ οΏ½ NOUN iajs-2560 237 6 ) ) PUNCT iajs-2560 237 7 ) ) PUNCT iajs-2560 238 1 + + PUNCT iajs-2560 238 2 𝐽(∁ 𝐽(∁ ADJ iajs-2560 238 3 ) ) PUNCT iajs-2560 238 4 or or CCONJ iajs-2560 238 5 π‘Ÿπ‘ βˆ π‘Ÿπ‘ βˆ ADJ iajs-2560 238 6 βŠ† βŠ† NUM iajs-2560 238 7 π‘“βˆ’1(𝐾 π‘“βˆ’1(𝐾 NUM iajs-2560 238 8 ) ) PUNCT iajs-2560 238 9 + + SYM iajs-2560 238 10 𝐽(∁ 𝐽(∁ ADJ iajs-2560 238 11 ) ) PUNCT iajs-2560 238 12 . . PUNCT iajs-2560 239 1 hence hence ADV iajs-2560 239 2 π‘“βˆ’1( π‘“βˆ’1( PROPN iajs-2560 239 3 οΏ½ οΏ½ PROPN iajs-2560 239 4 Μ… Μ… NOUN iajs-2560 239 5 οΏ½ οΏ½ NOUN iajs-2560 239 6 ) ) PUNCT iajs-2560 239 7 is be AUX iajs-2560 239 8 a a DET iajs-2560 239 9 nearly nearly ADV iajs-2560 239 10 primary-2 primary-2 NOUN iajs-2560 239 11 - - PUNCT iajs-2560 239 12 absorbing absorbing ADJ iajs-2560 239 13 submodule submodule NOUN iajs-2560 239 14 of of ADP iajs-2560 239 15 ∁. ∁. NOUN iajs-2560 239 16 proposition proposition NOUN iajs-2560 239 17 13 13 NUM iajs-2560 239 18 let let VERB iajs-2560 239 19 𝑓 𝑓 PRON iajs-2560 239 20 : : PUNCT iajs-2560 239 21 ∁⟢ ∁⟢ PROPN iajs-2560 239 22 βˆΜ… βˆΜ… PROPN iajs-2560 239 23 be be AUX iajs-2560 239 24 a a DET iajs-2560 239 25 small small ADJ iajs-2560 239 26 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2560 239 27 and and CCONJ iajs-2560 239 28 𝐾 𝐾 PROPN iajs-2560 239 29 is be AUX iajs-2560 239 30 a a DET iajs-2560 239 31 nearly nearly ADV iajs-2560 239 32 primary-2 primary-2 NOUN iajs-2560 239 33 - - PUNCT iajs-2560 239 34 absorbing absorb VERB iajs-2560 239 35 submodule submodule NOUN iajs-2560 239 36 of of ADP iajs-2560 239 37 ∁ ∁ PROPN iajs-2560 239 38 with with ADP iajs-2560 239 39 ker ker PROPN iajs-2560 239 40 ( ( PUNCT iajs-2560 239 41 𝑓 𝑓 PROPN iajs-2560 239 42 ) ) PUNCT iajs-2560 239 43 βŠ† βŠ† PROPN iajs-2560 239 44 𝐾 𝐾 PROPN iajs-2560 239 45 . . PUNCT iajs-2560 240 1 then then ADV iajs-2560 240 2 𝑓(𝐾 𝑓(𝐾 NUM iajs-2560 240 3 ) ) PUNCT iajs-2560 240 4 is be AUX iajs-2560 240 5 a a DET iajs-2560 240 6 nearly nearly ADV iajs-2560 240 7 primary-2 primary-2 NOUN iajs-2560 240 8 - - PUNCT iajs-2560 240 9 absorbing absorb VERB iajs-2560 240 10 submodule submodule NOUN iajs-2560 240 11 of of ADP iajs-2560 240 12 βˆΜ…. βˆΜ…. ADP iajs-2560 240 13 proof proof NOUN iajs-2560 240 14 : : PUNCT iajs-2560 240 15 let let VERB iajs-2560 240 16 π‘Ÿπ‘  π‘Ÿπ‘  PRON iajs-2560 240 17 οΏ½ οΏ½ NOUN iajs-2560 240 18 Μ… Μ… NOUN iajs-2560 240 19 οΏ½ οΏ½ NOUN iajs-2560 240 20 ∈ ∈ PROPN iajs-2560 240 21 𝑓(𝐾 𝑓(𝐾 PROPN iajs-2560 240 22 ) ) PUNCT iajs-2560 240 23 , , PUNCT iajs-2560 240 24 where where SCONJ iajs-2560 240 25 π‘Ÿ π‘Ÿ X iajs-2560 240 26 , , PUNCT iajs-2560 240 27 𝑠 𝑠 PROPN iajs-2560 240 28 ∈ ∈ PROPN iajs-2560 240 29 𝑅 𝑅 PROPN iajs-2560 240 30 , , PUNCT iajs-2560 240 31 οΏ½ οΏ½ NOUN iajs-2560 240 32 Μ… Μ… NOUN iajs-2560 240 33 οΏ½ οΏ½ PROPN iajs-2560 240 34 ∈ ∈ PROPN iajs-2560 240 35 βˆΜ…. βˆΜ…. AUX iajs-2560 240 36 since since SCONJ iajs-2560 240 37 𝑓 𝑓 PROPN iajs-2560 240 38 is be AUX iajs-2560 240 39 onto onto ADP iajs-2560 240 40 , , PUNCT iajs-2560 240 41 then then ADV iajs-2560 240 42 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-2560 240 43 ) ) PUNCT iajs-2560 240 44 = = SYM iajs-2560 240 45 οΏ½ οΏ½ NOUN iajs-2560 240 46 Μ… Μ… NOUN iajs-2560 240 47 οΏ½ οΏ½ PROPN iajs-2560 240 48 for for ADP iajs-2560 240 49 someπ‘₯ someπ‘₯ NOUN iajs-2560 240 50 ∈ ∈ PROPN iajs-2560 240 51 ∁. ∁. NOUN iajs-2560 240 52 thus thus ADV iajs-2560 240 53 π‘Ÿπ‘ π‘“(π‘₯ π‘Ÿπ‘ π‘“(π‘₯ NOUN iajs-2560 240 54 ) ) PUNCT iajs-2560 240 55 ∈ ∈ NOUN iajs-2560 240 56 𝑓(𝐾 𝑓(𝐾 PROPN iajs-2560 240 57 ) ) PUNCT iajs-2560 240 58 , , PUNCT iajs-2560 240 59 implies imply VERB iajs-2560 240 60 that that SCONJ iajs-2560 240 61 π‘Ÿπ‘ π‘“(π‘₯ π‘Ÿπ‘ π‘“(π‘₯ NOUN iajs-2560 240 62 ) ) PUNCT iajs-2560 240 63 = = SYM iajs-2560 240 64 𝑓(π‘˜ 𝑓(π‘˜ PROPN iajs-2560 240 65 ) ) PUNCT iajs-2560 240 66 for for ADP iajs-2560 240 67 some some DET iajs-2560 240 68 π‘˜ π‘˜ PRON iajs-2560 240 69 ∈ ∈ PROPN iajs-2560 240 70 𝐾 𝐾 PROPN iajs-2560 240 71 , , PUNCT iajs-2560 240 72 it it PRON iajs-2560 240 73 follows follow VERB iajs-2560 240 74 that that DET iajs-2560 240 75 𝑓(π‘Ÿπ‘ π‘₯ 𝑓(π‘Ÿπ‘ π‘₯ NOUN iajs-2560 240 76 βˆ’ βˆ’ NOUN iajs-2560 240 77 π‘˜ π‘˜ NOUN iajs-2560 240 78 ) ) PUNCT iajs-2560 241 1 = = SYM iajs-2560 241 2 π‘œ π‘œ NOUN iajs-2560 241 3 , , PUNCT iajs-2560 241 4 implies imply VERB iajs-2560 241 5 that that SCONJ iajs-2560 241 6 π‘Ÿπ‘ π‘₯ π‘Ÿπ‘ π‘₯ VERB iajs-2560 241 7 βˆ’ βˆ’ PROPN iajs-2560 241 8 π‘˜ π‘˜ PRON iajs-2560 241 9 ∈ ∈ PROPN iajs-2560 241 10 ker ker NOUN iajs-2560 241 11 ( ( PUNCT iajs-2560 241 12 𝑓 𝑓 X iajs-2560 241 13 ) ) PUNCT iajs-2560 241 14 βŠ† βŠ† PROPN iajs-2560 241 15 𝐾 𝐾 PROPN iajs-2560 241 16 , , PUNCT iajs-2560 241 17 then then ADV iajs-2560 241 18 π‘Ÿπ‘ π‘₯ π‘Ÿπ‘ π‘₯ NUM iajs-2560 241 19 ∈ ∈ NOUN iajs-2560 241 20 𝐾. 𝐾. PROPN iajs-2560 241 21 but but CCONJ iajs-2560 241 22 𝐾 𝐾 PROPN iajs-2560 241 23 is be AUX iajs-2560 241 24 a a DET iajs-2560 241 25 nearly nearly ADV iajs-2560 241 26 primary-2absorbing primary-2absorbe VERB iajs-2560 241 27 submodule submodule NOUN iajs-2560 241 28 of of ADP iajs-2560 241 29 ∁ ∁ PROPN iajs-2560 241 30 , , PUNCT iajs-2560 241 31 then then ADV iajs-2560 241 32 either either CCONJ iajs-2560 241 33 π‘Ÿπ‘₯ π‘Ÿπ‘₯ PROPN iajs-2560 241 34 ∈ ∈ PROPN iajs-2560 241 35 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 241 36 ) ) PUNCT iajs-2560 241 37 + + PUNCT iajs-2560 241 38 𝐽(∁ 𝐽(∁ ADJ iajs-2560 241 39 ) ) PUNCT iajs-2560 241 40 or or CCONJ iajs-2560 241 41 𝑠π‘₯ 𝑠π‘₯ ADP iajs-2560 241 42 ∈ ∈ PROPN iajs-2560 241 43 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 π‘Ÿπ‘Žπ‘‘βˆ(𝐾 NOUN iajs-2560 241 44 ) ) PUNCT iajs-2560 241 45 + + PUNCT iajs-2560 241 46 𝐽(∁ 𝐽(∁ ADJ iajs-2560 241 47 ) ) PUNCT iajs-2560 241 48 or or CCONJ iajs-2560 241 49 π‘Ÿπ‘ βˆβŠ† π‘Ÿπ‘ βˆβŠ† NUM iajs-2560 241 50 𝐾 𝐾 PROPN iajs-2560 241 51 + + CCONJ iajs-2560 241 52 𝐽(∁ 𝐽(∁ PROPN iajs-2560 241 53 ) ) PUNCT iajs-2560 241 54 , , PUNCT iajs-2560 241 55 it it PRON iajs-2560 241 56 follows follow VERB iajs-2560 241 57 that that SCONJ iajs-2560 241 58 by by ADP iajs-2560 241 59 lemma(11 lemma(11 PROPN iajs-2560 241 60 ) ) PUNCT iajs-2560 241 61 either either CCONJ iajs-2560 241 62 π‘Ÿπ‘“(π‘₯ π‘Ÿπ‘“(π‘₯ NOUN iajs-2560 241 63 ) ) PUNCT iajs-2560 241 64 ∈ ∈ PROPN iajs-2560 241 65 𝑓(π‘Ÿπ‘Žπ‘‘βˆ(𝐾 𝑓(π‘Ÿπ‘Žπ‘‘βˆ(𝐾 PROPN iajs-2560 241 66 ) ) PUNCT iajs-2560 241 67 ) ) PUNCT iajs-2560 242 1 + + PUNCT iajs-2560 242 2 𝑓(𝐽(∁ 𝑓(𝐽(∁ NOUN iajs-2560 242 3 ) ) PUNCT iajs-2560 242 4 ) ) PUNCT iajs-2560 243 1 βŠ† βŠ† NUM iajs-2560 243 2 π‘Ÿπ‘Žπ‘‘βˆΜ…(𝑓(𝐾 π‘Ÿπ‘Žπ‘‘βˆΜ…(𝑓(𝐾 NUM iajs-2560 243 3 ) ) PUNCT iajs-2560 243 4 ) ) PUNCT iajs-2560 244 1 + + PUNCT iajs-2560 244 2 𝑓(𝐽(∁ 𝑓(𝐽(∁ NOUN iajs-2560 244 3 ) ) PUNCT iajs-2560 244 4 ) ) PUNCT iajs-2560 244 5 or or CCONJ iajs-2560 244 6 𝑠𝑓(π‘₯ 𝑠𝑓(π‘₯ NOUN iajs-2560 244 7 ) ) PUNCT iajs-2560 244 8 ∈ ∈ PROPN iajs-2560 244 9 𝑓(π‘Ÿπ‘Žπ‘‘βˆ(𝐾 𝑓(π‘Ÿπ‘Žπ‘‘βˆ(𝐾 PROPN iajs-2560 244 10 ) ) PUNCT iajs-2560 244 11 ) ) PUNCT iajs-2560 245 1 + + PUNCT iajs-2560 245 2 𝑓(𝐽(∁ 𝑓(𝐽(∁ NOUN iajs-2560 245 3 ) ) PUNCT iajs-2560 245 4 ) ) PUNCT iajs-2560 246 1 βŠ† βŠ† NUM iajs-2560 246 2 π‘Ÿπ‘Žπ‘‘βˆΜ…(𝑓(𝐾 π‘Ÿπ‘Žπ‘‘βˆΜ…(𝑓(𝐾 NUM iajs-2560 246 3 ) ) PUNCT iajs-2560 246 4 ) ) PUNCT iajs-2560 247 1 + + PUNCT iajs-2560 247 2 𝑓(𝐽(∁ 𝑓(𝐽(∁ NOUN iajs-2560 247 3 ) ) PUNCT iajs-2560 247 4 ) ) PUNCT iajs-2560 247 5 or or CCONJ iajs-2560 247 6 π‘Ÿπ‘ π‘“(∁ π‘Ÿπ‘ π‘“(∁ PROPN iajs-2560 247 7 ) ) PUNCT iajs-2560 247 8 βŠ† βŠ† NUM iajs-2560 247 9 𝑓(𝐾 𝑓(𝐾 NUM iajs-2560 247 10 ) ) PUNCT iajs-2560 248 1 + + SYM iajs-2560 248 2 𝑓(𝐽(∁ 𝑓(𝐽(∁ NOUN iajs-2560 248 3 ) ) PUNCT iajs-2560 248 4 ) ) PUNCT iajs-2560 249 1 βŠ† βŠ† NUM iajs-2560 249 2 𝑓(𝐾 𝑓(𝐾 NOUN iajs-2560 249 3 ) ) PUNCT iajs-2560 249 4 + + NUM iajs-2560 249 5 𝐽(βˆΜ… 𝐽(βˆΜ… NOUN iajs-2560 249 6 ) ) PUNCT iajs-2560 249 7 . . PUNCT iajs-2560 250 1 also also ADV iajs-2560 250 2 , , PUNCT iajs-2560 250 3 by by ADP iajs-2560 250 4 lemma lemma PROPN iajs-2560 250 5 ( ( PUNCT iajs-2560 250 6 12 12 NUM iajs-2560 250 7 ) ) PUNCT iajs-2560 250 8 either either CCONJ iajs-2560 250 9 π‘Ÿ π‘Ÿ NOUN iajs-2560 250 10 οΏ½ οΏ½ NOUN iajs-2560 250 11 Μ… Μ… NOUN iajs-2560 250 12 οΏ½ οΏ½ NOUN iajs-2560 250 13 ∈ ∈ PROPN iajs-2560 250 14 π‘Ÿπ‘Žπ‘‘βˆΜ…(𝑓(𝐾 π‘Ÿπ‘Žπ‘‘βˆΜ…(𝑓(𝐾 PROPN iajs-2560 250 15 ) ) PUNCT iajs-2560 250 16 ) ) PUNCT iajs-2560 251 1 + + CCONJ iajs-2560 251 2 𝐽(βˆΜ… 𝐽(βˆΜ… NOUN iajs-2560 251 3 ) ) PUNCT iajs-2560 251 4 or or CCONJ iajs-2560 251 5 𝑠 𝑠 ADP iajs-2560 251 6 οΏ½ οΏ½ NOUN iajs-2560 251 7 Μ… Μ… NOUN iajs-2560 251 8 οΏ½ οΏ½ PROPN iajs-2560 251 9 ∈ ∈ PROPN iajs-2560 251 10 π‘Ÿπ‘Žπ‘‘βˆΜ…(𝑓(𝐾 π‘Ÿπ‘Žπ‘‘βˆΜ…(𝑓(𝐾 PROPN iajs-2560 251 11 ) ) PUNCT iajs-2560 251 12 ) ) PUNCT iajs-2560 252 1 + + CCONJ iajs-2560 252 2 𝐽(βˆΜ… 𝐽(βˆΜ… NOUN iajs-2560 252 3 ) ) PUNCT iajs-2560 252 4 or or CCONJ iajs-2560 252 5 π‘Ÿπ‘ βˆΜ… π‘Ÿπ‘ βˆΜ… PROPN iajs-2560 252 6 βŠ† βŠ† NUM iajs-2560 252 7 𝑓(𝐾 𝑓(𝐾 NUM iajs-2560 252 8 ) ) PUNCT iajs-2560 253 1 + + NUM iajs-2560 253 2 𝐽(βˆΜ… 𝐽(βˆΜ… NOUN iajs-2560 253 3 ) ) PUNCT iajs-2560 253 4 . . PUNCT iajs-2560 254 1 hence hence ADV iajs-2560 254 2 𝑓(𝐾 𝑓(𝐾 NUM iajs-2560 254 3 ) ) PUNCT iajs-2560 254 4 is be AUX iajs-2560 254 5 a a DET iajs-2560 254 6 nearly nearly ADV iajs-2560 254 7 primary-2absorbing primary-2absorbe VERB iajs-2560 254 8 submodule submodule NOUN iajs-2560 254 9 of of ADP iajs-2560 254 10 βˆΜ…. βˆΜ…. INTJ iajs-2560 254 11 lemma lemma PROPN iajs-2560 254 12 14 14 NUM iajs-2560 255 1 [ [ X iajs-2560 255 2 9 9 NUM iajs-2560 255 3 , , PUNCT iajs-2560 255 4 theorem(2.12 theorem(2.12 NUM iajs-2560 255 5 ) ) PUNCT iajs-2560 255 6 ] ] PUNCT iajs-2560 255 7 let let VERB iajs-2560 255 8 𝑅 𝑅 PROPN iajs-2560 255 9 be be AUX iajs-2560 255 10 a a DET iajs-2560 255 11 commutative commutative ADJ iajs-2560 255 12 ring ring NOUN iajs-2560 255 13 with with ADP iajs-2560 255 14 identity identity NOUN iajs-2560 255 15 , , PUNCT iajs-2560 255 16 𝐿 𝐿 PROPN iajs-2560 255 17 be be VERB iajs-2560 255 18 a a DET iajs-2560 255 19 proper proper ADJ iajs-2560 255 20 submodule submodule NOUN iajs-2560 255 21 of of ADP iajs-2560 255 22 a a DET iajs-2560 255 23 multiplication multiplication NOUN iajs-2560 255 24 𝑅-module 𝑅-module PROPN iajs-2560 255 25 ∁ ∁ PROPN iajs-2560 255 26 and and CCONJ iajs-2560 255 27 𝐽 𝐽 NOUN iajs-2560 255 28 = = PUNCT iajs-2560 256 1 [ [ X iajs-2560 256 2 𝐿:𝑅 𝐿:𝑅 X iajs-2560 256 3 ∁ ∁ PROPN iajs-2560 256 4 ] ] PUNCT iajs-2560 256 5 . . PUNCT iajs-2560 257 1 then then ADV iajs-2560 257 2 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 257 3 ) ) PUNCT iajs-2560 258 1 = = SYM iajs-2560 258 2 √𝐽. √𝐽. PROPN iajs-2560 258 3 ∁= ∁= PROPN iajs-2560 258 4 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 258 5 ∁ ∁ PROPN iajs-2560 258 6 ] ] PUNCT iajs-2560 258 7 . . PUNCT iajs-2560 259 1 ∁. ∁. PROPN iajs-2560 259 2 lemma lemma PROPN iajs-2560 259 3 15 15 NUM iajs-2560 260 1 [ [ SYM iajs-2560 260 2 2 2 NUM iajs-2560 260 3 , , PUNCT iajs-2560 260 4 theorem theorem ADJ iajs-2560 260 5 1 1 NUM iajs-2560 260 6 , , PUNCT iajs-2560 260 7 ( ( PUNCT iajs-2560 260 8 1 1 NUM iajs-2560 260 9 ) ) PUNCT iajs-2560 260 10 ] ] PUNCT iajs-2560 261 1 let let VERB iajs-2560 261 2 ∁ ∁ PROPN iajs-2560 261 3 is be AUX iajs-2560 261 4 a a DET iajs-2560 261 5 module module NOUN iajs-2560 261 6 over over ADP iajs-2560 261 7 artinian artinian ADJ iajs-2560 261 8 ring ring PROPN iajs-2560 261 9 𝑅 𝑅 PROPN iajs-2560 261 10 , , PUNCT iajs-2560 261 11 then then ADV iajs-2560 261 12 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 261 13 ) ) PUNCT iajs-2560 261 14 . . PUNCT iajs-2560 262 1 ∁= ∁= PROPN iajs-2560 262 2 𝐽(∁ 𝐽(∁ PROPN iajs-2560 262 3 ) ) PUNCT iajs-2560 262 4 . . PUNCT iajs-2560 263 1 proposition proposition NOUN iajs-2560 263 2 16 16 NUM iajs-2560 263 3 let let VERB iajs-2560 263 4 ∁ ∁ NOUN iajs-2560 263 5 be be AUX iajs-2560 263 6 a a DET iajs-2560 263 7 multiplication multiplication NOUN iajs-2560 263 8 module module NOUN iajs-2560 263 9 over over ADP iajs-2560 263 10 an an DET iajs-2560 263 11 artinian artinian ADJ iajs-2560 263 12 ring ring NOUN iajs-2560 263 13 𝑅 𝑅 PROPN iajs-2560 263 14 and and CCONJ iajs-2560 263 15 𝐿 𝐿 PROPN iajs-2560 263 16 be be VERB iajs-2560 263 17 a a DET iajs-2560 263 18 proper proper ADJ iajs-2560 263 19 submodule submodule NOUN iajs-2560 263 20 of of ADP iajs-2560 263 21 ∁. ∁. NOUN iajs-2560 263 22 then then ADV iajs-2560 263 23 𝐿 𝐿 PROPN iajs-2560 263 24 is be AUX iajs-2560 263 25 a a DET iajs-2560 263 26 nearly nearly ADV iajs-2560 263 27 primary-2 primary-2 NOUN iajs-2560 263 28 - - PUNCT iajs-2560 263 29 absorbing absorb VERB iajs-2560 263 30 submodule submodule NOUN iajs-2560 263 31 of of ADP iajs-2560 263 32 ∁ ∁ PROPN iajs-2560 263 33 if if SCONJ iajs-2560 263 34 and and CCONJ iajs-2560 263 35 only only ADV iajs-2560 263 36 if if SCONJ iajs-2560 263 37 [ [ X iajs-2560 263 38 𝐿:𝑅 𝐿:𝑅 X iajs-2560 263 39 ∁ ∁ PROPN iajs-2560 263 40 ] ] PUNCT iajs-2560 263 41 is be AUX iajs-2560 263 42 a a DET iajs-2560 263 43 nearly nearly ADV iajs-2560 263 44 primary-2 primary-2 NOUN iajs-2560 263 45 - - PUNCT iajs-2560 263 46 absorbing absorbing ADJ iajs-2560 263 47 ideal ideal NOUN iajs-2560 263 48 of of ADP iajs-2560 263 49 𝑅. 𝑅. SYM iajs-2560 263 50 122 122 NUM iajs-2560 263 51 ibn ibn PROPN iajs-2560 263 52 al al PROPN iajs-2560 263 53 - - PUNCT iajs-2560 263 54 haitham haitham PROPN iajs-2560 263 55 jour jour X iajs-2560 263 56 . . PROPN iajs-2560 263 57 for for ADP iajs-2560 263 58 pure pure ADJ iajs-2560 263 59 & & CCONJ iajs-2560 263 60 appl appl PROPN iajs-2560 263 61 . . PUNCT iajs-2560 264 1 sci sci PROPN iajs-2560 264 2 . . PROPN iajs-2560 265 1 34 34 NUM iajs-2560 265 2 ( ( PUNCT iajs-2560 265 3 1 1 NUM iajs-2560 265 4 ) ) PUNCT iajs-2560 265 5 2021 2021 NUM iajs-2560 265 6 proof proof NOUN iajs-2560 265 7 : : PUNCT iajs-2560 265 8 ( ( PUNCT iajs-2560 265 9 β‡’ β‡’ NOUN iajs-2560 265 10 ) ) PUNCT iajs-2560 265 11 let let VERB iajs-2560 265 12 π‘Žπ‘πΌ π‘Žπ‘πΌ NOUN iajs-2560 266 1 βŠ† βŠ† NUM iajs-2560 266 2 [ [ X iajs-2560 266 3 𝐿:𝑅 𝐿:𝑅 X iajs-2560 266 4 ∁ ∁ PROPN iajs-2560 266 5 ] ] PUNCT iajs-2560 266 6 for for ADP iajs-2560 266 7 π‘Ž π‘Ž PROPN iajs-2560 266 8 , , PUNCT iajs-2560 266 9 𝑐 𝑐 PROPN iajs-2560 266 10 ∈ ∈ PROPN iajs-2560 266 11 𝑅 𝑅 PROPN iajs-2560 266 12 , , PUNCT iajs-2560 266 13 𝐼 𝐼 PROPN iajs-2560 266 14 is be AUX iajs-2560 266 15 an an DET iajs-2560 266 16 ideal ideal NOUN iajs-2560 266 17 of of ADP iajs-2560 266 18 𝑅 𝑅 PROPN iajs-2560 266 19 with with ADP iajs-2560 266 20 π‘Žπ‘ π‘Žπ‘ PROPN iajs-2560 266 21 βˆ‰ βˆ‰ PROPN iajs-2560 267 1 [ [ X iajs-2560 267 2 [ [ X iajs-2560 267 3 [ [ X iajs-2560 267 4 𝐿:𝑅 𝐿:𝑅 X iajs-2560 267 5 ∁ ∁ PROPN iajs-2560 267 6 ] ] X iajs-2560 268 1 + + CCONJ iajs-2560 268 2 j(r):𝑅 j(r):𝑅 PROPN iajs-2560 268 3 r r X iajs-2560 268 4 ] ] X iajs-2560 268 5 ] ] X iajs-2560 268 6 = = PUNCT iajs-2560 269 1 [ [ X iajs-2560 269 2 𝐿:𝑅 𝐿:𝑅 X iajs-2560 269 3 ∁ ∁ PROPN iajs-2560 269 4 ] ] X iajs-2560 270 1 + + CCONJ iajs-2560 270 2 j(r j(r NOUN iajs-2560 270 3 ) ) PUNCT iajs-2560 270 4 , , PUNCT iajs-2560 270 5 that that PRON iajs-2560 270 6 is be AUX iajs-2560 270 7 π‘Žπ‘βˆβŠˆ π‘Žπ‘βˆβŠˆ NUM iajs-2560 270 8 [ [ X iajs-2560 270 9 𝐿:𝑅 𝐿:𝑅 X iajs-2560 270 10 ∁ ∁ PROPN iajs-2560 270 11 ] ] PUNCT iajs-2560 270 12 . . PUNCT iajs-2560 271 1 ∁ ∁ PROPN iajs-2560 271 2 + + CCONJ iajs-2560 271 3 j(r j(r NOUN iajs-2560 271 4 ) ) PUNCT iajs-2560 271 5 . . PUNCT iajs-2560 272 1 ∁. ∁. NOUN iajs-2560 273 1 but but CCONJ iajs-2560 273 2 ∁ ∁ PROPN iajs-2560 273 3 is be AUX iajs-2560 273 4 a a DET iajs-2560 273 5 module module NOUN iajs-2560 273 6 over over ADP iajs-2560 273 7 artinian artinian ADJ iajs-2560 273 8 ring ring NOUN iajs-2560 273 9 , , PUNCT iajs-2560 273 10 then then ADV iajs-2560 273 11 by by ADP iajs-2560 273 12 lemma lemma PROPN iajs-2560 273 13 ( ( PUNCT iajs-2560 273 14 15 15 NUM iajs-2560 273 15 ) ) PUNCT iajs-2560 273 16 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 273 17 ) ) PUNCT iajs-2560 273 18 . . PUNCT iajs-2560 274 1 ∁= ∁= PROPN iajs-2560 274 2 𝐽(∁ 𝐽(∁ PROPN iajs-2560 274 3 ) ) PUNCT iajs-2560 274 4 , , PUNCT iajs-2560 274 5 that that PRON iajs-2560 274 6 is be AUX iajs-2560 274 7 π‘Žπ‘βˆβŠˆ π‘Žπ‘βˆβŠˆ NUM iajs-2560 274 8 𝐿 𝐿 PROPN iajs-2560 274 9 + + CCONJ iajs-2560 274 10 j(∁ j(∁ PROPN iajs-2560 274 11 ) ) PUNCT iajs-2560 275 1 i.e. i.e. X iajs-2560 275 2 π‘Žπ‘ π‘Žπ‘ ADP iajs-2560 275 3 βˆ‰ βˆ‰ PROPN iajs-2560 276 1 [ [ X iajs-2560 276 2 𝐿 𝐿 PROPN iajs-2560 276 3 + + NUM iajs-2560 276 4 j(∁):𝑅 j(∁):𝑅 PROPN iajs-2560 276 5 ∁ ∁ PROPN iajs-2560 276 6 ] ] PUNCT iajs-2560 276 7 . . PUNCT iajs-2560 277 1 now now ADV iajs-2560 277 2 , , PUNCT iajs-2560 277 3 since since SCONJ iajs-2560 277 4 π‘Žπ‘πΌ π‘Žπ‘πΌ NOUN iajs-2560 277 5 βŠ† βŠ† NUM iajs-2560 277 6 [ [ X iajs-2560 277 7 𝐿:𝑅 𝐿:𝑅 X iajs-2560 277 8 ∁ ∁ PROPN iajs-2560 277 9 ] ] X iajs-2560 277 10 , , PUNCT iajs-2560 277 11 then then ADV iajs-2560 277 12 π‘Žπ‘(𝐼∁ π‘Žπ‘(𝐼∁ NOUN iajs-2560 277 13 ) ) PUNCT iajs-2560 277 14 βŠ† βŠ† NUM iajs-2560 277 15 𝐿. 𝐿. NOUN iajs-2560 277 16 but but CCONJ iajs-2560 277 17 𝐿 𝐿 PROPN iajs-2560 277 18 is be AUX iajs-2560 277 19 a a DET iajs-2560 277 20 nearly nearly ADV iajs-2560 277 21 primary-2 primary-2 NOUN iajs-2560 277 22 - - PUNCT iajs-2560 277 23 absorbing absorb VERB iajs-2560 277 24 submodule submodule NOUN iajs-2560 277 25 of of ADP iajs-2560 277 26 ∁ ∁ PROPN iajs-2560 277 27 , , PUNCT iajs-2560 277 28 then then ADV iajs-2560 277 29 by by ADP iajs-2560 277 30 proposition proposition NOUN iajs-2560 277 31 ( ( PUNCT iajs-2560 277 32 4 4 NUM iajs-2560 277 33 ) ) PUNCT iajs-2560 277 34 either either CCONJ iajs-2560 277 35 π‘Ž(𝐼∁ π‘Ž(𝐼∁ NOUN iajs-2560 277 36 ) ) PUNCT iajs-2560 277 37 βŠ† βŠ† NUM iajs-2560 277 38 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 277 39 ) ) PUNCT iajs-2560 277 40 + + SYM iajs-2560 277 41 𝐽(∁ 𝐽(∁ ADJ iajs-2560 277 42 ) ) PUNCT iajs-2560 277 43 or or CCONJ iajs-2560 277 44 𝑐(𝐼∁ 𝑐(𝐼∁ ADP iajs-2560 277 45 ) ) PUNCT iajs-2560 277 46 βŠ† βŠ† NUM iajs-2560 277 47 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 277 48 ) ) PUNCT iajs-2560 278 1 + + SYM iajs-2560 278 2 𝐽(∁ 𝐽(∁ ADJ iajs-2560 278 3 ) ) PUNCT iajs-2560 278 4 . . PUNCT iajs-2560 279 1 but but CCONJ iajs-2560 279 2 by by ADP iajs-2560 279 3 lemma(14 lemma(14 NOUN iajs-2560 279 4 ) ) PUNCT iajs-2560 279 5 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 279 6 ) ) PUNCT iajs-2560 279 7 = = SYM iajs-2560 279 8 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 279 9 ∁ ∁ PROPN iajs-2560 279 10 ] ] PUNCT iajs-2560 279 11 . . PUNCT iajs-2560 280 1 ∁ ∁ PROPN iajs-2560 280 2 and and CCONJ iajs-2560 280 3 by by ADP iajs-2560 280 4 lemma(15 lemma(15 NOUN iajs-2560 280 5 ) ) PUNCT iajs-2560 280 6 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 280 7 ) ) PUNCT iajs-2560 280 8 . . PUNCT iajs-2560 281 1 ∁= ∁= PROPN iajs-2560 281 2 𝐽(∁ 𝐽(∁ PROPN iajs-2560 281 3 ) ) PUNCT iajs-2560 281 4 . . PUNCT iajs-2560 282 1 thus thus ADV iajs-2560 282 2 either either CCONJ iajs-2560 282 3 π‘ŽπΌβˆ π‘ŽπΌβˆ PROPN iajs-2560 282 4 βŠ† βŠ† NUM iajs-2560 282 5 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 282 6 ∁ ∁ PROPN iajs-2560 282 7 ] ] PUNCT iajs-2560 282 8 . . PUNCT iajs-2560 283 1 ∁ ∁ PROPN iajs-2560 283 2 + + NUM iajs-2560 283 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 283 4 ) ) PUNCT iajs-2560 283 5 . . PUNCT iajs-2560 284 1 ∁ ∁ PROPN iajs-2560 284 2 or or CCONJ iajs-2560 284 3 π‘πΌβˆ π‘πΌβˆ VERB iajs-2560 284 4 βŠ† βŠ† NUM iajs-2560 284 5 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 284 6 ∁ ∁ PROPN iajs-2560 284 7 ] ] PUNCT iajs-2560 284 8 . . PUNCT iajs-2560 285 1 ∁ ∁ PROPN iajs-2560 285 2 + + CCONJ iajs-2560 285 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 285 4 ) ) PUNCT iajs-2560 285 5 . . PUNCT iajs-2560 286 1 ∁ ∁ PROPN iajs-2560 286 2 , , PUNCT iajs-2560 286 3 it it PRON iajs-2560 286 4 follows follow VERB iajs-2560 286 5 that that SCONJ iajs-2560 286 6 either either CCONJ iajs-2560 286 7 π‘ŽπΌ π‘ŽπΌ PROPN iajs-2560 286 8 βŠ† βŠ† NUM iajs-2560 286 9 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 286 10 ∁ ∁ PROPN iajs-2560 286 11 ] ] PUNCT iajs-2560 286 12 . . PUNCT iajs-2560 287 1 ∁ ∁ PROPN iajs-2560 287 2 + + CCONJ iajs-2560 287 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 287 4 ) ) PUNCT iajs-2560 287 5 or or CCONJ iajs-2560 287 6 𝑐𝐼 𝑐𝐼 PROPN iajs-2560 287 7 βŠ† βŠ† NUM iajs-2560 287 8 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 287 9 ∁ ∁ PROPN iajs-2560 287 10 ] ] PUNCT iajs-2560 287 11 . . PUNCT iajs-2560 288 1 ∁ ∁ PROPN iajs-2560 288 2 + + CCONJ iajs-2560 288 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 288 4 ) ) PUNCT iajs-2560 288 5 . . PUNCT iajs-2560 289 1 therefore therefore ADV iajs-2560 290 1 [ [ X iajs-2560 290 2 𝐿:𝑅 𝐿:𝑅 X iajs-2560 290 3 ∁ ∁ PROPN iajs-2560 290 4 ] ] PUNCT iajs-2560 290 5 is be AUX iajs-2560 290 6 a a DET iajs-2560 290 7 nearly nearly ADV iajs-2560 290 8 primary-2 primary-2 NOUN iajs-2560 290 9 - - PUNCT iajs-2560 290 10 absorbing absorbing ADJ iajs-2560 290 11 ideal ideal NOUN iajs-2560 290 12 of of ADP iajs-2560 290 13 𝑅. 𝑅. NOUN iajs-2560 290 14 ( ( PUNCT iajs-2560 290 15 ⇐ ⇐ ADJ iajs-2560 290 16 ) ) PUNCT iajs-2560 290 17 assume assume VERB iajs-2560 290 18 that that SCONJ iajs-2560 290 19 [ [ X iajs-2560 290 20 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2560 290 21 ∁ ∁ PROPN iajs-2560 290 22 ] ] PUNCT iajs-2560 290 23 is be AUX iajs-2560 290 24 a a DET iajs-2560 290 25 nearly nearly ADV iajs-2560 290 26 primary-2 primary-2 NOUN iajs-2560 290 27 - - PUNCT iajs-2560 290 28 absorbing absorbing ADJ iajs-2560 290 29 ideal ideal NOUN iajs-2560 290 30 of of ADP iajs-2560 290 31 𝑅 𝑅 PROPN iajs-2560 290 32 , , PUNCT iajs-2560 290 33 and and CCONJ iajs-2560 290 34 π‘Ÿπ‘ πΎ π‘Ÿπ‘ πΎ PROPN iajs-2560 290 35 βŠ† βŠ† NUM iajs-2560 290 36 𝐿 𝐿 PROPN iajs-2560 290 37 , , PUNCT iajs-2560 290 38 for for ADP iajs-2560 290 39 π‘Ÿ,𝑠 π‘Ÿ,𝑠 PROPN iajs-2560 290 40 ∈ ∈ PROPN iajs-2560 290 41 𝑅 𝑅 PROPN iajs-2560 290 42 , , PUNCT iajs-2560 290 43 𝐾 𝐾 PROPN iajs-2560 290 44 is be AUX iajs-2560 290 45 a a DET iajs-2560 290 46 submodule submodule NOUN iajs-2560 290 47 of of ADP iajs-2560 290 48 ∁ ∁ PROPN iajs-2560 290 49 with with ADP iajs-2560 290 50 π‘Ÿπ‘  π‘Ÿπ‘  PROPN iajs-2560 290 51 βˆ‰ βˆ‰ PROPN iajs-2560 291 1 [ [ X iajs-2560 291 2 𝐿 𝐿 PROPN iajs-2560 291 3 + + CCONJ iajs-2560 291 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 291 5 ∁ ∁ NOUN iajs-2560 291 6 ] ] PUNCT iajs-2560 291 7 , , PUNCT iajs-2560 291 8 it it PRON iajs-2560 291 9 follows follow VERB iajs-2560 291 10 that that SCONJ iajs-2560 291 11 π‘Ÿπ‘ βˆ π‘Ÿπ‘ βˆ ADJ iajs-2560 291 12 ⊈ ⊈ PROPN iajs-2560 291 13 𝐿 𝐿 PROPN iajs-2560 291 14 + + PROPN iajs-2560 291 15 𝐽(∁ 𝐽(∁ PROPN iajs-2560 291 16 ) ) PUNCT iajs-2560 291 17 . . PUNCT iajs-2560 292 1 but but CCONJ iajs-2560 292 2 ∁ ∁ PROPN iajs-2560 292 3 is be AUX iajs-2560 292 4 a a DET iajs-2560 292 5 module module NOUN iajs-2560 292 6 over over ADP iajs-2560 292 7 artinian artinian ADJ iajs-2560 292 8 ring ring NOUN iajs-2560 292 9 , , PUNCT iajs-2560 292 10 then then ADV iajs-2560 292 11 by by ADP iajs-2560 292 12 lemma lemma PROPN iajs-2560 292 13 ( ( PUNCT iajs-2560 292 14 15 15 NUM iajs-2560 292 15 ) ) PUNCT iajs-2560 292 16 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 292 17 ) ) PUNCT iajs-2560 292 18 . . PUNCT iajs-2560 293 1 ∁= ∁= PROPN iajs-2560 293 2 𝐽(∁ 𝐽(∁ PROPN iajs-2560 293 3 ) ) PUNCT iajs-2560 293 4 . . PUNCT iajs-2560 294 1 thus thus ADV iajs-2560 294 2 π‘Ÿπ‘ βˆ π‘Ÿπ‘ βˆ ADJ iajs-2560 294 3 ⊈ ⊈ PUNCT iajs-2560 295 1 [ [ X iajs-2560 295 2 𝐿:𝑅 𝐿:𝑅 X iajs-2560 295 3 ∁ ∁ PROPN iajs-2560 295 4 ] ] PUNCT iajs-2560 295 5 . . PUNCT iajs-2560 296 1 ∁ ∁ PROPN iajs-2560 296 2 + + CCONJ iajs-2560 296 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 296 4 ) ) PUNCT iajs-2560 296 5 . . PUNCT iajs-2560 297 1 ∁ ∁ PROPN iajs-2560 297 2 , , PUNCT iajs-2560 297 3 it it PRON iajs-2560 297 4 follows follow VERB iajs-2560 297 5 that that SCONJ iajs-2560 297 6 π‘Ÿπ‘  π‘Ÿπ‘  NUM iajs-2560 297 7 ⊈ ⊈ PROPN iajs-2560 298 1 [ [ X iajs-2560 298 2 𝐿:𝑅 𝐿:𝑅 X iajs-2560 298 3 ∁ ∁ PROPN iajs-2560 298 4 ] ] X iajs-2560 298 5 + + NUM iajs-2560 298 6 𝐽(𝑅 𝐽(𝑅 X iajs-2560 298 7 ) ) PUNCT iajs-2560 298 8 = = PUNCT iajs-2560 299 1 [ [ X iajs-2560 299 2 [ [ X iajs-2560 299 3 [ [ X iajs-2560 299 4 𝐿:𝑅 𝐿:𝑅 X iajs-2560 299 5 ∁ ∁ PROPN iajs-2560 299 6 ] ] X iajs-2560 300 1 + + CCONJ iajs-2560 300 2 j(r):𝑅 j(r):𝑅 NOUN iajs-2560 300 3 r r X iajs-2560 300 4 ] ] X iajs-2560 300 5 ] ] PUNCT iajs-2560 300 6 . . PUNCT iajs-2560 301 1 now now ADV iajs-2560 301 2 , , PUNCT iajs-2560 301 3 since since SCONJ iajs-2560 301 4 π‘Ÿπ‘ πΎ π‘Ÿπ‘ πΎ PROPN iajs-2560 301 5 βŠ† βŠ† NUM iajs-2560 301 6 𝐿 𝐿 PROPN iajs-2560 301 7 and and CCONJ iajs-2560 301 8 ∁ ∁ PROPN iajs-2560 301 9 is be AUX iajs-2560 301 10 a a DET iajs-2560 301 11 multiplication multiplication NOUN iajs-2560 301 12 , , PUNCT iajs-2560 301 13 then then ADV iajs-2560 301 14 𝐾 𝐾 PROPN iajs-2560 301 15 = = PUNCT iajs-2560 301 16 𝐼∁ 𝐼∁ X iajs-2560 301 17 for for ADP iajs-2560 301 18 some some DET iajs-2560 301 19 ideal ideal NOUN iajs-2560 301 20 𝐼 𝐼 ADP iajs-2560 301 21 of of ADP iajs-2560 301 22 𝑅. 𝑅. NOUN iajs-2560 301 23 hence hence ADV iajs-2560 301 24 π‘Ÿπ‘ πΌβˆ π‘Ÿπ‘ πΌβˆ PROPN iajs-2560 301 25 βŠ† βŠ† NUM iajs-2560 301 26 𝐿 𝐿 PROPN iajs-2560 301 27 , , PUNCT iajs-2560 301 28 implies imply VERB iajs-2560 301 29 that that SCONJ iajs-2560 301 30 π‘Ÿπ‘ πΌ π‘Ÿπ‘ πΌ PRON iajs-2560 301 31 βŠ† βŠ† NUM iajs-2560 301 32 [ [ X iajs-2560 301 33 𝐿:𝑅 𝐿:𝑅 X iajs-2560 301 34 ∁ ∁ PROPN iajs-2560 301 35 ] ] PUNCT iajs-2560 301 36 . . PUNCT iajs-2560 302 1 but but CCONJ iajs-2560 302 2 [ [ X iajs-2560 302 3 𝐿:𝑅 𝐿:𝑅 X iajs-2560 302 4 ∁ ∁ PROPN iajs-2560 302 5 ] ] PUNCT iajs-2560 302 6 is be AUX iajs-2560 302 7 a a DET iajs-2560 302 8 nearly nearly ADV iajs-2560 302 9 primary-2 primary-2 NOUN iajs-2560 302 10 - - PUNCT iajs-2560 302 11 absorbing absorbing ADJ iajs-2560 302 12 ideal ideal NOUN iajs-2560 302 13 of of ADP iajs-2560 302 14 𝑅 𝑅 PROPN iajs-2560 302 15 and and CCONJ iajs-2560 302 16 π‘Ÿπ‘  π‘Ÿπ‘  PROPN iajs-2560 302 17 βˆ‰ βˆ‰ PROPN iajs-2560 303 1 [ [ X iajs-2560 303 2 [ [ X iajs-2560 303 3 [ [ X iajs-2560 303 4 𝐿:𝑅 𝐿:𝑅 X iajs-2560 303 5 ∁ ∁ PROPN iajs-2560 303 6 ] ] X iajs-2560 304 1 + + CCONJ iajs-2560 305 1 j(r):𝑅 j(r):𝑅 NOUN iajs-2560 305 2 r r X iajs-2560 305 3 ] ] X iajs-2560 305 4 ] ] PUNCT iajs-2560 305 5 , , PUNCT iajs-2560 305 6 then then ADV iajs-2560 305 7 by by ADP iajs-2560 305 8 proposition proposition NOUN iajs-2560 305 9 ( ( PUNCT iajs-2560 305 10 4 4 X iajs-2560 305 11 ) ) PUNCT iajs-2560 305 12 we we PRON iajs-2560 305 13 have have VERB iajs-2560 305 14 either either CCONJ iajs-2560 305 15 π‘ŸπΌ π‘ŸπΌ NUM iajs-2560 305 16 βŠ† βŠ† NUM iajs-2560 305 17 √[𝐿:𝑅 √[𝐿:𝑅 X iajs-2560 305 18 ∁ ∁ PROPN iajs-2560 305 19 ] ] X iajs-2560 306 1 + + CCONJ iajs-2560 306 2 𝐽(𝑅 𝐽(𝑅 NOUN iajs-2560 306 3 ) ) PUNCT iajs-2560 306 4 or or CCONJ iajs-2560 306 5 𝑠𝐼 𝑠𝐼 PROPN iajs-2560 306 6 βŠ† βŠ† NUM iajs-2560 306 7 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 306 8 ∁ ∁ PROPN iajs-2560 306 9 ] ] X iajs-2560 306 10 + + NUM iajs-2560 306 11 𝐽(𝑅 𝐽(𝑅 NOUN iajs-2560 306 12 ) ) PUNCT iajs-2560 306 13 . . PUNCT iajs-2560 307 1 that that PRON iajs-2560 307 2 is be AUX iajs-2560 307 3 π‘ŸπΌβˆ π‘ŸπΌβˆ NUM iajs-2560 307 4 βŠ† βŠ† NUM iajs-2560 307 5 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 307 6 ∁ ∁ PROPN iajs-2560 307 7 ] ] PUNCT iajs-2560 307 8 . . PUNCT iajs-2560 308 1 ∁ ∁ PROPN iajs-2560 308 2 + + NUM iajs-2560 308 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 308 4 ) ) PUNCT iajs-2560 308 5 . . PUNCT iajs-2560 309 1 ∁ ∁ PROPN iajs-2560 309 2 or or CCONJ iajs-2560 309 3 π‘ πΌβˆ π‘ πΌβˆ NUM iajs-2560 309 4 βŠ† βŠ† NUM iajs-2560 309 5 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 309 6 ∁ ∁ PROPN iajs-2560 309 7 ] ] PUNCT iajs-2560 309 8 . . PUNCT iajs-2560 310 1 ∁ ∁ PROPN iajs-2560 310 2 + + NUM iajs-2560 310 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 310 4 ) ) PUNCT iajs-2560 310 5 . . PUNCT iajs-2560 311 1 ∁ ∁ PROPN iajs-2560 311 2 . . PUNCT iajs-2560 312 1 but but CCONJ iajs-2560 312 2 ∁ ∁ PROPN iajs-2560 312 3 is be AUX iajs-2560 312 4 a a DET iajs-2560 312 5 module module NOUN iajs-2560 312 6 over over ADP iajs-2560 312 7 artinian artinian ADJ iajs-2560 312 8 ring ring NOUN iajs-2560 312 9 , , PUNCT iajs-2560 312 10 then then ADV iajs-2560 312 11 by by ADP iajs-2560 312 12 lemma lemma PROPN iajs-2560 312 13 ( ( PUNCT iajs-2560 312 14 15 15 NUM iajs-2560 312 15 ) ) PUNCT iajs-2560 312 16 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 312 17 ) ) PUNCT iajs-2560 312 18 . . PUNCT iajs-2560 313 1 ∁= ∁= PROPN iajs-2560 313 2 𝐽(∁ 𝐽(∁ PROPN iajs-2560 313 3 ) ) PUNCT iajs-2560 313 4 and and CCONJ iajs-2560 313 5 by by ADP iajs-2560 313 6 lemma(14 lemma(14 NOUN iajs-2560 313 7 ) ) PUNCT iajs-2560 313 8 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 313 9 ) ) PUNCT iajs-2560 314 1 = = SYM iajs-2560 314 2 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 314 3 ∁ ∁ PROPN iajs-2560 314 4 ] ] PUNCT iajs-2560 314 5 . . PUNCT iajs-2560 315 1 ∁ ∁ NOUN iajs-2560 315 2 we we PRON iajs-2560 315 3 get get VERB iajs-2560 315 4 either either CCONJ iajs-2560 315 5 π‘ŸπΎ π‘ŸπΎ ADJ iajs-2560 315 6 βŠ† βŠ† NUM iajs-2560 315 7 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 315 8 ) ) PUNCT iajs-2560 316 1 + + SYM iajs-2560 316 2 𝐽(∁ 𝐽(∁ ADJ iajs-2560 316 3 ) ) PUNCT iajs-2560 316 4 or or CCONJ iajs-2560 316 5 𝑠𝐾 𝑠𝐾 PROPN iajs-2560 316 6 βŠ† βŠ† NUM iajs-2560 316 7 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 316 8 ) ) PUNCT iajs-2560 316 9 + + SYM iajs-2560 316 10 𝐽(∁ 𝐽(∁ ADJ iajs-2560 316 11 ) ) PUNCT iajs-2560 316 12 . . PUNCT iajs-2560 317 1 hence hence ADV iajs-2560 317 2 by by ADP iajs-2560 317 3 proposition proposition NOUN iajs-2560 317 4 ( ( PUNCT iajs-2560 317 5 4 4 X iajs-2560 317 6 ) ) PUNCT iajs-2560 317 7 𝐿 𝐿 PROPN iajs-2560 317 8 is be AUX iajs-2560 317 9 a a DET iajs-2560 317 10 nearly nearly ADV iajs-2560 317 11 primary-2 primary-2 NOUN iajs-2560 317 12 - - PUNCT iajs-2560 317 13 absorbing absorb VERB iajs-2560 317 14 submodule submodule NOUN iajs-2560 317 15 of of ADP iajs-2560 317 16 ∁. ∁. PROPN iajs-2560 317 17 lemma lemma PROPN iajs-2560 317 18 17 17 NUM iajs-2560 318 1 [ [ SYM iajs-2560 318 2 10 10 NUM iajs-2560 318 3 , , PUNCT iajs-2560 318 4 proposition proposition NOUN iajs-2560 318 5 ( ( PUNCT iajs-2560 318 6 17.10 17.10 NUM iajs-2560 318 7 ) ) PUNCT iajs-2560 318 8 ] ] PUNCT iajs-2560 318 9 if if SCONJ iajs-2560 318 10 𝑃 𝑃 NOUN iajs-2560 318 11 is be AUX iajs-2560 318 12 projective projective ADJ iajs-2560 318 13 𝑅-module 𝑅-module PROPN iajs-2560 318 14 , , PUNCT iajs-2560 318 15 then then ADV iajs-2560 318 16 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 318 17 ) ) PUNCT iajs-2560 318 18 . . PUNCT iajs-2560 319 1 𝑃 𝑃 NOUN iajs-2560 319 2 = = SYM iajs-2560 319 3 𝐽(𝑃 𝐽(𝑃 NOUN iajs-2560 319 4 ) ) PUNCT iajs-2560 319 5 . . PUNCT iajs-2560 320 1 proposition proposition NOUN iajs-2560 320 2 18 18 NUM iajs-2560 320 3 let let VERB iajs-2560 320 4 ∁ ∁ NOUN iajs-2560 320 5 be be AUX iajs-2560 320 6 a a DET iajs-2560 320 7 multiplication multiplication NOUN iajs-2560 320 8 projective projective ADJ iajs-2560 320 9 𝑅-module 𝑅-module PROPN iajs-2560 320 10 and and CCONJ iajs-2560 320 11 𝐿 𝐿 PROPN iajs-2560 320 12 is be AUX iajs-2560 320 13 a a DET iajs-2560 320 14 proper proper ADJ iajs-2560 320 15 submodule submodule NOUN iajs-2560 320 16 of of ADP iajs-2560 320 17 ∁. ∁. NOUN iajs-2560 320 18 then then ADV iajs-2560 320 19 𝐿 𝐿 PROPN iajs-2560 320 20 is be AUX iajs-2560 320 21 a a DET iajs-2560 320 22 nearly nearly ADV iajs-2560 320 23 primary-2 primary-2 NOUN iajs-2560 320 24 - - PUNCT iajs-2560 320 25 absorbing absorb VERB iajs-2560 320 26 submodule submodule NOUN iajs-2560 320 27 of of ADP iajs-2560 320 28 ∁ ∁ PROPN iajs-2560 320 29 if if SCONJ iajs-2560 320 30 and and CCONJ iajs-2560 320 31 only only ADV iajs-2560 320 32 if if SCONJ iajs-2560 320 33 [ [ X iajs-2560 320 34 𝐿:𝑅 𝐿:𝑅 X iajs-2560 320 35 ∁ ∁ PROPN iajs-2560 320 36 ] ] PUNCT iajs-2560 320 37 is be AUX iajs-2560 320 38 a a DET iajs-2560 320 39 nearly nearly ADV iajs-2560 320 40 primary-2absorbing primary-2absorbing NOUN iajs-2560 320 41 ideal ideal NOUN iajs-2560 320 42 of of ADP iajs-2560 320 43 𝑅. 𝑅. ADJ iajs-2560 320 44 proof proof NOUN iajs-2560 320 45 : : PUNCT iajs-2560 320 46 ( ( PUNCT iajs-2560 320 47 β‡’ β‡’ NOUN iajs-2560 320 48 ) ) PUNCT iajs-2560 320 49 let let VERB iajs-2560 320 50 π‘Žπ‘πΌ π‘Žπ‘πΌ NOUN iajs-2560 321 1 βŠ† βŠ† NUM iajs-2560 321 2 [ [ X iajs-2560 321 3 𝐿:𝑅 𝐿:𝑅 X iajs-2560 321 4 ∁ ∁ PROPN iajs-2560 321 5 ] ] PUNCT iajs-2560 321 6 for for ADP iajs-2560 321 7 π‘Ž π‘Ž PROPN iajs-2560 321 8 , , PUNCT iajs-2560 321 9 𝑐 𝑐 PROPN iajs-2560 321 10 ∈ ∈ PROPN iajs-2560 321 11 𝑅 𝑅 PROPN iajs-2560 321 12 , , PUNCT iajs-2560 321 13 𝐼 𝐼 PROPN iajs-2560 321 14 is be AUX iajs-2560 321 15 an an DET iajs-2560 321 16 ideal ideal NOUN iajs-2560 321 17 of of ADP iajs-2560 321 18 𝑅 𝑅 PROPN iajs-2560 321 19 with with ADP iajs-2560 321 20 π‘Žπ‘ π‘Žπ‘ PROPN iajs-2560 321 21 βˆ‰ βˆ‰ PROPN iajs-2560 322 1 [ [ X iajs-2560 322 2 [ [ X iajs-2560 322 3 [ [ X iajs-2560 322 4 𝐿:𝑅 𝐿:𝑅 X iajs-2560 322 5 ∁ ∁ PROPN iajs-2560 322 6 ] ] X iajs-2560 323 1 + + CCONJ iajs-2560 323 2 j(r):𝑅 j(r):𝑅 PROPN iajs-2560 323 3 r r X iajs-2560 323 4 ] ] X iajs-2560 323 5 ] ] X iajs-2560 323 6 = = PUNCT iajs-2560 324 1 [ [ X iajs-2560 324 2 𝐿:𝑅 𝐿:𝑅 X iajs-2560 324 3 ∁ ∁ PROPN iajs-2560 324 4 ] ] X iajs-2560 325 1 + + CCONJ iajs-2560 325 2 j(r j(r NOUN iajs-2560 325 3 ) ) PUNCT iajs-2560 325 4 , , PUNCT iajs-2560 325 5 that that PRON iajs-2560 325 6 is be AUX iajs-2560 325 7 π‘Žπ‘βˆβŠˆ π‘Žπ‘βˆβŠˆ NUM iajs-2560 325 8 [ [ X iajs-2560 325 9 𝐿:𝑅 𝐿:𝑅 X iajs-2560 325 10 ∁ ∁ PROPN iajs-2560 325 11 ] ] PUNCT iajs-2560 325 12 . . PUNCT iajs-2560 326 1 ∁ ∁ PROPN iajs-2560 326 2 + + CCONJ iajs-2560 326 3 j(r j(r NOUN iajs-2560 326 4 ) ) PUNCT iajs-2560 326 5 . . PUNCT iajs-2560 327 1 ∁. ∁. NOUN iajs-2560 328 1 but but CCONJ iajs-2560 328 2 ∁ ∁ PROPN iajs-2560 328 3 is be AUX iajs-2560 328 4 projective projective ADJ iajs-2560 328 5 𝑅-module 𝑅-module NOUN iajs-2560 328 6 , , PUNCT iajs-2560 328 7 then then ADV iajs-2560 328 8 by by ADP iajs-2560 328 9 lemma lemma PROPN iajs-2560 328 10 ( ( PUNCT iajs-2560 328 11 17 17 NUM iajs-2560 328 12 ) ) PUNCT iajs-2560 328 13 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 328 14 ) ) PUNCT iajs-2560 328 15 . . PUNCT iajs-2560 329 1 ∁= ∁= PROPN iajs-2560 329 2 𝐽(∁ 𝐽(∁ PROPN iajs-2560 329 3 ) ) PUNCT iajs-2560 329 4 , , PUNCT iajs-2560 329 5 that that PRON iajs-2560 329 6 is be AUX iajs-2560 329 7 π‘Žπ‘βˆβŠˆ π‘Žπ‘βˆβŠˆ NUM iajs-2560 329 8 𝐿 𝐿 PROPN iajs-2560 329 9 + + CCONJ iajs-2560 329 10 j(∁ j(∁ PROPN iajs-2560 329 11 ) ) PUNCT iajs-2560 330 1 i.e. i.e. X iajs-2560 330 2 π‘Žπ‘ π‘Žπ‘ ADP iajs-2560 330 3 βˆ‰ βˆ‰ PROPN iajs-2560 331 1 [ [ X iajs-2560 331 2 𝐿 𝐿 PROPN iajs-2560 331 3 + + NUM iajs-2560 331 4 j(∁):𝑅 j(∁):𝑅 PROPN iajs-2560 331 5 ∁ ∁ PROPN iajs-2560 331 6 ] ] PUNCT iajs-2560 331 7 . . PUNCT iajs-2560 332 1 now now ADV iajs-2560 332 2 , , PUNCT iajs-2560 332 3 since since SCONJ iajs-2560 332 4 π‘Žπ‘πΌ π‘Žπ‘πΌ NOUN iajs-2560 332 5 βŠ† βŠ† NUM iajs-2560 332 6 [ [ X iajs-2560 332 7 𝐿:𝑅 𝐿:𝑅 X iajs-2560 332 8 ∁ ∁ PROPN iajs-2560 332 9 ] ] X iajs-2560 332 10 , , PUNCT iajs-2560 332 11 then then ADV iajs-2560 332 12 π‘Žπ‘(𝐼∁ π‘Žπ‘(𝐼∁ NOUN iajs-2560 332 13 ) ) PUNCT iajs-2560 332 14 βŠ† βŠ† NUM iajs-2560 332 15 𝐿. 𝐿. NOUN iajs-2560 332 16 but but CCONJ iajs-2560 332 17 𝐿 𝐿 PROPN iajs-2560 332 18 is be AUX iajs-2560 332 19 a a DET iajs-2560 332 20 nearly nearly ADV iajs-2560 332 21 primary-2 primary-2 NOUN iajs-2560 332 22 - - PUNCT iajs-2560 332 23 absorbing absorb VERB iajs-2560 332 24 submodule submodule NOUN iajs-2560 332 25 of of ADP iajs-2560 332 26 ∁ ∁ PROPN iajs-2560 332 27 , , PUNCT iajs-2560 332 28 then then ADV iajs-2560 332 29 by by ADP iajs-2560 332 30 proposition proposition NOUN iajs-2560 332 31 ( ( PUNCT iajs-2560 332 32 4 4 NUM iajs-2560 332 33 ) ) PUNCT iajs-2560 332 34 either either CCONJ iajs-2560 332 35 π‘Ž(𝐼∁ π‘Ž(𝐼∁ NOUN iajs-2560 332 36 ) ) PUNCT iajs-2560 332 37 βŠ† βŠ† NUM iajs-2560 332 38 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 332 39 ) ) PUNCT iajs-2560 332 40 + + SYM iajs-2560 332 41 𝐽(∁ 𝐽(∁ ADJ iajs-2560 332 42 ) ) PUNCT iajs-2560 332 43 or or CCONJ iajs-2560 332 44 𝑐(𝐼∁ 𝑐(𝐼∁ ADP iajs-2560 332 45 ) ) PUNCT iajs-2560 332 46 βŠ† βŠ† NUM iajs-2560 332 47 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 332 48 ) ) PUNCT iajs-2560 333 1 + + SYM iajs-2560 333 2 𝐽(∁ 𝐽(∁ ADJ iajs-2560 333 3 ) ) PUNCT iajs-2560 333 4 . . PUNCT iajs-2560 334 1 but but CCONJ iajs-2560 334 2 by by ADP iajs-2560 334 3 lemma(14 lemma(14 NOUN iajs-2560 334 4 ) ) PUNCT iajs-2560 334 5 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 334 6 ) ) PUNCT iajs-2560 334 7 = = SYM iajs-2560 334 8 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 334 9 ∁ ∁ PROPN iajs-2560 334 10 ] ] PUNCT iajs-2560 334 11 . . PUNCT iajs-2560 335 1 ∁ ∁ PROPN iajs-2560 335 2 and and CCONJ iajs-2560 335 3 by by ADP iajs-2560 335 4 lemma(17 lemma(17 NOUN iajs-2560 335 5 ) ) PUNCT iajs-2560 335 6 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 335 7 ) ) PUNCT iajs-2560 335 8 . . PUNCT iajs-2560 336 1 ∁= ∁= PROPN iajs-2560 336 2 𝐽(∁ 𝐽(∁ PROPN iajs-2560 336 3 ) ) PUNCT iajs-2560 336 4 . . PUNCT iajs-2560 337 1 thus thus ADV iajs-2560 337 2 either either CCONJ iajs-2560 337 3 π‘ŽπΌβˆ π‘ŽπΌβˆ PROPN iajs-2560 337 4 βŠ† βŠ† NUM iajs-2560 337 5 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 337 6 ∁ ∁ PROPN iajs-2560 337 7 ] ] PUNCT iajs-2560 337 8 . . PUNCT iajs-2560 338 1 ∁ ∁ PROPN iajs-2560 338 2 + + NUM iajs-2560 338 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 338 4 ) ) PUNCT iajs-2560 338 5 . . PUNCT iajs-2560 339 1 ∁ ∁ PROPN iajs-2560 339 2 or or CCONJ iajs-2560 339 3 π‘πΌβˆ π‘πΌβˆ VERB iajs-2560 339 4 βŠ† βŠ† NUM iajs-2560 339 5 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 339 6 ∁ ∁ PROPN iajs-2560 339 7 ] ] PUNCT iajs-2560 339 8 . . PUNCT iajs-2560 340 1 ∁ ∁ PROPN iajs-2560 340 2 + + CCONJ iajs-2560 340 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 340 4 ) ) PUNCT iajs-2560 340 5 . . PUNCT iajs-2560 341 1 ∁. ∁. X iajs-2560 342 1 it it PRON iajs-2560 342 2 follows follow VERB iajs-2560 342 3 that that SCONJ iajs-2560 342 4 either either CCONJ iajs-2560 342 5 π‘ŽπΌ π‘ŽπΌ PROPN iajs-2560 342 6 βŠ† βŠ† NUM iajs-2560 342 7 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 342 8 ∁ ∁ PROPN iajs-2560 342 9 ] ] PUNCT iajs-2560 342 10 . . PUNCT iajs-2560 343 1 ∁ ∁ PROPN iajs-2560 343 2 + + CCONJ iajs-2560 343 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 343 4 ) ) PUNCT iajs-2560 343 5 or or CCONJ iajs-2560 343 6 𝑐𝐼 𝑐𝐼 PROPN iajs-2560 343 7 βŠ† βŠ† NUM iajs-2560 343 8 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 343 9 ∁ ∁ PROPN iajs-2560 343 10 ] ] PUNCT iajs-2560 343 11 . . PUNCT iajs-2560 344 1 ∁ ∁ PROPN iajs-2560 344 2 + + CCONJ iajs-2560 344 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 344 4 ) ) PUNCT iajs-2560 344 5 . . PUNCT iajs-2560 345 1 therefore therefore ADV iajs-2560 345 2 , , PUNCT iajs-2560 345 3 [ [ X iajs-2560 345 4 𝐿:𝑅 𝐿:𝑅 X iajs-2560 345 5 ∁ ∁ PROPN iajs-2560 345 6 ] ] PUNCT iajs-2560 345 7 is be AUX iajs-2560 345 8 a a DET iajs-2560 345 9 nearly nearly ADV iajs-2560 345 10 primary-2 primary-2 NOUN iajs-2560 345 11 - - PUNCT iajs-2560 345 12 absorbing absorbing ADJ iajs-2560 345 13 ideal ideal NOUN iajs-2560 345 14 of of ADP iajs-2560 345 15 𝑅. 𝑅. NOUN iajs-2560 345 16 ( ( PUNCT iajs-2560 345 17 ⇐ ⇐ ADJ iajs-2560 345 18 ) ) PUNCT iajs-2560 345 19 assume assume VERB iajs-2560 345 20 that that SCONJ iajs-2560 345 21 [ [ X iajs-2560 345 22 𝐿:𝑅 𝐿:𝑅 PROPN iajs-2560 345 23 ∁ ∁ PROPN iajs-2560 345 24 ] ] PUNCT iajs-2560 345 25 is be AUX iajs-2560 345 26 a a DET iajs-2560 345 27 nearly nearly ADV iajs-2560 345 28 primary-2 primary-2 NOUN iajs-2560 345 29 - - PUNCT iajs-2560 345 30 absorbing absorbing ADJ iajs-2560 345 31 ideal ideal NOUN iajs-2560 345 32 of of ADP iajs-2560 345 33 𝑅 𝑅 PROPN iajs-2560 345 34 , , PUNCT iajs-2560 345 35 and and CCONJ iajs-2560 345 36 π‘Ÿπ‘ πΎ π‘Ÿπ‘ πΎ PROPN iajs-2560 345 37 βŠ† βŠ† NUM iajs-2560 345 38 𝐿 𝐿 PROPN iajs-2560 345 39 , , PUNCT iajs-2560 345 40 for for ADP iajs-2560 345 41 π‘Ÿ,𝑠 π‘Ÿ,𝑠 PROPN iajs-2560 345 42 ∈ ∈ PROPN iajs-2560 345 43 𝑅 𝑅 PROPN iajs-2560 345 44 , , PUNCT iajs-2560 345 45 𝐾 𝐾 PROPN iajs-2560 345 46 is be AUX iajs-2560 345 47 a a DET iajs-2560 345 48 submodule submodule NOUN iajs-2560 345 49 of of ADP iajs-2560 345 50 ∁ ∁ PROPN iajs-2560 345 51 with with ADP iajs-2560 345 52 π‘Ÿπ‘  π‘Ÿπ‘  PROPN iajs-2560 345 53 βˆ‰ βˆ‰ PROPN iajs-2560 346 1 [ [ X iajs-2560 346 2 𝐿 𝐿 PROPN iajs-2560 346 3 + + CCONJ iajs-2560 346 4 𝐽(∁):𝑅 𝐽(∁):𝑅 PROPN iajs-2560 346 5 ∁ ∁ NOUN iajs-2560 346 6 ] ] PUNCT iajs-2560 346 7 , , PUNCT iajs-2560 346 8 it it PRON iajs-2560 346 9 follows follow VERB iajs-2560 346 10 that that SCONJ iajs-2560 346 11 π‘Ÿπ‘ βˆ π‘Ÿπ‘ βˆ ADJ iajs-2560 346 12 ⊈ ⊈ PROPN iajs-2560 346 13 𝐿 𝐿 PROPN iajs-2560 346 14 + + PROPN iajs-2560 346 15 𝐽(∁ 𝐽(∁ PROPN iajs-2560 346 16 ) ) PUNCT iajs-2560 346 17 . . PUNCT iajs-2560 347 1 but but CCONJ iajs-2560 347 2 ∁ ∁ PROPN iajs-2560 347 3 is be AUX iajs-2560 347 4 a a DET iajs-2560 347 5 project project NOUN iajs-2560 347 6 𝑅-module 𝑅-module NOUN iajs-2560 347 7 , , PUNCT iajs-2560 347 8 then then ADV iajs-2560 347 9 by by ADP iajs-2560 347 10 lemma lemma PROPN iajs-2560 347 11 ( ( PUNCT iajs-2560 347 12 17 17 NUM iajs-2560 347 13 ) ) PUNCT iajs-2560 347 14 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 347 15 ) ) PUNCT iajs-2560 347 16 . . PUNCT iajs-2560 348 1 ∁= ∁= PROPN iajs-2560 348 2 𝐽(∁ 𝐽(∁ PROPN iajs-2560 348 3 ) ) PUNCT iajs-2560 348 4 . . PUNCT iajs-2560 349 1 thus thus ADV iajs-2560 349 2 π‘Ÿπ‘ βˆ π‘Ÿπ‘ βˆ ADJ iajs-2560 349 3 ⊈ ⊈ PUNCT iajs-2560 350 1 [ [ X iajs-2560 350 2 𝐿:𝑅 𝐿:𝑅 X iajs-2560 350 3 ∁ ∁ PROPN iajs-2560 350 4 ] ] PUNCT iajs-2560 350 5 . . PUNCT iajs-2560 351 1 ∁ ∁ PROPN iajs-2560 351 2 + + CCONJ iajs-2560 351 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 351 4 ) ) PUNCT iajs-2560 351 5 . . PUNCT iajs-2560 352 1 ∁. ∁. X iajs-2560 353 1 it it PRON iajs-2560 353 2 follows follow VERB iajs-2560 353 3 that that PRON iajs-2560 353 4 π‘Ÿπ‘  π‘Ÿπ‘  NUM iajs-2560 353 5 ⊈ ⊈ PROPN iajs-2560 354 1 [ [ X iajs-2560 354 2 𝐿:𝑅 𝐿:𝑅 X iajs-2560 354 3 ∁ ∁ PROPN iajs-2560 354 4 ] ] X iajs-2560 354 5 + + NUM iajs-2560 354 6 𝐽(𝑅 𝐽(𝑅 X iajs-2560 354 7 ) ) PUNCT iajs-2560 354 8 = = PUNCT iajs-2560 355 1 [ [ X iajs-2560 355 2 [ [ X iajs-2560 355 3 [ [ X iajs-2560 355 4 𝐿:𝑅 𝐿:𝑅 X iajs-2560 355 5 ∁ ∁ PROPN iajs-2560 355 6 ] ] X iajs-2560 356 1 + + CCONJ iajs-2560 356 2 j(r):𝑅 j(r):𝑅 NOUN iajs-2560 356 3 r r X iajs-2560 356 4 ] ] X iajs-2560 356 5 ] ] PUNCT iajs-2560 356 6 . . PUNCT iajs-2560 357 1 now now ADV iajs-2560 357 2 , , PUNCT iajs-2560 357 3 since since SCONJ iajs-2560 357 4 π‘Ÿπ‘ πΎ π‘Ÿπ‘ πΎ PROPN iajs-2560 357 5 βŠ† βŠ† NUM iajs-2560 357 6 𝐿 𝐿 PROPN iajs-2560 357 7 and and CCONJ iajs-2560 357 8 ∁ ∁ PROPN iajs-2560 357 9 is be AUX iajs-2560 357 10 a a DET iajs-2560 357 11 123 123 NUM iajs-2560 357 12 ibn ibn PROPN iajs-2560 357 13 al al PROPN iajs-2560 357 14 - - PUNCT iajs-2560 357 15 haitham haitham PROPN iajs-2560 357 16 jour jour X iajs-2560 357 17 . . PROPN iajs-2560 358 1 for for ADP iajs-2560 358 2 pure pure ADJ iajs-2560 358 3 & & CCONJ iajs-2560 358 4 appl appl PROPN iajs-2560 358 5 . . PUNCT iajs-2560 359 1 sci sci PROPN iajs-2560 359 2 . . PROPN iajs-2560 360 1 34 34 NUM iajs-2560 360 2 ( ( PUNCT iajs-2560 360 3 1 1 NUM iajs-2560 360 4 ) ) PUNCT iajs-2560 360 5 2021 2021 NUM iajs-2560 360 6 multiplication multiplication NOUN iajs-2560 360 7 , , PUNCT iajs-2560 360 8 then then ADV iajs-2560 360 9 𝐾 𝐾 PROPN iajs-2560 360 10 = = PUNCT iajs-2560 360 11 𝐼∁ 𝐼∁ X iajs-2560 360 12 for for ADP iajs-2560 360 13 some some DET iajs-2560 360 14 ideal ideal NOUN iajs-2560 360 15 𝐼 𝐼 ADP iajs-2560 360 16 of of ADP iajs-2560 360 17 𝑅. 𝑅. NOUN iajs-2560 360 18 hence hence ADV iajs-2560 360 19 π‘Ÿπ‘ πΌβˆ π‘Ÿπ‘ πΌβˆ PROPN iajs-2560 360 20 βŠ† βŠ† NUM iajs-2560 360 21 𝐿 𝐿 PROPN iajs-2560 360 22 , , PUNCT iajs-2560 360 23 implies imply VERB iajs-2560 360 24 that that SCONJ iajs-2560 360 25 π‘Ÿπ‘ πΌ π‘Ÿπ‘ πΌ PRON iajs-2560 360 26 βŠ† βŠ† NUM iajs-2560 360 27 [ [ X iajs-2560 360 28 𝐿:𝑅 𝐿:𝑅 X iajs-2560 360 29 ∁ ∁ PROPN iajs-2560 360 30 ] ] PUNCT iajs-2560 360 31 . . PUNCT iajs-2560 361 1 but but CCONJ iajs-2560 361 2 [ [ X iajs-2560 361 3 𝐿:𝑅 𝐿:𝑅 X iajs-2560 361 4 ∁ ∁ PROPN iajs-2560 361 5 ] ] PUNCT iajs-2560 361 6 is be AUX iajs-2560 361 7 a a DET iajs-2560 361 8 nearly nearly ADV iajs-2560 361 9 primary-2 primary-2 NOUN iajs-2560 361 10 - - PUNCT iajs-2560 361 11 absorbing absorbing ADJ iajs-2560 361 12 ideal ideal NOUN iajs-2560 361 13 of of ADP iajs-2560 361 14 𝑅 𝑅 PROPN iajs-2560 361 15 and and CCONJ iajs-2560 361 16 π‘Ÿπ‘  π‘Ÿπ‘  PROPN iajs-2560 361 17 βˆ‰ βˆ‰ PROPN iajs-2560 362 1 [ [ X iajs-2560 362 2 [ [ X iajs-2560 362 3 [ [ X iajs-2560 362 4 𝐿:𝑅 𝐿:𝑅 X iajs-2560 362 5 ∁ ∁ PROPN iajs-2560 362 6 ] ] X iajs-2560 363 1 + + CCONJ iajs-2560 364 1 j(r):𝑅 j(r):𝑅 NOUN iajs-2560 364 2 r r X iajs-2560 364 3 ] ] X iajs-2560 364 4 ] ] PUNCT iajs-2560 364 5 , , PUNCT iajs-2560 364 6 then then ADV iajs-2560 364 7 by by ADP iajs-2560 364 8 proposition proposition NOUN iajs-2560 364 9 ( ( PUNCT iajs-2560 364 10 4 4 X iajs-2560 364 11 ) ) PUNCT iajs-2560 364 12 we we PRON iajs-2560 364 13 have have VERB iajs-2560 364 14 either either CCONJ iajs-2560 364 15 π‘ŸπΌ π‘ŸπΌ NUM iajs-2560 364 16 βŠ† βŠ† NUM iajs-2560 364 17 √[𝐿:𝑅 √[𝐿:𝑅 X iajs-2560 364 18 ∁ ∁ PROPN iajs-2560 364 19 ] ] X iajs-2560 365 1 + + CCONJ iajs-2560 365 2 𝐽(𝑅 𝐽(𝑅 NOUN iajs-2560 365 3 ) ) PUNCT iajs-2560 365 4 or or CCONJ iajs-2560 365 5 𝑠𝐼 𝑠𝐼 PROPN iajs-2560 365 6 βŠ† βŠ† NUM iajs-2560 365 7 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 365 8 ∁ ∁ PROPN iajs-2560 365 9 ] ] X iajs-2560 365 10 + + NUM iajs-2560 365 11 𝐽(𝑅 𝐽(𝑅 NOUN iajs-2560 365 12 ) ) PUNCT iajs-2560 365 13 . . PUNCT iajs-2560 366 1 that that PRON iajs-2560 366 2 is be AUX iajs-2560 366 3 π‘ŸπΌβˆ π‘ŸπΌβˆ NUM iajs-2560 366 4 βŠ† βŠ† NUM iajs-2560 366 5 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 366 6 ∁ ∁ PROPN iajs-2560 366 7 ] ] PUNCT iajs-2560 366 8 . . PUNCT iajs-2560 367 1 ∁ ∁ PROPN iajs-2560 367 2 + + NUM iajs-2560 367 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 367 4 ) ) PUNCT iajs-2560 367 5 . . PUNCT iajs-2560 368 1 ∁ ∁ PROPN iajs-2560 368 2 or or CCONJ iajs-2560 368 3 π‘ πΌβˆ π‘ πΌβˆ NUM iajs-2560 368 4 βŠ† βŠ† NUM iajs-2560 368 5 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 368 6 ∁ ∁ PROPN iajs-2560 368 7 ] ] PUNCT iajs-2560 368 8 . . PUNCT iajs-2560 369 1 ∁ ∁ PROPN iajs-2560 369 2 + + NUM iajs-2560 369 3 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 369 4 ) ) PUNCT iajs-2560 369 5 . . PUNCT iajs-2560 370 1 ∁ ∁ PROPN iajs-2560 370 2 . . PUNCT iajs-2560 371 1 but but CCONJ iajs-2560 371 2 ∁ ∁ PROPN iajs-2560 371 3 is be AUX iajs-2560 371 4 projective projective ADJ iajs-2560 371 5 𝑅-module 𝑅-module NOUN iajs-2560 371 6 , , PUNCT iajs-2560 371 7 then then ADV iajs-2560 371 8 by by ADP iajs-2560 371 9 lemma lemma PROPN iajs-2560 371 10 ( ( PUNCT iajs-2560 371 11 17 17 NUM iajs-2560 371 12 ) ) PUNCT iajs-2560 371 13 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 371 14 ) ) PUNCT iajs-2560 371 15 . . PUNCT iajs-2560 372 1 ∁= ∁= PROPN iajs-2560 372 2 𝐽(∁ 𝐽(∁ PROPN iajs-2560 372 3 ) ) PUNCT iajs-2560 372 4 and and CCONJ iajs-2560 372 5 by by ADP iajs-2560 372 6 lemma(14 lemma(14 NOUN iajs-2560 372 7 ) ) PUNCT iajs-2560 372 8 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 372 9 ) ) PUNCT iajs-2560 373 1 = = SYM iajs-2560 373 2 √[𝐿:𝑅 √[𝐿:𝑅 PROPN iajs-2560 373 3 ∁ ∁ PROPN iajs-2560 373 4 ] ] PUNCT iajs-2560 373 5 . . PUNCT iajs-2560 374 1 ∁ ∁ NOUN iajs-2560 374 2 we we PRON iajs-2560 374 3 get get VERB iajs-2560 374 4 either either CCONJ iajs-2560 374 5 π‘ŸπΎ π‘ŸπΎ ADJ iajs-2560 374 6 βŠ† βŠ† NUM iajs-2560 374 7 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 374 8 ) ) PUNCT iajs-2560 375 1 + + SYM iajs-2560 375 2 𝐽(∁ 𝐽(∁ ADJ iajs-2560 375 3 ) ) PUNCT iajs-2560 375 4 or or CCONJ iajs-2560 375 5 𝑠𝐾 𝑠𝐾 PROPN iajs-2560 375 6 βŠ† βŠ† NUM iajs-2560 375 7 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 π‘Ÿπ‘Žπ‘‘βˆ(𝐿 NOUN iajs-2560 375 8 ) ) PUNCT iajs-2560 375 9 + + SYM iajs-2560 375 10 𝐽(∁ 𝐽(∁ ADJ iajs-2560 375 11 ) ) PUNCT iajs-2560 375 12 . . PUNCT iajs-2560 376 1 hence hence ADV iajs-2560 376 2 by by ADP iajs-2560 376 3 proposition proposition NOUN iajs-2560 376 4 ( ( PUNCT iajs-2560 376 5 4 4 X iajs-2560 376 6 ) ) PUNCT iajs-2560 376 7 𝐿 𝐿 PROPN iajs-2560 376 8 is be AUX iajs-2560 376 9 a a DET iajs-2560 376 10 nearly nearly ADV iajs-2560 376 11 primary-2 primary-2 NOUN iajs-2560 376 12 - - PUNCT iajs-2560 376 13 absorbing absorb VERB iajs-2560 376 14 submodule submodule NOUN iajs-2560 376 15 of of ADP iajs-2560 376 16 ∁. ∁. NOUN iajs-2560 376 17 we we PRON iajs-2560 376 18 need need VERB iajs-2560 376 19 to to PART iajs-2560 376 20 invite invite VERB iajs-2560 376 21 the the DET iajs-2560 376 22 following follow VERB iajs-2560 376 23 finding finding NOUN iajs-2560 376 24 before before SCONJ iajs-2560 376 25 we we PRON iajs-2560 376 26 study study VERB iajs-2560 376 27 the the DET iajs-2560 376 28 next next ADJ iajs-2560 376 29 propositions proposition NOUN iajs-2560 376 30 . . PUNCT iajs-2560 377 1 lemma lemma PROPN iajs-2560 377 2 19[14 19[14 NUM iajs-2560 377 3 , , PUNCT iajs-2560 377 4 corollary corollary NOUN iajs-2560 377 5 of of ADP iajs-2560 377 6 theorem theorem NOUN iajs-2560 377 7 . . PROPN iajs-2560 377 8 9 9 NUM iajs-2560 377 9 ] ] PUNCT iajs-2560 377 10 let let VERB iajs-2560 377 11 𝐼1 𝐼1 NOUN iajs-2560 377 12 and and CCONJ iajs-2560 377 13 𝐼2 𝐼2 NOUN iajs-2560 377 14 are be AUX iajs-2560 377 15 ideals ideal NOUN iajs-2560 377 16 of of ADP iajs-2560 377 17 a a DET iajs-2560 377 18 ring ring NOUN iajs-2560 377 19 𝑅 𝑅 PROPN iajs-2560 377 20 and and CCONJ iajs-2560 377 21 ∁ ∁ PROPN iajs-2560 377 22 is be AUX iajs-2560 377 23 a a DET iajs-2560 377 24 finitely finitely ADV iajs-2560 377 25 generated generate VERB iajs-2560 377 26 multiplication multiplication NOUN iajs-2560 377 27 𝑅-module 𝑅-module PROPN iajs-2560 377 28 . . PUNCT iajs-2560 378 1 then then ADV iajs-2560 378 2 𝐼1∁ 𝐼1∁ X iajs-2560 378 3 βŠ† βŠ† NUM iajs-2560 378 4 𝐼2∁ 𝐼2∁ NOUN iajs-2560 378 5 if if SCONJ iajs-2560 378 6 and and CCONJ iajs-2560 378 7 only only ADV iajs-2560 378 8 if if SCONJ iajs-2560 378 9 𝐼1 𝐼1 ADJ iajs-2560 378 10 βŠ† βŠ† NUM iajs-2560 378 11 𝐼2 𝐼2 NOUN iajs-2560 378 12 + + CCONJ iajs-2560 378 13 π‘Žπ‘›π‘›π‘…(∁ π‘Žπ‘›π‘›π‘…(∁ PROPN iajs-2560 378 14 ) ) PUNCT iajs-2560 378 15 . . PUNCT iajs-2560 379 1 lemma lemma PROPN iajs-2560 379 2 20[15 20[15 PROPN iajs-2560 379 3 , , PUNCT iajs-2560 379 4 proposition proposition NOUN iajs-2560 379 5 . . PUNCT iajs-2560 380 1 ( ( PUNCT iajs-2560 380 2 2.4 2.4 NUM iajs-2560 380 3 ) ) PUNCT iajs-2560 380 4 ] ] PUNCT iajs-2560 380 5 let let VERB iajs-2560 380 6 ∁ ∁ NOUN iajs-2560 380 7 be be AUX iajs-2560 380 8 a a DET iajs-2560 380 9 multiplication multiplication NOUN iajs-2560 380 10 𝑅-module 𝑅-module PROPN iajs-2560 380 11 and and CCONJ iajs-2560 380 12 𝐼 𝐼 PROPN iajs-2560 380 13 is be AUX iajs-2560 380 14 an an DET iajs-2560 380 15 ideal ideal NOUN iajs-2560 380 16 of of ADP iajs-2560 380 17 𝑅 𝑅 PROPN iajs-2560 380 18 such such ADJ iajs-2560 380 19 that that DET iajs-2560 380 20 π‘Žπ‘›π‘›π‘…(∁ π‘Žπ‘›π‘›π‘…(∁ PROPN iajs-2560 380 21 ) ) PUNCT iajs-2560 380 22 βŠ† βŠ† NUM iajs-2560 380 23 𝐼 𝐼 PROPN iajs-2560 380 24 , , PUNCT iajs-2560 380 25 then then ADV iajs-2560 380 26 π‘Ÿπ‘Žπ‘‘βˆ(𝐼∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐼∁ NUM iajs-2560 380 27 ) ) PUNCT iajs-2560 381 1 = = SYM iajs-2560 381 2 √𝐼∁. √𝐼∁. NOUN iajs-2560 381 3 proposition proposition NOUN iajs-2560 381 4 21 21 NUM iajs-2560 381 5 let let VERB iajs-2560 381 6 ∁ ∁ NOUN iajs-2560 381 7 be be AUX iajs-2560 381 8 a a DET iajs-2560 381 9 faithful faithful ADJ iajs-2560 381 10 finitely finitely ADV iajs-2560 381 11 generated generate VERB iajs-2560 381 12 multiplication multiplication NOUN iajs-2560 381 13 𝑅-module 𝑅-module PROPN iajs-2560 381 14 over over ADP iajs-2560 381 15 an an DET iajs-2560 381 16 artinian artinian ADJ iajs-2560 381 17 ring ring NOUN iajs-2560 381 18 𝑅 𝑅 PROPN iajs-2560 381 19 and and CCONJ iajs-2560 381 20 𝐼 𝐼 PROPN iajs-2560 381 21 is be AUX iajs-2560 381 22 a a DET iajs-2560 381 23 nearly nearly ADV iajs-2560 381 24 primary-2 primary-2 NOUN iajs-2560 381 25 - - PUNCT iajs-2560 381 26 absorbing absorbing ADJ iajs-2560 381 27 ideal ideal NOUN iajs-2560 381 28 of of ADP iajs-2560 381 29 𝑅 𝑅 PROPN iajs-2560 381 30 and and CCONJ iajs-2560 381 31 πΌβˆβ‰  πΌβˆβ‰  ADP iajs-2560 381 32 ∁. ∁. NOUN iajs-2560 381 33 then then ADV iajs-2560 381 34 𝐼∁ 𝐼∁ PUNCT iajs-2560 381 35 is be AUX iajs-2560 381 36 a a DET iajs-2560 381 37 nearly nearly ADV iajs-2560 381 38 primary-2 primary-2 NOUN iajs-2560 381 39 - - PUNCT iajs-2560 381 40 absorbing absorb VERB iajs-2560 381 41 submodule submodule NOUN iajs-2560 381 42 of of ADP iajs-2560 381 43 ∁. ∁. NOUN iajs-2560 381 44 proof proof NOUN iajs-2560 381 45 : : PUNCT iajs-2560 381 46 let let VERB iajs-2560 381 47 π‘Žπ‘π‘₯ π‘Žπ‘π‘₯ VERB iajs-2560 381 48 ∈ ∈ PROPN iajs-2560 381 49 𝐼∁ 𝐼∁ X iajs-2560 381 50 for for ADP iajs-2560 381 51 π‘Ž,𝑐 π‘Ž,𝑐 PROPN iajs-2560 381 52 ∈ ∈ PROPN iajs-2560 381 53 𝑅 𝑅 PROPN iajs-2560 381 54 , , PUNCT iajs-2560 381 55 π‘₯ π‘₯ PRON iajs-2560 381 56 ∈ ∈ ADJ iajs-2560 381 57 ∁ ∁ NOUN iajs-2560 381 58 , , PUNCT iajs-2560 381 59 then then ADV iajs-2560 381 60 π‘Žπ‘(π‘₯ π‘Žπ‘(π‘₯ NUM iajs-2560 381 61 ) ) PUNCT iajs-2560 381 62 βŠ† βŠ† NUM iajs-2560 381 63 𝐼∁ 𝐼∁ X iajs-2560 381 64 , , PUNCT iajs-2560 381 65 implies imply VERB iajs-2560 381 66 that that SCONJ iajs-2560 381 67 π‘Žπ‘π½βˆ π‘Žπ‘π½βˆ PROPN iajs-2560 381 68 βŠ† βŠ† X iajs-2560 381 69 𝐼∁ 𝐼∁ X iajs-2560 381 70 for for ADP iajs-2560 381 71 some some DET iajs-2560 381 72 ideal ideal ADJ iajs-2560 381 73 𝐽 𝐽 PROPN iajs-2560 381 74 of of ADP iajs-2560 381 75 𝑅 𝑅 PROPN iajs-2560 381 76 since since SCONJ iajs-2560 381 77 ∁ ∁ PROPN iajs-2560 381 78 is be AUX iajs-2560 381 79 a a DET iajs-2560 381 80 multiplication multiplication NOUN iajs-2560 381 81 . . PUNCT iajs-2560 382 1 hence hence ADV iajs-2560 382 2 by by ADP iajs-2560 382 3 lemma(19 lemma(19 NOUN iajs-2560 382 4 ) ) PUNCT iajs-2560 382 5 π‘Žπ‘π½ π‘Žπ‘π½ NOUN iajs-2560 382 6 βŠ† βŠ† NUM iajs-2560 382 7 𝐼 𝐼 PROPN iajs-2560 382 8 + + X iajs-2560 382 9 π‘Žπ‘›π‘›π‘…(∁ π‘Žπ‘›π‘›π‘…(∁ PROPN iajs-2560 382 10 ) ) PUNCT iajs-2560 382 11 , , PUNCT iajs-2560 382 12 but but CCONJ iajs-2560 382 13 ∁ ∁ PROPN iajs-2560 382 14 is be AUX iajs-2560 382 15 a a DET iajs-2560 382 16 faithful faithful NOUN iajs-2560 382 17 . . PUNCT iajs-2560 383 1 it it PRON iajs-2560 383 2 follows follow VERB iajs-2560 383 3 that that SCONJ iajs-2560 383 4 π‘Žπ‘›π‘›π‘…(∁ π‘Žπ‘›π‘›π‘…(∁ PROPN iajs-2560 383 5 ) ) PUNCT iajs-2560 383 6 = = PUNCT iajs-2560 384 1 ( ( PUNCT iajs-2560 384 2 0 0 NUM iajs-2560 384 3 ) ) PUNCT iajs-2560 384 4 , , PUNCT iajs-2560 384 5 that that PRON iajs-2560 384 6 is be AUX iajs-2560 384 7 π‘Žπ‘π½ π‘Žπ‘π½ PRON iajs-2560 384 8 βŠ† βŠ† NUM iajs-2560 384 9 𝐼. 𝐼. NOUN iajs-2560 384 10 since since SCONJ iajs-2560 384 11 𝐼 𝐼 PROPN iajs-2560 384 12 is be AUX iajs-2560 384 13 a a DET iajs-2560 384 14 nearly nearly ADV iajs-2560 384 15 primary-2 primary-2 NOUN iajs-2560 384 16 - - PUNCT iajs-2560 384 17 absorbing absorbing ADJ iajs-2560 384 18 ideal ideal NOUN iajs-2560 384 19 of of ADP iajs-2560 384 20 𝑅 𝑅 PROPN iajs-2560 384 21 , , PUNCT iajs-2560 384 22 then then ADV iajs-2560 384 23 by by ADP iajs-2560 384 24 proposition(4 proposition(4 PROPN iajs-2560 384 25 ) ) PUNCT iajs-2560 385 1 either either CCONJ iajs-2560 385 2 π‘Žπ½ π‘Žπ½ ADV iajs-2560 385 3 βŠ† βŠ† NUM iajs-2560 385 4 √𝐼 √𝐼 NOUN iajs-2560 385 5 + + NUM iajs-2560 385 6 𝐽(𝑅 𝐽(𝑅 NOUN iajs-2560 385 7 ) ) PUNCT iajs-2560 385 8 or or CCONJ iajs-2560 385 9 𝑐𝐽 𝑐𝐽 NOUN iajs-2560 385 10 βŠ† βŠ† NUM iajs-2560 385 11 √𝐼 √𝐼 NOUN iajs-2560 385 12 + + CCONJ iajs-2560 385 13 𝐽(𝑅 𝐽(𝑅 NOUN iajs-2560 385 14 ) ) PUNCT iajs-2560 385 15 or or CCONJ iajs-2560 385 16 π‘Žπ‘ π‘Žπ‘ ADP iajs-2560 385 17 ∈ ∈ PROPN iajs-2560 386 1 [ [ X iajs-2560 386 2 𝐼 𝐼 PROPN iajs-2560 386 3 + + CCONJ iajs-2560 386 4 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 386 5 ): ): PUNCT iajs-2560 386 6 𝑅 𝑅 NOUN iajs-2560 386 7 ] ] PUNCT iajs-2560 386 8 = = PUNCT iajs-2560 386 9 𝐼 𝐼 PROPN iajs-2560 386 10 + + CCONJ iajs-2560 386 11 𝐽(𝑅 𝐽(𝑅 PROPN iajs-2560 386 12 ) ) PUNCT iajs-2560 386 13 . . PUNCT iajs-2560 387 1 it it PRON iajs-2560 387 2 follows follow VERB iajs-2560 387 3 that that SCONJ iajs-2560 387 4 π‘Žπ½βˆ π‘Žπ½βˆ NOUN iajs-2560 387 5 βŠ† βŠ† NUM iajs-2560 387 6 √𝐼∁ √𝐼∁ PROPN iajs-2560 387 7 + + NUM iajs-2560 387 8 𝐽(𝑅)∁ 𝐽(𝑅)∁ NOUN iajs-2560 387 9 or or CCONJ iajs-2560 387 10 π‘π½βˆ π‘π½βˆ VERB iajs-2560 387 11 βŠ† βŠ† NUM iajs-2560 387 12 √𝐼∁ √𝐼∁ PROPN iajs-2560 387 13 + + CCONJ iajs-2560 387 14 𝐽(𝑅)∁ 𝐽(𝑅)∁ NOUN iajs-2560 387 15 or or CCONJ iajs-2560 387 16 π‘Žπ‘βˆ π‘Žπ‘βˆ PRON iajs-2560 387 17 βŠ† βŠ† NUM iajs-2560 387 18 𝐼∁ 𝐼∁ X iajs-2560 387 19 + + CCONJ iajs-2560 387 20 𝐽(𝑅)∁. 𝐽(𝑅)∁. PROPN iajs-2560 387 21 but but CCONJ iajs-2560 387 22 by by ADP iajs-2560 387 23 lemma(15 lemma(15 NOUN iajs-2560 387 24 ) ) PUNCT iajs-2560 387 25 𝐽(𝑅)∁= 𝐽(𝑅)∁= PROPN iajs-2560 387 26 𝐽(∁ 𝐽(∁ ADP iajs-2560 387 27 ) ) PUNCT iajs-2560 387 28 and and CCONJ iajs-2560 387 29 by by ADP iajs-2560 387 30 lemma(20 lemma(20 NOUN iajs-2560 387 31 ) ) PUNCT iajs-2560 387 32 √𝐼∁ √𝐼∁ PROPN iajs-2560 387 33 = = PUNCT iajs-2560 387 34 π‘Ÿπ‘Žπ‘‘βˆ(𝐼∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐼∁ NUM iajs-2560 387 35 ) ) PUNCT iajs-2560 387 36 . . PUNCT iajs-2560 388 1 thus thus ADV iajs-2560 388 2 either either CCONJ iajs-2560 388 3 π‘Žπ‘₯ π‘Žπ‘₯ ADP iajs-2560 388 4 ∈ ∈ PROPN iajs-2560 388 5 π‘Ÿπ‘Žπ‘‘βˆ(𝐼∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐼∁ NUM iajs-2560 388 6 ) ) PUNCT iajs-2560 388 7 + + PUNCT iajs-2560 388 8 𝐽(∁ 𝐽(∁ ADJ iajs-2560 388 9 ) ) PUNCT iajs-2560 388 10 or or CCONJ iajs-2560 388 11 𝑐π‘₯ 𝑐π‘₯ ADV iajs-2560 388 12 ∈ ∈ NOUN iajs-2560 388 13 π‘Ÿπ‘Žπ‘‘βˆ(𝐼∁ π‘Ÿπ‘Žπ‘‘βˆ(𝐼∁ NUM iajs-2560 388 14 ) ) PUNCT iajs-2560 388 15 + + PUNCT iajs-2560 388 16 𝐽(∁ 𝐽(∁ ADJ iajs-2560 388 17 ) ) PUNCT iajs-2560 388 18 or or CCONJ iajs-2560 388 19 π‘Žπ‘βˆβŠ† π‘Žπ‘βˆβŠ† NOUN iajs-2560 388 20 𝐼∁ 𝐼∁ PUNCT iajs-2560 389 1 + + X iajs-2560 389 2 𝐽(∁ 𝐽(∁ ADJ iajs-2560 389 3 ) ) PUNCT iajs-2560 389 4 . . PUNCT iajs-2560 390 1 hence hence ADV iajs-2560 390 2 𝐼∁ 𝐼∁ PUNCT iajs-2560 390 3 is be AUX iajs-2560 390 4 a a DET iajs-2560 390 5 nearly nearly ADV iajs-2560 390 6 primary-2 primary-2 NOUN iajs-2560 390 7 - - PUNCT iajs-2560 390 8 absorbing absorbing ADJ iajs-2560 390 9 submodule submodule NOUN iajs-2560 390 10 of of ADP iajs-2560 390 11 ∁. ∁. NOUN iajs-2560 390 12 proposition proposition NOUN iajs-2560 390 13 22 22 NUM iajs-2560 390 14 let let VERB iajs-2560 390 15 ∁ ∁ NOUN iajs-2560 390 16 be be AUX iajs-2560 390 17 a a DET iajs-2560 390 18 faithful faithful ADJ iajs-2560 390 19 finitely finitely ADV iajs-2560 390 20 generated generate VERB iajs-2560 390 21 multiplication multiplication NOUN iajs-2560 390 22 𝑅-module 𝑅-module PROPN iajs-2560 390 23 over over ADP iajs-2560 390 24 an an DET iajs-2560 390 25 artinian artinian ADJ iajs-2560 390 26 ring ring NOUN iajs-2560 390 27 𝑅 𝑅 PROPN iajs-2560 390 28 and and CCONJ iajs-2560 390 29 𝐾 𝐾 PROPN iajs-2560 390 30 be be VERB iajs-2560 390 31 a a DET iajs-2560 390 32 proper proper ADJ iajs-2560 390 33 submodule submodule NOUN iajs-2560 390 34 of of ADP iajs-2560 390 35 ∁. ∁. NOUN iajs-2560 390 36 then then ADV iajs-2560 390 37 the the DET iajs-2560 390 38 next next ADJ iajs-2560 390 39 statements statement NOUN iajs-2560 390 40 are be AUX iajs-2560 390 41 equivalent equivalent ADJ iajs-2560 390 42 . . PUNCT iajs-2560 391 1 1 1 X iajs-2560 391 2 . . X iajs-2560 391 3 l l NOUN iajs-2560 391 4 is be AUX iajs-2560 391 5 a a DET iajs-2560 391 6 nearly nearly ADV iajs-2560 391 7 primary-2 primary-2 NOUN iajs-2560 391 8 - - PUNCT iajs-2560 391 9 absorbing absorb VERB iajs-2560 391 10 submodule submodule NOUN iajs-2560 391 11 of of ADP iajs-2560 391 12 ∁. ∁. NOUN iajs-2560 391 13 2 2 NUM iajs-2560 391 14 . . PUNCT iajs-2560 392 1 [ [ X iajs-2560 392 2 l:𝑅 l:𝑅 NOUN iajs-2560 392 3 ∁ ∁ X iajs-2560 392 4 ] ] PUNCT iajs-2560 392 5 is be AUX iajs-2560 392 6 a a DET iajs-2560 392 7 nearly nearly ADV iajs-2560 392 8 primary-2 primary-2 NOUN iajs-2560 392 9 - - PUNCT iajs-2560 392 10 absorbing absorbing ADJ iajs-2560 392 11 ideal ideal NOUN iajs-2560 392 12 of of ADP iajs-2560 392 13 𝑅. 𝑅. NOUN iajs-2560 392 14 3 3 NUM iajs-2560 392 15 . . PUNCT iajs-2560 393 1 l l NOUN iajs-2560 393 2 = = PUNCT iajs-2560 393 3 j∁ j∁ ADJ iajs-2560 393 4 for for ADP iajs-2560 393 5 some some DET iajs-2560 393 6 nearly nearly ADV iajs-2560 393 7 primary-2 primary-2 NOUN iajs-2560 393 8 - - PUNCT iajs-2560 393 9 absorbing absorbing ADJ iajs-2560 393 10 ideal ideal NOUN iajs-2560 393 11 of of ADP iajs-2560 393 12 𝑅. 𝑅. ADJ iajs-2560 393 13 proof proof NOUN iajs-2560 393 14 : : PUNCT iajs-2560 393 15 1 1 NUM iajs-2560 393 16 ⇔ ⇔ X iajs-2560 393 17 2 2 NUM iajs-2560 393 18 via via ADP iajs-2560 393 19 proposition(16 proposition(16 NOUN iajs-2560 393 20 ) ) PUNCT iajs-2560 393 21 . . PUNCT iajs-2560 394 1 2 2 NUM iajs-2560 394 2 β‡’ β‡’ NOUN iajs-2560 394 3 3 3 NUM iajs-2560 394 4 since since SCONJ iajs-2560 394 5 [ [ X iajs-2560 394 6 l:𝑅 l:𝑅 PROPN iajs-2560 394 7 ∁ ∁ X iajs-2560 394 8 ] ] PUNCT iajs-2560 394 9 is be AUX iajs-2560 394 10 a a DET iajs-2560 394 11 nearly nearly ADV iajs-2560 394 12 primary-2 primary-2 NOUN iajs-2560 394 13 - - PUNCT iajs-2560 394 14 absorbing absorbing ADJ iajs-2560 394 15 ideal ideal NOUN iajs-2560 394 16 of of ADP iajs-2560 394 17 𝑅 𝑅 PROPN iajs-2560 394 18 with with ADP iajs-2560 394 19 π‘Žπ‘›π‘›π‘…(∁ π‘Žπ‘›π‘›π‘…(∁ PROPN iajs-2560 394 20 ) ) PUNCT iajs-2560 394 21 = = PUNCT iajs-2560 395 1 [ [ X iajs-2560 395 2 0 0 NUM iajs-2560 395 3 : : PUNCT iajs-2560 395 4 ∁ ∁ NUM iajs-2560 395 5 ] ] PUNCT iajs-2560 396 1 βŠ† βŠ† NUM iajs-2560 396 2 [ [ X iajs-2560 396 3 l:𝑅 l:𝑅 NOUN iajs-2560 396 4 ∁ ∁ X iajs-2560 396 5 ] ] PUNCT iajs-2560 396 6 and and CCONJ iajs-2560 396 7 l l NOUN iajs-2560 396 8 = = PUNCT iajs-2560 397 1 [ [ X iajs-2560 397 2 l:𝑅 l:𝑅 NOUN iajs-2560 397 3 ∁]∁ ∁]∁ NOUN iajs-2560 397 4 , , PUNCT iajs-2560 397 5 implies imply VERB iajs-2560 397 6 that that SCONJ iajs-2560 397 7 l l NOUN iajs-2560 397 8 = = NOUN iajs-2560 397 9 i∁ i∁ NOUN iajs-2560 397 10 where where SCONJ iajs-2560 397 11 i i PRON iajs-2560 397 12 = = PUNCT iajs-2560 398 1 [ [ X iajs-2560 398 2 l:𝑅 l:𝑅 NOUN iajs-2560 398 3 ∁ ∁ X iajs-2560 398 4 ] ] PUNCT iajs-2560 398 5 is be AUX iajs-2560 398 6 a a DET iajs-2560 398 7 nearly nearly ADV iajs-2560 398 8 primary-2 primary-2 NOUN iajs-2560 398 9 - - PUNCT iajs-2560 398 10 absorbing absorbing ADJ iajs-2560 398 11 ideal ideal NOUN iajs-2560 398 12 of of ADP iajs-2560 398 13 𝑅. 𝑅. SYM iajs-2560 398 14 3 3 NUM iajs-2560 398 15 ⟹ ⟹ NUM iajs-2560 398 16 2 2 NUM iajs-2560 398 17 suppose suppose VERB iajs-2560 398 18 that that SCONJ iajs-2560 398 19 𝐿 𝐿 PROPN iajs-2560 398 20 = = PUNCT iajs-2560 398 21 𝐽∁ 𝐽∁ X iajs-2560 398 22 for for ADP iajs-2560 398 23 some some DET iajs-2560 398 24 nearly nearly ADV iajs-2560 398 25 primary-2 primary-2 NOUN iajs-2560 398 26 - - PUNCT iajs-2560 398 27 absorbing absorbing ADJ iajs-2560 398 28 ideal ideal NOUN iajs-2560 398 29 𝐽 𝐽 PROPN iajs-2560 398 30 of of ADP iajs-2560 398 31 𝑅. 𝑅. NOUN iajs-2560 398 32 since since SCONJ iajs-2560 398 33 ∁ ∁ PROPN iajs-2560 398 34 is be AUX iajs-2560 398 35 multiplication multiplication NOUN iajs-2560 398 36 , , PUNCT iajs-2560 398 37 then then ADV iajs-2560 398 38 l l NOUN iajs-2560 398 39 = = PUNCT iajs-2560 399 1 [ [ X iajs-2560 399 2 l:𝑅 l:𝑅 NOUN iajs-2560 399 3 ∁]∁= ∁]∁= PROPN iajs-2560 399 4 i∁. i∁. PROPN iajs-2560 399 5 since since SCONJ iajs-2560 399 6 ∁ ∁ PROPN iajs-2560 399 7 is be AUX iajs-2560 399 8 a a DET iajs-2560 399 9 faithful faithful ADJ iajs-2560 399 10 finitely finitely ADV iajs-2560 399 11 generated generate VERB iajs-2560 399 12 multiplication multiplication NOUN iajs-2560 399 13 𝑅module 𝑅module PROPN iajs-2560 399 14 over over ADP iajs-2560 399 15 an an DET iajs-2560 399 16 artinian artinian ADJ iajs-2560 399 17 ring ring PROPN iajs-2560 399 18 𝑅 𝑅 PROPN iajs-2560 399 19 , , PUNCT iajs-2560 399 20 then then ADV iajs-2560 399 21 we we PRON iajs-2560 399 22 have have VERB iajs-2560 399 23 [ [ X iajs-2560 399 24 l:𝑅 l:𝑅 NOUN iajs-2560 399 25 ∁ ∁ PROPN iajs-2560 399 26 ] ] X iajs-2560 399 27 = = PUNCT iajs-2560 399 28 j. j. X iajs-2560 400 1 thus thus ADV iajs-2560 400 2 [ [ X iajs-2560 400 3 l:𝑅 l:𝑅 PROPN iajs-2560 400 4 ∁ ∁ X iajs-2560 400 5 ] ] PUNCT iajs-2560 400 6 is be AUX iajs-2560 400 7 a a DET iajs-2560 400 8 nearly nearly ADV iajs-2560 400 9 primary-2absorbing primary-2absorbing NOUN iajs-2560 400 10 ideal ideal NOUN iajs-2560 400 11 of of ADP iajs-2560 400 12 𝑅. 𝑅. NOUN iajs-2560 400 13 124 124 NUM iajs-2560 400 14 ibn ibn PROPN iajs-2560 400 15 al al PROPN iajs-2560 400 16 - - PUNCT iajs-2560 400 17 haitham haitham PROPN iajs-2560 400 18 jour jour X iajs-2560 400 19 . . PROPN iajs-2560 400 20 for for ADP iajs-2560 400 21 pure pure ADJ iajs-2560 400 22 & & CCONJ iajs-2560 400 23 appl appl PROPN iajs-2560 400 24 . . PUNCT iajs-2560 401 1 sci sci PROPN iajs-2560 401 2 . . PROPN iajs-2560 402 1 34 34 NUM iajs-2560 402 2 ( ( PUNCT iajs-2560 402 3 1 1 NUM iajs-2560 402 4 ) ) PUNCT iajs-2560 402 5 2021 2021 NUM iajs-2560 402 6 3 3 NUM iajs-2560 402 7 . . PUNCT iajs-2560 402 8 conclusion conclusion NOUN iajs-2560 402 9 in in ADP iajs-2560 402 10 this this DET iajs-2560 402 11 article article NOUN iajs-2560 402 12 we we PRON iajs-2560 402 13 introduce introduce VERB iajs-2560 402 14 a a DET iajs-2560 402 15 new new ADJ iajs-2560 402 16 generalization generalization NOUN iajs-2560 402 17 of of ADP iajs-2560 402 18 ( ( PUNCT iajs-2560 402 19 prime prime ADJ iajs-2560 402 20 , , PUNCT iajs-2560 402 21 primary primary ADJ iajs-2560 402 22 , , PUNCT iajs-2560 402 23 2 2 NUM iajs-2560 402 24 - - PUNCT iajs-2560 402 25 absorbing absorbing ADJ iajs-2560 402 26 ) ) PUNCT iajs-2560 402 27 submodules submodule NOUN iajs-2560 402 28 called call VERB iajs-2560 402 29 a a DET iajs-2560 402 30 nearly nearly ADV iajs-2560 402 31 primary-2 primary-2 NOUN iajs-2560 402 32 - - PUNCT iajs-2560 402 33 absorbing absorbing ADJ iajs-2560 402 34 submodules submodule NOUN iajs-2560 402 35 and and CCONJ iajs-2560 402 36 we we PRON iajs-2560 402 37 explain explain VERB iajs-2560 402 38 the the DET iajs-2560 402 39 converse converse NOUN iajs-2560 402 40 implication implication NOUN iajs-2560 402 41 of of ADP iajs-2560 402 42 the the DET iajs-2560 402 43 above above ADJ iajs-2560 402 44 by by ADP iajs-2560 402 45 examples example NOUN iajs-2560 402 46 . . PUNCT iajs-2560 403 1 many many ADJ iajs-2560 403 2 characterizations characterization NOUN iajs-2560 403 3 of of ADP iajs-2560 403 4 this this DET iajs-2560 403 5 generalization generalization NOUN iajs-2560 403 6 are be AUX iajs-2560 403 7 introduced introduce VERB iajs-2560 403 8 . . PUNCT iajs-2560 404 1 relationships relationship NOUN iajs-2560 404 2 of of ADP iajs-2560 404 3 this this DET iajs-2560 404 4 generalization generalization NOUN iajs-2560 404 5 with with ADP iajs-2560 404 6 other other ADJ iajs-2560 404 7 classes class NOUN iajs-2560 404 8 of of ADP iajs-2560 404 9 modules module NOUN iajs-2560 404 10 are be AUX iajs-2560 404 11 given give VERB iajs-2560 404 12 . . PUNCT iajs-2560 405 1 references reference NOUN iajs-2560 405 2 1 1 NUM iajs-2560 405 3 . . X iajs-2560 406 1 lu lu PROPN iajs-2560 406 2 , , PUNCT iajs-2560 406 3 c.p c.p PROPN iajs-2560 406 4 . . 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