id sid tid token lemma pos iajs-2513 1 1 microsoft microsoft PROPN iajs-2513 1 2 word word NOUN iajs-2513 1 3 92 92 NUM iajs-2513 1 4 - - SYM iajs-2513 1 5 101 101 NUM iajs-2513 1 6 ibn ibn PROPN iajs-2513 1 7 al al PROPN iajs-2513 1 8 - - PUNCT iajs-2513 1 9 haitham haitham PROPN iajs-2513 1 10 jour jour X iajs-2513 1 11 . . PROPN iajs-2513 2 1 for for ADP iajs-2513 2 2 pure pure ADJ iajs-2513 2 3 & & CCONJ iajs-2513 2 4 appl appl PROPN iajs-2513 2 5 . . PUNCT iajs-2513 3 1 sci sci PROPN iajs-2513 3 2 . . PROPN iajs-2513 4 1 33 33 NUM iajs-2513 4 2 ( ( PUNCT iajs-2513 4 3 4 4 NUM iajs-2513 4 4 ) ) PUNCT iajs-2513 4 5 2020 2020 NUM iajs-2513 5 1 92 92 NUM iajs-2513 5 2                     SPACE iajs-2513 5 3 approximaitly approximaitly ADV iajs-2513 5 4 quasi quasi ADJ iajs-2513 5 5 - - ADJ iajs-2513 5 6 primary primary ADJ iajs-2513 5 7 submodules submodule NOUN iajs-2513 5 8       SPACE iajs-2513 5 9 ali ali PROPN iajs-2513 5 10 sh sh PROPN iajs-2513 5 11 . . PROPN iajs-2513 5 12 ajeel ajeel PROPN iajs-2513 5 13 omar omar PROPN iajs-2513 5 14 a. a. PROPN iajs-2513 5 15 abdulla abdulla PROPN iajs-2513 5 16 haibat haibat PROPN iajs-2513 5 17 k. k. PROPN iajs-2513 5 18 mohammadali mohammadali PROPN iajs-2513 5 19         SPACE iajs-2513 5 20 ali.shebl@st.tu.edu.iq ali.shebl@st.tu.edu.iq ADJ iajs-2513 5 21 omar.aldoori87@gmail.com omar.aldoori87@gmail.com NOUN iajs-2513 5 22 h.mohammadali@tu.edu.iq h.mohammadali@tu.edu.iq NOUN iajs-2513 5 23 abstract abstract NOUN iajs-2513 5 24 in in ADP iajs-2513 5 25 this this DET iajs-2513 5 26 paper paper NOUN iajs-2513 5 27 , , PUNCT iajs-2513 5 28 we we PRON iajs-2513 5 29 introduce introduce VERB iajs-2513 5 30 and and CCONJ iajs-2513 5 31 study study VERB iajs-2513 5 32 the the DET iajs-2513 5 33 notation notation NOUN iajs-2513 5 34 of of ADP iajs-2513 5 35 approximaitly approximaitly ADV iajs-2513 5 36 quasi quasi ADJ iajs-2513 5 37 - - ADJ iajs-2513 5 38 primary primary ADJ iajs-2513 5 39 submodules submodule NOUN iajs-2513 5 40 of of ADP iajs-2513 5 41 a a DET iajs-2513 5 42 unitary unitary ADJ iajs-2513 5 43 left leave VERB iajs-2513 6 1 𝑅-module 𝑅-module PROPN iajs-2513 6 2 𝑄 𝑄 PRON iajs-2513 6 3 over over ADP iajs-2513 6 4 a a DET iajs-2513 6 5 commutative commutative ADJ iajs-2513 6 6 ring ring NOUN iajs-2513 6 7 𝑅 𝑅 PROPN iajs-2513 6 8 with with ADP iajs-2513 6 9 identity identity NOUN iajs-2513 6 10 . . PUNCT iajs-2513 7 1 this this DET iajs-2513 7 2 concept concept NOUN iajs-2513 7 3 is be AUX iajs-2513 7 4 a a DET iajs-2513 7 5 generalization generalization NOUN iajs-2513 7 6 of of ADP iajs-2513 7 7 prime prime ADJ iajs-2513 7 8 and and CCONJ iajs-2513 7 9 primary primary ADJ iajs-2513 7 10 submodules submodule NOUN iajs-2513 7 11 , , PUNCT iajs-2513 7 12 where where SCONJ iajs-2513 7 13 a a DET iajs-2513 7 14 proper proper ADJ iajs-2513 7 15 submodule submodule NOUN iajs-2513 7 16 𝐸 𝐸 PROPN iajs-2513 7 17 of of ADP iajs-2513 7 18 an an DET iajs-2513 7 19 𝑅-module 𝑅-module PROPN iajs-2513 7 20 𝑄 𝑄 PROPN iajs-2513 7 21 is be AUX iajs-2513 7 22 called call VERB iajs-2513 7 23 an an DET iajs-2513 7 24 approximaitly approximaitly ADV iajs-2513 7 25 quasi quasi ADJ iajs-2513 7 26 - - NOUN iajs-2513 7 27 primary primary ADJ iajs-2513 7 28 ( ( PUNCT iajs-2513 7 29 for for ADP iajs-2513 7 30 short short ADJ iajs-2513 7 31 app app PROPN iajs-2513 7 32 - - PUNCT iajs-2513 7 33 qp qp NOUN iajs-2513 7 34 ) ) PUNCT iajs-2513 7 35 submodule submodule NOUN iajs-2513 7 36 of of ADP iajs-2513 7 37 𝑄 𝑄 PROPN iajs-2513 7 38 , , PUNCT iajs-2513 7 39 if if SCONJ iajs-2513 7 40 𝑟𝑞 𝑟𝑞 PROPN iajs-2513 7 41 ∈ ∈ PROPN iajs-2513 7 42 𝐸 𝐸 PROPN iajs-2513 7 43 , , PUNCT iajs-2513 7 44 for for ADP iajs-2513 7 45 𝑟 𝑟 DET iajs-2513 7 46 ∈ ∈ PROPN iajs-2513 7 47 𝑅 𝑅 PROPN iajs-2513 7 48 , , PUNCT iajs-2513 7 49 𝑞 𝑞 X iajs-2513 7 50 ∈ ∈ PROPN iajs-2513 7 51 𝑄 𝑄 PROPN iajs-2513 7 52 , , PUNCT iajs-2513 7 53 implies imply VERB iajs-2513 7 54 that that SCONJ iajs-2513 7 55 either either CCONJ iajs-2513 7 56 𝑞 𝑞 PROPN iajs-2513 7 57 ∈ ∈ PROPN iajs-2513 7 58 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 7 59 𝐸 𝐸 PROPN iajs-2513 7 60 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 7 61 𝑄 𝑄 PROPN iajs-2513 7 62 or or CCONJ iajs-2513 7 63 𝑟 𝑟 PRON iajs-2513 7 64 𝑄 𝑄 PROPN iajs-2513 7 65 ⊆ ⊆ NUM iajs-2513 7 66 𝐸 𝐸 PROPN iajs-2513 7 67 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 7 68 𝑄 𝑄 PROPN iajs-2513 7 69 , , PUNCT iajs-2513 7 70 for for ADP iajs-2513 7 71 some some PRON iajs-2513 7 72 𝑛 𝑛 DET iajs-2513 7 73 ∈ ∈ PROPN iajs-2513 7 74 𝑍 𝑍 NOUN iajs-2513 7 75 . . PUNCT iajs-2513 8 1 many many ADJ iajs-2513 8 2 basic basic ADJ iajs-2513 8 3 properties property NOUN iajs-2513 8 4 , , PUNCT iajs-2513 8 5 examples example NOUN iajs-2513 8 6 and and CCONJ iajs-2513 8 7 characterizations characterization NOUN iajs-2513 8 8 of of ADP iajs-2513 8 9 this this DET iajs-2513 8 10 concept concept NOUN iajs-2513 8 11 are be AUX iajs-2513 8 12 introduced introduce VERB iajs-2513 8 13 . . PUNCT iajs-2513 9 1 keywords keyword NOUN iajs-2513 9 2 : : PUNCT iajs-2513 9 3 prime prime ADJ iajs-2513 9 4 submodules submodule NOUN iajs-2513 9 5 , , PUNCT iajs-2513 9 6 primary primary ADJ iajs-2513 9 7 submodules submodule NOUN iajs-2513 9 8 , , PUNCT iajs-2513 9 9 socle socle NOUN iajs-2513 9 10 of of ADP iajs-2513 9 11 modules module NOUN iajs-2513 9 12 , , PUNCT iajs-2513 9 13 radical radical ADJ iajs-2513 9 14 of of ADP iajs-2513 9 15 submodules submodule NOUN iajs-2513 9 16 , , PUNCT iajs-2513 9 17 multiplication multiplication NOUN iajs-2513 9 18 modules module NOUN iajs-2513 9 19 , , PUNCT iajs-2513 9 20 nonsingular nonsingular ADJ iajs-2513 9 21 modules module NOUN iajs-2513 9 22 . . PUNCT iajs-2513 10 1 1 1 X iajs-2513 10 2 . . X iajs-2513 10 3 introduction introduction NOUN iajs-2513 10 4 in in ADP iajs-2513 10 5 this this DET iajs-2513 10 6 article article NOUN iajs-2513 10 7 all all DET iajs-2513 10 8 rings ring NOUN iajs-2513 10 9 are be AUX iajs-2513 10 10 commutative commutative ADJ iajs-2513 10 11 with with ADP iajs-2513 10 12 identity identity NOUN iajs-2513 10 13 , , PUNCT iajs-2513 10 14 and and CCONJ iajs-2513 10 15 all all DET iajs-2513 10 16 modules module NOUN iajs-2513 10 17 are be AUX iajs-2513 10 18 left leave VERB iajs-2513 10 19 unitary unitary ADJ iajs-2513 10 20 𝑅modules 𝑅modules PROPN iajs-2513 10 21 . . PUNCT iajs-2513 11 1 dauns daun NOUN iajs-2513 11 2 , , PUNCT iajs-2513 11 3 j. j. PROPN iajs-2513 11 4 in in ADP iajs-2513 11 5 1978 1978 NUM iajs-2513 11 6 introduced introduce VERB iajs-2513 11 7 and and CCONJ iajs-2513 11 8 studied study VERB iajs-2513 11 9 the the DET iajs-2513 11 10 concept concept NOUN iajs-2513 11 11 of of ADP iajs-2513 11 12 prime prime ADJ iajs-2513 11 13 submodule submodule NOUN iajs-2513 11 14 , , PUNCT iajs-2513 11 15 where where SCONJ iajs-2513 11 16 a a DET iajs-2513 11 17 proper proper ADJ iajs-2513 11 18 submodule submodule NOUN iajs-2513 11 19 𝐸 𝐸 PROPN iajs-2513 11 20 of of ADP iajs-2513 11 21 an an DET iajs-2513 11 22 𝑅module 𝑅module PROPN iajs-2513 11 23 𝑄 𝑄 PROPN iajs-2513 11 24 was be AUX iajs-2513 11 25 prime prime ADJ iajs-2513 11 26 if if SCONJ iajs-2513 11 27 𝑟𝑞 𝑟𝑞 PROPN iajs-2513 11 28 ∈ ∈ PROPN iajs-2513 11 29 𝐸 𝐸 PROPN iajs-2513 11 30 , , PUNCT iajs-2513 11 31 for for ADP iajs-2513 11 32 𝑟 𝑟 DET iajs-2513 11 33 ∈ ∈ PROPN iajs-2513 11 34 𝑅 𝑅 PROPN iajs-2513 11 35 , , PUNCT iajs-2513 11 36 𝑞 𝑞 X iajs-2513 11 37 ∈ ∈ PROPN iajs-2513 11 38 𝑄 𝑄 PROPN iajs-2513 11 39 , , PUNCT iajs-2513 11 40 implying imply VERB iajs-2513 11 41 that that SCONJ iajs-2513 11 42 either either CCONJ iajs-2513 11 43 𝑞 𝑞 PROPN iajs-2513 11 44 ∈ ∈ PROPN iajs-2513 11 45 𝐸 𝐸 PROPN iajs-2513 11 46 or or CCONJ iajs-2513 11 47 𝑟𝑄 𝑟𝑄 VERB iajs-2513 11 48 ⊆ ⊆ NUM iajs-2513 11 49 𝐸 𝐸 PROPN iajs-2513 12 1 [ [ X iajs-2513 12 2 1 1 NUM iajs-2513 12 3 ] ] PUNCT iajs-2513 12 4 . . PUNCT iajs-2513 13 1 recently recently ADV iajs-2513 13 2 many many ADJ iajs-2513 13 3 generalizations generalization NOUN iajs-2513 13 4 of of ADP iajs-2513 13 5 prime prime ADJ iajs-2513 13 6 submodule submodule NOUN iajs-2513 13 7 have have AUX iajs-2513 13 8 been be AUX iajs-2513 13 9 introduced introduce VERB iajs-2513 13 10 for for ADP iajs-2513 13 11 example example NOUN iajs-2513 13 12 , , PUNCT iajs-2513 13 13 see see VERB iajs-2513 13 14 [ [ X iajs-2513 13 15 2 2 NUM iajs-2513 13 16 - - SYM iajs-2513 13 17 5 5 NUM iajs-2513 13 18 ] ] PUNCT iajs-2513 13 19 . . PUNCT iajs-2513 14 1 primary primary ADJ iajs-2513 14 2 submodules submodule NOUN iajs-2513 14 3 as as ADP iajs-2513 14 4 a a DET iajs-2513 14 5 generalization generalization NOUN iajs-2513 14 6 of of ADP iajs-2513 14 7 prime prime ADJ iajs-2513 14 8 submodules submodule NOUN iajs-2513 14 9 was be AUX iajs-2513 14 10 first first ADV iajs-2513 14 11 introduced introduce VERB iajs-2513 14 12 in in ADP iajs-2513 14 13 [ [ X iajs-2513 14 14 6 6 NUM iajs-2513 14 15 ] ] PUNCT iajs-2513 14 16 , , PUNCT iajs-2513 14 17 where where SCONJ iajs-2513 14 18 a a DET iajs-2513 14 19 proper proper ADJ iajs-2513 14 20 submodule submodule NOUN iajs-2513 14 21 𝐸 𝐸 PROPN iajs-2513 14 22 of of ADP iajs-2513 14 23 𝑄 𝑄 PRON iajs-2513 14 24 was be AUX iajs-2513 14 25 called call VERB iajs-2513 14 26 primary primary ADJ iajs-2513 14 27 submodule submodule NOUN iajs-2513 14 28 if if SCONJ iajs-2513 14 29 whenever whenever ADV iajs-2513 14 30 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 14 31 ∈ ∈ PROPN iajs-2513 14 32 𝐸 𝐸 PROPN iajs-2513 14 33 , , PUNCT iajs-2513 14 34 for for ADP iajs-2513 14 35 𝑟 𝑟 DET iajs-2513 14 36 ∈ ∈ PROPN iajs-2513 14 37 𝑅 𝑅 PROPN iajs-2513 14 38 , , PUNCT iajs-2513 14 39 𝑞 𝑞 X iajs-2513 14 40 ∈ ∈ PROPN iajs-2513 14 41 𝑄 𝑄 PROPN iajs-2513 14 42 , , PUNCT iajs-2513 14 43 implying imply VERB iajs-2513 14 44 that that SCONJ iajs-2513 14 45 either either CCONJ iajs-2513 14 46 𝑞 𝑞 PROPN iajs-2513 14 47 ∈ ∈ PROPN iajs-2513 14 48 𝐸 𝐸 PROPN iajs-2513 14 49 or or CCONJ iajs-2513 14 50 𝑟 𝑟 PRON iajs-2513 14 51 𝑄 𝑄 PROPN iajs-2513 14 52 ⊆ ⊆ NUM iajs-2513 14 53 𝐸 𝐸 PROPN iajs-2513 14 54 , , PUNCT iajs-2513 14 55 for for ADP iajs-2513 14 56 some some PRON iajs-2513 14 57 𝑛 𝑛 DET iajs-2513 14 58 ∈ ∈ PROPN iajs-2513 14 59 𝑍 𝑍 NOUN iajs-2513 14 60 . . PUNCT iajs-2513 15 1 the the DET iajs-2513 15 2 concept concept NOUN iajs-2513 15 3 of of ADP iajs-2513 15 4 quasi quasi ADJ iajs-2513 15 5 - - ADJ iajs-2513 15 6 primary primary ADJ iajs-2513 15 7 ideal ideal NOUN iajs-2513 15 8 which which PRON iajs-2513 15 9 was be AUX iajs-2513 15 10 introduced introduce VERB iajs-2513 15 11 and and CCONJ iajs-2513 15 12 studied study VERB iajs-2513 15 13 by by ADP iajs-2513 15 14 fuchs fuch NOUN iajs-2513 15 15 , , PUNCT iajs-2513 15 16 l. l. PROPN iajs-2513 16 1 [ [ X iajs-2513 16 2 7 7 NUM iajs-2513 16 3 ] ] PUNCT iajs-2513 16 4 , , PUNCT iajs-2513 16 5 where where SCONJ iajs-2513 16 6 a a DET iajs-2513 16 7 proper proper ADJ iajs-2513 16 8 ideal ideal NOUN iajs-2513 16 9 𝐼 𝐼 PROPN iajs-2513 16 10 of of ADP iajs-2513 16 11 a a DET iajs-2513 16 12 ring ring NOUN iajs-2513 16 13 𝑅 𝑅 PROPN iajs-2513 16 14 was be AUX iajs-2513 16 15 called call VERB iajs-2513 16 16 quasi quasi ADJ iajs-2513 16 17 - - ADJ iajs-2513 16 18 primary primary ADJ iajs-2513 16 19 ideal ideal NOUN iajs-2513 16 20 if if SCONJ iajs-2513 16 21 𝑟𝑠 𝑟𝑠 PROPN iajs-2513 16 22 ∈ ∈ PROPN iajs-2513 16 23 𝐼 𝐼 PROPN iajs-2513 16 24 , , PUNCT iajs-2513 16 25 for for ADP iajs-2513 16 26 𝑟 𝑟 NOUN iajs-2513 16 27 , , PUNCT iajs-2513 16 28 𝑠 𝑠 PROPN iajs-2513 16 29 ∈ ∈ PROPN iajs-2513 16 30 𝑅 𝑅 PROPN iajs-2513 16 31 , , PUNCT iajs-2513 16 32 implying imply VERB iajs-2513 16 33 that that SCONJ iajs-2513 16 34 𝑟 𝑟 X iajs-2513 16 35 ∈ ∈ PROPN iajs-2513 16 36 √𝐼 √𝐼 X iajs-2513 16 37 or or CCONJ iajs-2513 16 38 𝑠 𝑠 PROPN iajs-2513 16 39 ∈ ∈ PROPN iajs-2513 16 40 √𝐼 √𝐼 PROPN iajs-2513 16 41 , , PUNCT iajs-2513 16 42 where where SCONJ iajs-2513 16 43 √𝐼 √𝐼 VERB iajs-2513 16 44 𝑟 𝑟 X iajs-2513 16 45 ∈ ∈ PROPN iajs-2513 16 46 𝑅 𝑅 PROPN iajs-2513 16 47 : : PUNCT iajs-2513 16 48 𝑟 𝑟 X iajs-2513 16 49 ∈ ∈ NOUN iajs-2513 16 50 𝐼 𝐼 ADP iajs-2513 16 51 for for ADP iajs-2513 16 52 some some DET iajs-2513 16 53 𝑛 𝑛 DET iajs-2513 16 54 ∈ ∈ PROPN iajs-2513 16 55 𝑍 𝑍 NOUN iajs-2513 16 56 . . PUNCT iajs-2513 17 1 in in ADP iajs-2513 17 2 ibn ibn PROPN iajs-2513 17 3 al al PROPN iajs-2513 17 4 haitham haitham PROPN iajs-2513 17 5 journal journal PROPN iajs-2513 17 6 for for ADP iajs-2513 17 7 pure pure ADJ iajs-2513 17 8 and and CCONJ iajs-2513 17 9 applied apply VERB iajs-2513 17 10 science science NOUN iajs-2513 17 11 journal journal PROPN iajs-2513 17 12 homepage homepage NOUN iajs-2513 17 13 : : PUNCT iajs-2513 17 14 http://jih.uobaghdad.edu.iq/index.php/j/index http://jih.uobaghdad.edu.iq/index.php/j/index NOUN iajs-2513 17 15 directorate directorate ADJ iajs-2513 17 16 general general NOUN iajs-2513 17 17 of of ADP iajs-2513 17 18 education education NOUN iajs-2513 17 19 salahaddin salahaddin PROPN iajs-2513 17 20 , , PUNCT iajs-2513 17 21 the the DET iajs-2513 17 22 ministry ministry PROPN iajs-2513 17 23 of of ADP iajs-2513 17 24 education education PROPN iajs-2513 17 25 , , PUNCT iajs-2513 17 26 tikrit tikrit NOUN iajs-2513 17 27 , , PUNCT iajs-2513 17 28 iraq iraq PROPN iajs-2513 17 29 . . PUNCT iajs-2513 17 30     SPACE iajs-2513 17 31 directorate directorate ADJ iajs-2513 17 32 general general NOUN iajs-2513 17 33 of of ADP iajs-2513 17 34 education education NOUN iajs-2513 17 35 salahaddin salahaddin PROPN iajs-2513 17 36 , , PUNCT iajs-2513 17 37 the the DET iajs-2513 17 38 ministry ministry PROPN iajs-2513 17 39 of of ADP iajs-2513 17 40 education education PROPN iajs-2513 17 41 , , PUNCT iajs-2513 17 42 tikrit tikrit NOUN iajs-2513 17 43 , , PUNCT iajs-2513 17 44 iraq iraq PROPN iajs-2513 17 45 . . PUNCT iajs-2513 17 46     SPACE iajs-2513 17 47 department department NOUN iajs-2513 17 48 of of ADP iajs-2513 17 49 mathematics mathematics PROPN iajs-2513 17 50 , , PUNCT iajs-2513 17 51 college college NOUN iajs-2513 17 52 of of ADP iajs-2513 17 53 computer computer NOUN iajs-2513 17 54 sciences sciences PROPN iajs-2513 17 55 and and CCONJ iajs-2513 17 56 mathematics mathematic NOUN iajs-2513 17 57 , , PUNCT iajs-2513 17 58 tikrit tikrit NOUN iajs-2513 17 59 university university NOUN iajs-2513 17 60 , , PUNCT iajs-2513 17 61 tikrit tikrit NOUN iajs-2513 17 62 , , PUNCT iajs-2513 17 63 iraq iraq PROPN iajs-2513 17 64 . . PUNCT iajs-2513 18 1 doi doi PROPN iajs-2513 18 2 : : PUNCT iajs-2513 18 3 10.30526/33.4.2513 10.30526/33.4.2513 PROPN iajs-2513 18 4 article article NOUN iajs-2513 18 5 history history NOUN iajs-2513 18 6 : : PUNCT iajs-2513 18 7 received receive VERB iajs-2513 18 8 7 7 NUM iajs-2513 18 9 january january NOUN iajs-2513 18 10 2020 2020 NUM iajs-2513 18 11 , , PUNCT iajs-2513 18 12 accepted accept VERB iajs-2513 18 13 12 12 NUM iajs-2513 18 14 february february NOUN iajs-2513 18 15 2020 2020 NUM iajs-2513 18 16 , , PUNCT iajs-2513 18 17 published publish VERB iajs-2513 18 18 in in ADP iajs-2513 18 19 october october PROPN iajs-2513 18 20 2020 2020 NUM iajs-2513 18 21     SPACE iajs-2513 18 22 93 93 NUM iajs-2513 18 23     SPACE iajs-2513 18 24 ibn ibn PROPN iajs-2513 18 25 al al PROPN iajs-2513 18 26 - - PUNCT iajs-2513 18 27 haitham haitham PROPN iajs-2513 18 28 jour jour X iajs-2513 18 29 . . PROPN iajs-2513 19 1 for for ADP iajs-2513 19 2 pure pure ADJ iajs-2513 19 3 & & CCONJ iajs-2513 19 4 appl appl PROPN iajs-2513 19 5 . . PUNCT iajs-2513 20 1 sci sci PROPN iajs-2513 20 2 . . PROPN iajs-2513 21 1 33 33 NUM iajs-2513 21 2 ( ( PUNCT iajs-2513 21 3 4 4 NUM iajs-2513 21 4 ) ) PUNCT iajs-2513 21 5 2020 2020 NUM iajs-2513 21 6 particular particular ADJ iajs-2513 21 7 𝐼 𝐼 PROPN iajs-2513 21 8 is be AUX iajs-2513 21 9 quasi quasi ADJ iajs-2513 21 10 - - ADJ iajs-2513 21 11 primary primary ADJ iajs-2513 21 12 ideal ideal NOUN iajs-2513 21 13 of of ADP iajs-2513 21 14 𝑅 𝑅 PROPN iajs-2513 21 15 if if SCONJ iajs-2513 21 16 and and CCONJ iajs-2513 21 17 only only ADV iajs-2513 21 18 if if SCONJ iajs-2513 21 19 √𝐼 √𝐼 PRON iajs-2513 21 20 is be AUX iajs-2513 21 21 a a DET iajs-2513 21 22 prime prime ADJ iajs-2513 21 23 ideal ideal NOUN iajs-2513 21 24 of of ADP iajs-2513 21 25 𝑅 𝑅 PROPN iajs-2513 21 26 [ [ X iajs-2513 21 27 7 7 NUM iajs-2513 21 28 , , PUNCT iajs-2513 21 29 p. p. NOUN iajs-2513 21 30 176 176 NUM iajs-2513 21 31 ] ] PUNCT iajs-2513 21 32 . . PUNCT iajs-2513 22 1 in in ADP iajs-2513 22 2 2016 2016 NUM iajs-2513 22 3 hosein hosein PROPN iajs-2513 22 4 , , PUNCT iajs-2513 22 5 f. f. PROPN iajs-2513 22 6 et et PROPN iajs-2513 22 7 . . PROPN iajs-2513 22 8 extended extend VERB iajs-2513 22 9 the the DET iajs-2513 22 10 notation notation NOUN iajs-2513 22 11 of of ADP iajs-2513 22 12 quasi quasi ADJ iajs-2513 22 13 - - ADJ iajs-2513 22 14 primary primary ADJ iajs-2513 22 15 ideal ideal NOUN iajs-2513 22 16 to to ADP iajs-2513 22 17 submodules submodule NOUN iajs-2513 22 18 , , PUNCT iajs-2513 22 19 where where SCONJ iajs-2513 22 20 a a DET iajs-2513 22 21 proper proper ADJ iajs-2513 22 22 submodule submodule NOUN iajs-2513 22 23 𝐸 𝐸 PROPN iajs-2513 22 24 of of ADP iajs-2513 22 25 an an DET iajs-2513 22 26 𝑅-module 𝑅-module PROPN iajs-2513 22 27 𝑄 𝑄 PROPN iajs-2513 22 28 was be AUX iajs-2513 22 29 called call VERB iajs-2513 22 30 quasi quasi ADJ iajs-2513 22 31 - - NOUN iajs-2513 22 32 primary primary ADJ iajs-2513 22 33 if if SCONJ iajs-2513 22 34 𝑟𝑞 𝑟𝑞 PROPN iajs-2513 22 35 ∈ ∈ PROPN iajs-2513 22 36 𝐸 𝐸 PROPN iajs-2513 22 37 , , PUNCT iajs-2513 22 38 for for ADP iajs-2513 22 39 𝑟 𝑟 DET iajs-2513 22 40 ∈ ∈ PROPN iajs-2513 22 41 𝑅 𝑅 PROPN iajs-2513 22 42 , , PUNCT iajs-2513 22 43 𝑞 𝑞 X iajs-2513 22 44 ∈ ∈ PROPN iajs-2513 22 45 𝑄 𝑄 PROPN iajs-2513 22 46 , , PUNCT iajs-2513 22 47 implying imply VERB iajs-2513 22 48 that that SCONJ iajs-2513 22 49 either either CCONJ iajs-2513 22 50 𝑞 𝑞 PROPN iajs-2513 22 51 ∈ ∈ PROPN iajs-2513 22 52 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 22 53 𝐸 𝐸 PROPN iajs-2513 22 54 or or CCONJ iajs-2513 22 55 𝑟 𝑟 PRON iajs-2513 22 56 ∈ ∈ PROPN iajs-2513 22 57 𝐸 𝐸 PROPN iajs-2513 22 58 : : PUNCT iajs-2513 22 59 𝑄 𝑄 PROPN iajs-2513 22 60 , , PUNCT iajs-2513 22 61 “ " PUNCT iajs-2513 22 62 where where SCONJ iajs-2513 22 63 𝑟𝑎𝑑 𝑟𝑎𝑑 PRON iajs-2513 22 64 𝐸 𝐸 PROPN iajs-2513 22 65 define define VERB iajs-2513 22 66 the the DET iajs-2513 22 67 intersection intersection NOUN iajs-2513 22 68 of of ADP iajs-2513 22 69 all all DET iajs-2513 22 70 prime prime ADJ iajs-2513 22 71 submodules submodule NOUN iajs-2513 22 72 of of ADP iajs-2513 22 73 𝑄 𝑄 PRON iajs-2513 22 74 contining contine VERB iajs-2513 22 75 𝐸 𝐸 PROPN iajs-2513 23 1 [ [ X iajs-2513 23 2 8 8 NUM iajs-2513 23 3 ] ] PUNCT iajs-2513 23 4 ” " PUNCT iajs-2513 23 5 . . PUNCT iajs-2513 24 1 those those DET iajs-2513 24 2 two two NUM iajs-2513 24 3 concepts concept NOUN iajs-2513 24 4 led lead VERB iajs-2513 24 5 us we PRON iajs-2513 24 6 to to PART iajs-2513 24 7 introduce introduce VERB iajs-2513 24 8 the the DET iajs-2513 24 9 notation notation NOUN iajs-2513 24 10 of of ADP iajs-2513 24 11 approximaitly approximaitly ADV iajs-2513 24 12 quasi quasi ADJ iajs-2513 24 13 - - ADJ iajs-2513 24 14 primary primary ADJ iajs-2513 24 15 submodule submodule NOUN iajs-2513 24 16 as as ADP iajs-2513 24 17 generalization generalization NOUN iajs-2513 24 18 of of ADP iajs-2513 24 19 prime prime ADJ iajs-2513 24 20 and and CCONJ iajs-2513 24 21 primary primary ADJ iajs-2513 24 22 submodules submodule NOUN iajs-2513 24 23 , , PUNCT iajs-2513 24 24 where where SCONJ iajs-2513 24 25 a a DET iajs-2513 24 26 proper proper ADJ iajs-2513 24 27 submodule submodule NOUN iajs-2513 24 28 𝐸 𝐸 PROPN iajs-2513 24 29 of of ADP iajs-2513 24 30 an an DET iajs-2513 24 31 𝑅-module 𝑅-module PROPN iajs-2513 24 32 𝑄 𝑄 PROPN iajs-2513 24 33 is be AUX iajs-2513 24 34 called call VERB iajs-2513 24 35 an an DET iajs-2513 24 36 approximaitly approximaitly ADV iajs-2513 24 37 quasiprimary quasiprimary ADJ iajs-2513 24 38 ( ( PUNCT iajs-2513 24 39 for for ADP iajs-2513 24 40 short short ADJ iajs-2513 24 41 app app PROPN iajs-2513 24 42 - - PUNCT iajs-2513 24 43 qp qp NOUN iajs-2513 24 44 ) ) PUNCT iajs-2513 24 45 submodule submodule NOUN iajs-2513 24 46 of of ADP iajs-2513 24 47 𝑄 𝑄 PROPN iajs-2513 24 48 , , PUNCT iajs-2513 24 49 if if SCONJ iajs-2513 24 50 𝑟𝑞 𝑟𝑞 PROPN iajs-2513 24 51 ∈ ∈ PROPN iajs-2513 24 52 𝐸 𝐸 PROPN iajs-2513 24 53 , , PUNCT iajs-2513 24 54 for for ADP iajs-2513 24 55 𝑟 𝑟 DET iajs-2513 24 56 ∈ ∈ PROPN iajs-2513 24 57 𝑅 𝑅 PROPN iajs-2513 24 58 , , PUNCT iajs-2513 24 59 𝑞 𝑞 X iajs-2513 24 60 ∈ ∈ PROPN iajs-2513 24 61 𝑄 𝑄 PROPN iajs-2513 24 62 , , PUNCT iajs-2513 24 63 implies imply VERB iajs-2513 24 64 that that SCONJ iajs-2513 24 65 either either CCONJ iajs-2513 24 66 𝑞 𝑞 PROPN iajs-2513 24 67 ∈ ∈ PROPN iajs-2513 24 68 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 24 69 𝐸 𝐸 PROPN iajs-2513 24 70 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 24 71 𝑄 𝑄 PROPN iajs-2513 24 72 or or CCONJ iajs-2513 24 73 𝑟 𝑟 PRON iajs-2513 24 74 𝑄 𝑄 PROPN iajs-2513 24 75 ⊆ ⊆ NUM iajs-2513 24 76 𝐸 𝐸 PROPN iajs-2513 24 77 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 24 78 𝑄 𝑄 PROPN iajs-2513 24 79 , , PUNCT iajs-2513 24 80 for for ADP iajs-2513 24 81 some some PRON iajs-2513 24 82 𝑛 𝑛 DET iajs-2513 24 83 ∈ ∈ PROPN iajs-2513 24 84 𝑍 𝑍 NOUN iajs-2513 24 85 . . PUNCT iajs-2513 25 1 the the DET iajs-2513 25 2 socle socle NOUN iajs-2513 25 3 of of ADP iajs-2513 25 4 a a DET iajs-2513 25 5 module module NOUN iajs-2513 25 6 𝑄 𝑄 PRON iajs-2513 25 7 denoted denote VERB iajs-2513 25 8 by by ADP iajs-2513 25 9 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 25 10 𝑄 𝑄 PROPN iajs-2513 25 11 is be AUX iajs-2513 25 12 the the DET iajs-2513 25 13 intersection intersection NOUN iajs-2513 25 14 of of ADP iajs-2513 25 15 all all DET iajs-2513 25 16 essential essential ADJ iajs-2513 25 17 submodules submodule NOUN iajs-2513 25 18 of of ADP iajs-2513 25 19 𝑄 𝑄 PRON iajs-2513 25 20 [ [ X iajs-2513 25 21 9 9 NUM iajs-2513 25 22 ] ] PUNCT iajs-2513 25 23 . . PUNCT iajs-2513 26 1 several several ADJ iajs-2513 26 2 results result NOUN iajs-2513 26 3 of of ADP iajs-2513 26 4 approximaitly approximaitly ADV iajs-2513 26 5 quasi quasi ADJ iajs-2513 26 6 - - ADJ iajs-2513 26 7 primary primary ADJ iajs-2513 26 8 are be AUX iajs-2513 26 9 introduced introduce VERB iajs-2513 26 10 . . PUNCT iajs-2513 27 1 2 2 X iajs-2513 27 2 . . X iajs-2513 27 3 approximaitly approximaitly ADV iajs-2513 27 4 quasi quasi ADJ iajs-2513 27 5 - - ADJ iajs-2513 27 6 primary primary ADJ iajs-2513 27 7 submodules submodule NOUN iajs-2513 27 8 in in ADP iajs-2513 27 9 this this DET iajs-2513 27 10 part part NOUN iajs-2513 27 11 of of ADP iajs-2513 27 12 the the DET iajs-2513 27 13 paper paper NOUN iajs-2513 27 14 , , PUNCT iajs-2513 27 15 we we PRON iajs-2513 27 16 introduce introduce VERB iajs-2513 27 17 the the DET iajs-2513 27 18 definition definition NOUN iajs-2513 27 19 of of ADP iajs-2513 27 20 approximaitly approximaitly ADV iajs-2513 27 21 quasi quasi ADJ iajs-2513 27 22 - - ADJ iajs-2513 27 23 primary primary ADJ iajs-2513 27 24 submodule submodule NOUN iajs-2513 27 25 and and CCONJ iajs-2513 27 26 give give VERB iajs-2513 27 27 it it PRON iajs-2513 27 28 some some DET iajs-2513 27 29 basic basic ADJ iajs-2513 27 30 properties property NOUN iajs-2513 27 31 and and CCONJ iajs-2513 27 32 characterizations characterization NOUN iajs-2513 27 33 . . PUNCT iajs-2513 28 1 definition definition NOUN iajs-2513 28 2 ( ( PUNCT iajs-2513 28 3 1 1 X iajs-2513 28 4 ) ) PUNCT iajs-2513 28 5 a a DET iajs-2513 28 6 proper proper ADJ iajs-2513 28 7 submodule submodule NOUN iajs-2513 28 8 𝐸 𝐸 PROPN iajs-2513 28 9 of of ADP iajs-2513 28 10 an an DET iajs-2513 28 11 𝑅-module 𝑅-module PROPN iajs-2513 28 12 𝑄 𝑄 PROPN iajs-2513 28 13 is be AUX iajs-2513 28 14 called call VERB iajs-2513 28 15 an an DET iajs-2513 28 16 approximaitly approximaitly ADV iajs-2513 28 17 quasi quasi ADJ iajs-2513 28 18 - - NOUN iajs-2513 28 19 primary primary ADJ iajs-2513 28 20 ( ( PUNCT iajs-2513 28 21 for for ADP iajs-2513 28 22 short short ADJ iajs-2513 28 23 app app PROPN iajs-2513 28 24 - - PUNCT iajs-2513 28 25 qp qp NOUN iajs-2513 28 26 ) ) PUNCT iajs-2513 28 27 submodule submodule NOUN iajs-2513 28 28 of of ADP iajs-2513 28 29 𝑄 𝑄 PROPN iajs-2513 28 30 , , PUNCT iajs-2513 28 31 if if SCONJ iajs-2513 28 32 𝑟𝑞 𝑟𝑞 PROPN iajs-2513 28 33 ∈ ∈ PROPN iajs-2513 28 34 𝐸 𝐸 PROPN iajs-2513 28 35 , , PUNCT iajs-2513 28 36 for for ADP iajs-2513 28 37 𝑟 𝑟 DET iajs-2513 28 38 ∈ ∈ PROPN iajs-2513 28 39 𝑅 𝑅 PROPN iajs-2513 28 40 , , PUNCT iajs-2513 28 41 𝑞 𝑞 X iajs-2513 28 42 ∈ ∈ PROPN iajs-2513 28 43 𝑄 𝑄 PROPN iajs-2513 28 44 , , PUNCT iajs-2513 28 45 implies imply VERB iajs-2513 28 46 that that SCONJ iajs-2513 28 47 either either CCONJ iajs-2513 28 48 𝑞 𝑞 PROPN iajs-2513 28 49 ∈ ∈ PROPN iajs-2513 28 50 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 28 51 𝐸 𝐸 PROPN iajs-2513 28 52 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 28 53 𝑄 𝑄 PROPN iajs-2513 28 54 or or CCONJ iajs-2513 28 55 𝑟 𝑟 PRON iajs-2513 28 56 𝑄 𝑄 PROPN iajs-2513 28 57 ⊆ ⊆ NUM iajs-2513 28 58 𝐸 𝐸 PROPN iajs-2513 28 59 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 28 60 𝑄 𝑄 PROPN iajs-2513 28 61 , , PUNCT iajs-2513 28 62 for for ADP iajs-2513 28 63 some some PRON iajs-2513 28 64 𝑛 𝑛 DET iajs-2513 28 65 ∈ ∈ PROPN iajs-2513 28 66 𝑍 𝑍 NOUN iajs-2513 28 67 . . PUNCT iajs-2513 28 68 and and CCONJ iajs-2513 28 69 an an DET iajs-2513 28 70 ideal ideal ADJ iajs-2513 28 71 𝐴 𝐴 PROPN iajs-2513 28 72 of of ADP iajs-2513 28 73 a a DET iajs-2513 28 74 ring ring NOUN iajs-2513 28 75 𝑅 𝑅 PROPN iajs-2513 28 76 is be AUX iajs-2513 28 77 called call VERB iajs-2513 28 78 app app PROPN iajs-2513 28 79 - - PUNCT iajs-2513 28 80 qp qp NOUN iajs-2513 28 81 ideal ideal NOUN iajs-2513 28 82 of of ADP iajs-2513 28 83 𝑅 𝑅 PROPN iajs-2513 28 84 if if SCONJ iajs-2513 28 85 𝐴 𝐴 PROPN iajs-2513 28 86 is be AUX iajs-2513 28 87 an an DET iajs-2513 28 88 app app PROPN iajs-2513 28 89 - - PUNCT iajs-2513 28 90 qp qp NOUN iajs-2513 28 91 submodule submodule NOUN iajs-2513 28 92 of of ADP iajs-2513 28 93 an an DET iajs-2513 28 94 𝑅-module 𝑅-module PROPN iajs-2513 28 95 𝑅. 𝑅. NOUN iajs-2513 28 96 remarks remark NOUN iajs-2513 28 97 and and CCONJ iajs-2513 28 98 examples example NOUN iajs-2513 28 99 ( ( PUNCT iajs-2513 28 100 2 2 NUM iajs-2513 28 101 ) ) PUNCT iajs-2513 28 102 1 1 NUM iajs-2513 28 103 ) ) PUNCT iajs-2513 28 104 it it PRON iajs-2513 28 105 is be AUX iajs-2513 28 106 clear clear ADJ iajs-2513 28 107 that that SCONJ iajs-2513 28 108 every every DET iajs-2513 28 109 primary primary ADJ iajs-2513 28 110 submodule submodule NOUN iajs-2513 28 111 is be AUX iajs-2513 28 112 an an DET iajs-2513 28 113 app app PROPN iajs-2513 28 114 - - PUNCT iajs-2513 28 115 qp qp NOUN iajs-2513 28 116 , , PUNCT iajs-2513 28 117 but but CCONJ iajs-2513 28 118 not not PART iajs-2513 28 119 conversely conversely ADV iajs-2513 28 120 . . PUNCT iajs-2513 29 1 the the DET iajs-2513 29 2 following follow VERB iajs-2513 29 3 example example NOUN iajs-2513 29 4 explains explain VERB iajs-2513 29 5 that that SCONJ iajs-2513 29 6 : : PUNCT iajs-2513 29 7 consider consider VERB iajs-2513 29 8 the the DET iajs-2513 29 9 𝑍-module 𝑍-module PROPN iajs-2513 29 10 𝑍 𝑍 PROPN iajs-2513 29 11 , , PUNCT iajs-2513 29 12 the the DET iajs-2513 29 13 submodule submodule NOUN iajs-2513 29 14 𝐸 𝐸 PROPN iajs-2513 29 15 〈 〈 NOUN iajs-2513 29 16 0 0 NUM iajs-2513 29 17 〉 〉 NOUN iajs-2513 29 18 is be AUX iajs-2513 29 19 not not PART iajs-2513 29 20 primary primary ADJ iajs-2513 29 21 submodule submodule NOUN iajs-2513 29 22 of of ADP iajs-2513 29 23 𝑍-module 𝑍-module PROPN iajs-2513 29 24 𝑍 𝑍 NOUN iajs-2513 29 25 , , PUNCT iajs-2513 29 26 since since SCONJ iajs-2513 29 27 4 4 NUM iajs-2513 29 28 . . SYM iajs-2513 29 29 3 3 NUM iajs-2513 29 30 ∈ ∈ PROPN iajs-2513 29 31 〈 〈 NOUN iajs-2513 29 32 0 0 NUM iajs-2513 29 33 〉 〉 NOUN iajs-2513 29 34 , , PUNCT iajs-2513 29 35 for for ADP iajs-2513 29 36 4 4 NUM iajs-2513 29 37 ∈ ∈ PROPN iajs-2513 29 38 𝑍 𝑍 NOUN iajs-2513 29 39 , , PUNCT iajs-2513 29 40 3 3 NUM iajs-2513 29 41 ∈ ∈ NOUN iajs-2513 29 42 𝑍 𝑍 NOUN iajs-2513 29 43 , , PUNCT iajs-2513 29 44 but but CCONJ iajs-2513 29 45 3 3 NUM iajs-2513 29 46 ∉ ∉ PROPN iajs-2513 29 47 〈 〈 PROPN iajs-2513 29 48 0 0 NUM iajs-2513 29 49 〉 〉 NOUN iajs-2513 29 50 and and CCONJ iajs-2513 29 51 4 4 NUM iajs-2513 29 52 ∉ ∉ PROPN iajs-2513 29 53 〈 〈 PROPN iajs-2513 29 54 0 0 NUM iajs-2513 29 55 〉 〉 NOUN iajs-2513 29 56 : : PUNCT iajs-2513 29 57 𝑍 𝑍 VERB iajs-2513 29 58 √12𝑍 √12𝑍 PROPN iajs-2513 29 59 6𝑍. 6𝑍. NUM iajs-2513 30 1 but but CCONJ iajs-2513 30 2 𝐸 𝐸 PROPN iajs-2513 30 3 〈 〈 NOUN iajs-2513 30 4 0 0 NUM iajs-2513 30 5 〉 〉 NOUN iajs-2513 30 6 is be AUX iajs-2513 30 7 an an DET iajs-2513 30 8 app app PROPN iajs-2513 30 9 - - PUNCT iajs-2513 30 10 qp qp NOUN iajs-2513 30 11 submodule submodule NOUN iajs-2513 30 12 of of ADP iajs-2513 30 13 the the DET iajs-2513 30 14 𝑍-module 𝑍-module PROPN iajs-2513 30 15 𝑍 𝑍 NOUN iajs-2513 30 16 , , PUNCT iajs-2513 30 17 since since SCONJ iajs-2513 30 18 for for SCONJ iajs-2513 30 19 all all DET iajs-2513 30 20 𝑟 𝑟 PRON iajs-2513 30 21 ∈ ∈ PROPN iajs-2513 30 22 𝑅 𝑅 PROPN iajs-2513 30 23 , , PUNCT iajs-2513 30 24 𝑞 𝑞 PROPN iajs-2513 30 25 ∈ ∈ PROPN iajs-2513 30 26 𝑍 𝑍 VERB iajs-2513 30 27 such such ADJ iajs-2513 30 28 that that PRON iajs-2513 30 29 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 30 30 ∈ ∈ PROPN iajs-2513 30 31 𝐸 𝐸 PROPN iajs-2513 30 32 , , PUNCT iajs-2513 30 33 implies imply VERB iajs-2513 30 34 that that SCONJ iajs-2513 30 35 either either CCONJ iajs-2513 30 36 𝑞 𝑞 PROPN iajs-2513 30 37 ∈ ∈ PROPN iajs-2513 30 38 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 30 39 〈 〈 PROPN iajs-2513 30 40 0 0 NUM iajs-2513 30 41 〉 〉 NOUN iajs-2513 30 42 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 30 43 𝑍 𝑍 PROPN iajs-2513 30 44 〈 〈 NOUN iajs-2513 30 45 6 6 NUM iajs-2513 30 46 〉 〉 NOUN iajs-2513 30 47 〈 〈 NOUN iajs-2513 30 48 2 2 NUM iajs-2513 30 49 〉 〉 NOUN iajs-2513 30 50 〈 〈 NOUN iajs-2513 30 51 2 2 NUM iajs-2513 30 52 〉 〉 NOUN iajs-2513 30 53 or or CCONJ iajs-2513 30 54 𝑟 𝑟 NOUN iajs-2513 30 55 ∈ ∈ NOUN iajs-2513 30 56 〈 〈 PROPN iajs-2513 30 57 0 0 NUM iajs-2513 30 58 〉 〉 NOUN iajs-2513 30 59 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 30 60 𝑍 𝑍 PROPN iajs-2513 30 61 : : PUNCT iajs-2513 30 62 𝑍 𝑍 PROPN iajs-2513 30 63 〈 〈 NOUN iajs-2513 30 64 2 2 NUM iajs-2513 30 65 〉 〉 NOUN iajs-2513 30 66 : : PUNCT iajs-2513 30 67 𝑍 𝑍 PROPN iajs-2513 30 68 √2𝑍 √2𝑍 NOUN iajs-2513 30 69 2𝑍. 2𝑍. NUM iajs-2513 30 70 that that PRON iajs-2513 30 71 is be AUX iajs-2513 30 72 if if SCONJ iajs-2513 30 73 4 4 NUM iajs-2513 30 74 . . SYM iajs-2513 30 75 3 3 NUM iajs-2513 30 76 ∈ ∈ PROPN iajs-2513 30 77 𝐸 𝐸 PROPN iajs-2513 30 78 , , PUNCT iajs-2513 30 79 for for ADP iajs-2513 30 80 4 4 NUM iajs-2513 30 81 ∈ ∈ PROPN iajs-2513 30 82 𝑍 𝑍 NOUN iajs-2513 30 83 , , PUNCT iajs-2513 30 84 3 3 NUM iajs-2513 30 85 ∈ ∈ NOUN iajs-2513 30 86 𝑍 𝑍 NOUN iajs-2513 30 87 , , PUNCT iajs-2513 30 88 and and CCONJ iajs-2513 30 89 3 3 NUM iajs-2513 30 90 ∉ ∉ X iajs-2513 30 91 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 30 92 〈 〈 PROPN iajs-2513 30 93 0 0 NUM iajs-2513 30 94 〉 〉 NOUN iajs-2513 30 95 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 30 96 𝑍 𝑍 PROPN iajs-2513 30 97 〈 〈 NOUN iajs-2513 30 98 2 2 NUM iajs-2513 30 99 〉 〉 NOUN iajs-2513 30 100 but but CCONJ iajs-2513 30 101 4 4 NUM iajs-2513 30 102 ∈ ∈ NOUN iajs-2513 30 103 〈 〈 NOUN iajs-2513 30 104 0 0 NUM iajs-2513 30 105 〉 〉 NOUN iajs-2513 30 106 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 30 107 𝑍 𝑍 PROPN iajs-2513 30 108 : : PUNCT iajs-2513 30 109 𝑍 𝑍 PROPN iajs-2513 30 110 2𝑍. 2𝑍. NUM iajs-2513 30 111 2 2 NUM iajs-2513 30 112 ) ) PUNCT iajs-2513 30 113 it it PRON iajs-2513 30 114 is be AUX iajs-2513 30 115 clear clear ADJ iajs-2513 30 116 that that SCONJ iajs-2513 30 117 every every DET iajs-2513 30 118 prime prime ADJ iajs-2513 30 119 submodule submodule NOUN iajs-2513 30 120 is be AUX iajs-2513 30 121 an an DET iajs-2513 30 122 app app PROPN iajs-2513 30 123 - - PUNCT iajs-2513 30 124 qp qp NOUN iajs-2513 30 125 submodule submodule NOUN iajs-2513 30 126 , , PUNCT iajs-2513 30 127 but but CCONJ iajs-2513 30 128 not not PART iajs-2513 30 129 conversely conversely ADV iajs-2513 30 130 . . PUNCT iajs-2513 31 1 the the DET iajs-2513 31 2 following follow VERB iajs-2513 31 3 example example NOUN iajs-2513 31 4 shows show VERB iajs-2513 31 5 that that SCONJ iajs-2513 31 6 : : PUNCT iajs-2513 31 7 consider consider VERB iajs-2513 31 8 the the DET iajs-2513 31 9 𝑍-module 𝑍-module PROPN iajs-2513 31 10 𝑍 𝑍 PROPN iajs-2513 31 11 , , PUNCT iajs-2513 31 12 the the DET iajs-2513 31 13 submodule submodule NOUN iajs-2513 31 14 𝐸 𝐸 PROPN iajs-2513 31 15 〈 〈 NOUN iajs-2513 31 16 0 0 NUM iajs-2513 31 17 〉 〉 NOUN iajs-2513 31 18 is be AUX iajs-2513 31 19 not not PART iajs-2513 31 20 prime prime ADJ iajs-2513 31 21 submodule submodule NOUN iajs-2513 31 22 of of ADP iajs-2513 31 23 the the DET iajs-2513 31 24 𝑍-module 𝑍-module PROPN iajs-2513 31 25 𝑍 𝑍 NOUN iajs-2513 31 26 , , PUNCT iajs-2513 31 27 since since SCONJ iajs-2513 31 28 2 2 NUM iajs-2513 31 29 . . SYM iajs-2513 31 30 2 2 NUM iajs-2513 31 31 ∈ ∈ PROPN iajs-2513 31 32 𝐸 𝐸 PROPN iajs-2513 31 33 , , PUNCT iajs-2513 31 34 for for ADP iajs-2513 31 35 2 2 NUM iajs-2513 31 36 ∈ ∈ PROPN iajs-2513 31 37 𝑍 𝑍 NOUN iajs-2513 31 38 , , PUNCT iajs-2513 31 39 2 2 NUM iajs-2513 31 40 ∈ ∈ NOUN iajs-2513 31 41 𝑍 𝑍 NOUN iajs-2513 31 42 , , PUNCT iajs-2513 31 43 but but CCONJ iajs-2513 31 44 2 2 NUM iajs-2513 31 45 ∉ ∉ PROPN iajs-2513 31 46 𝐸 𝐸 PROPN iajs-2513 31 47 and and CCONJ iajs-2513 31 48 2 2 NUM iajs-2513 31 49 ∉ ∉ PROPN iajs-2513 31 50 〈 〈 PROPN iajs-2513 31 51 0 0 NUM iajs-2513 31 52 〉 〉 NOUN iajs-2513 31 53 : : PUNCT iajs-2513 31 54 𝑍 𝑍 VERB iajs-2513 31 55 4𝑍. 4𝑍. NUM iajs-2513 31 56 while while SCONJ iajs-2513 31 57 𝐸 𝐸 PROPN iajs-2513 31 58 is be AUX iajs-2513 31 59 an an DET iajs-2513 31 60 app app PROPN iajs-2513 31 61 - - PUNCT iajs-2513 31 62 qp qp NOUN iajs-2513 31 63 submodule submodule NOUN iajs-2513 31 64 of of ADP iajs-2513 31 65 the the DET iajs-2513 31 66 𝑍-module 𝑍-module PROPN iajs-2513 31 67 𝑍 𝑍 NOUN iajs-2513 31 68 , , PUNCT iajs-2513 31 69 since since SCONJ iajs-2513 31 70 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 31 71 𝑍 𝑍 PROPN iajs-2513 31 72 〈 〈 NOUN iajs-2513 31 73 2 2 NUM iajs-2513 31 74 〉 〉 NOUN iajs-2513 31 75 and and CCONJ iajs-2513 31 76 for for ADP iajs-2513 31 77 all all DET iajs-2513 31 78 𝑟 𝑟 PRON iajs-2513 31 79 ∈ ∈ PROPN iajs-2513 31 80 𝑍 𝑍 NOUN iajs-2513 31 81 , , PUNCT iajs-2513 31 82 𝑞 𝑞 PROPN iajs-2513 31 83 ∈ ∈ PROPN iajs-2513 31 84 𝑍 𝑍 VERB iajs-2513 31 85 such such ADJ iajs-2513 31 86 that that PRON iajs-2513 31 87 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 31 88 ∈ ∈ PROPN iajs-2513 31 89 𝐸 𝐸 PROPN iajs-2513 31 90 , , PUNCT iajs-2513 31 91 implies imply VERB iajs-2513 31 92 that that SCONJ iajs-2513 31 93 either either CCONJ iajs-2513 31 94 𝑞 𝑞 PROPN iajs-2513 31 95 ∈ ∈ PROPN iajs-2513 31 96 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 31 97 〈 〈 PROPN iajs-2513 31 98 0 0 NUM iajs-2513 31 99 〉 〉 NOUN iajs-2513 31 100 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 31 101 𝑍 𝑍 PROPN iajs-2513 31 102 〈 〈 NOUN iajs-2513 31 103 2 2 NUM iajs-2513 31 104 〉 〉 NOUN iajs-2513 31 105 〈 〈 NOUN iajs-2513 31 106 2 2 NUM iajs-2513 31 107 〉 〉 NOUN iajs-2513 31 108 〈 〈 NOUN iajs-2513 31 109 2 2 NUM iajs-2513 31 110 〉 〉 NOUN iajs-2513 31 111 or or CCONJ iajs-2513 31 112 𝑟 𝑟 NOUN iajs-2513 31 113 ∈ ∈ NOUN iajs-2513 31 114 〈 〈 PROPN iajs-2513 31 115 0 0 NUM iajs-2513 31 116 〉 〉 NOUN iajs-2513 31 117 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 31 118 𝑍 𝑍 PROPN iajs-2513 31 119 : : PUNCT iajs-2513 31 120 𝑍 𝑍 PROPN iajs-2513 31 121 √2𝑍 √2𝑍 NOUN iajs-2513 31 122 2𝑍. 2𝑍. NUM iajs-2513 31 123 that that PRON iajs-2513 31 124 is be AUX iajs-2513 31 125 if if SCONJ iajs-2513 31 126 2 2 NUM iajs-2513 31 127 . . SYM iajs-2513 31 128 2 2 NUM iajs-2513 31 129 ∈ ∈ PROPN iajs-2513 31 130 𝐸 𝐸 PROPN iajs-2513 31 131 , , PUNCT iajs-2513 31 132 for for ADP iajs-2513 31 133 2 2 NUM iajs-2513 31 134 ∈ ∈ PROPN iajs-2513 31 135 𝑍 𝑍 NOUN iajs-2513 31 136 , , PUNCT iajs-2513 31 137 2 2 NUM iajs-2513 31 138 ∈ ∈ NOUN iajs-2513 31 139 𝑍 𝑍 NOUN iajs-2513 31 140 implies imply VERB iajs-2513 31 141 that that SCONJ iajs-2513 31 142 2 2 NUM iajs-2513 31 143 ∈ ∈ NOUN iajs-2513 31 144 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 31 145 〈 〈 PROPN iajs-2513 31 146 0 0 NUM iajs-2513 31 147 〉 〉 NOUN iajs-2513 31 148 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 31 149 𝑍 𝑍 PROPN iajs-2513 31 150 〈 〈 NOUN iajs-2513 31 151 2 2 NUM iajs-2513 31 152 〉 〉 NOUN iajs-2513 31 153 and and CCONJ iajs-2513 31 154 2 2 NUM iajs-2513 31 155 ∈ ∈ NOUN iajs-2513 31 156 〈 〈 NOUN iajs-2513 31 157 0 0 NUM iajs-2513 31 158 〉 〉 NOUN iajs-2513 31 159 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 31 160 𝑍 𝑍 PROPN iajs-2513 31 161 : : PUNCT iajs-2513 31 162 𝑍 𝑍 PROPN iajs-2513 31 163 2𝑍. 2𝑍. NUM iajs-2513 31 164 3 3 NUM iajs-2513 31 165 ) ) PUNCT iajs-2513 31 166 it it PRON iajs-2513 31 167 is be AUX iajs-2513 31 168 clear clear ADJ iajs-2513 31 169 that that SCONJ iajs-2513 31 170 every every DET iajs-2513 31 171 quasi quasi ADJ iajs-2513 31 172 - - ADJ iajs-2513 31 173 prime prime ADJ iajs-2513 31 174 submodule submodule NOUN iajs-2513 31 175 is be AUX iajs-2513 31 176 an an DET iajs-2513 31 177 app app PROPN iajs-2513 31 178 - - PUNCT iajs-2513 31 179 qp qp NOUN iajs-2513 31 180 submodule submodule NOUN iajs-2513 31 181 , , PUNCT iajs-2513 31 182 but but CCONJ iajs-2513 31 183 not not PART iajs-2513 31 184 conversely conversely ADV iajs-2513 31 185 , , PUNCT iajs-2513 31 186 where where SCONJ iajs-2513 31 187 a a DET iajs-2513 31 188 proper proper ADJ iajs-2513 31 189 submodule submodule NOUN iajs-2513 31 190 𝐸 𝐸 PROPN iajs-2513 31 191 of of ADP iajs-2513 31 192 𝑄 𝑄 PROPN iajs-2513 31 193 is be AUX iajs-2513 31 194 called call VERB iajs-2513 31 195 quasi quasi ADJ iajs-2513 31 196 - - NOUN iajs-2513 31 197 prime prime ADJ iajs-2513 31 198 if if SCONJ iajs-2513 31 199 𝑟𝑠𝑞 𝑟𝑠𝑞 NUM iajs-2513 31 200 ∈ ∈ PROPN iajs-2513 31 201 𝐸. 𝐸. PROPN iajs-2513 31 202 for for ADP iajs-2513 31 203 𝑟 𝑟 NOUN iajs-2513 31 204 , , PUNCT iajs-2513 31 205 𝑠 𝑠 PROPN iajs-2513 31 206 ∈ ∈ PROPN iajs-2513 31 207 𝑅 𝑅 PROPN iajs-2513 31 208 , , PUNCT iajs-2513 31 209 𝑞 𝑞 X iajs-2513 31 210 ∈ ∈ PROPN iajs-2513 31 211 𝑄 𝑄 PROPN iajs-2513 31 212 , , PUNCT iajs-2513 31 213 implies imply VERB iajs-2513 31 214 that that SCONJ iajs-2513 31 215 either either CCONJ iajs-2513 31 216 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 31 217 ∈ ∈ PROPN iajs-2513 31 218 𝐸 𝐸 PROPN iajs-2513 31 219 or or CCONJ iajs-2513 31 220 𝑠𝑞 𝑠𝑞 ADP iajs-2513 31 221 ∈ ∈ PROPN iajs-2513 31 222 𝐸 𝐸 PROPN iajs-2513 32 1 [ [ X iajs-2513 32 2 10 10 NUM iajs-2513 32 3 ] ] PUNCT iajs-2513 32 4 . . PUNCT iajs-2513 33 1 the the DET iajs-2513 33 2 following following ADJ iajs-2513 33 3 example example NOUN iajs-2513 33 4 explains explain VERB iajs-2513 33 5 that that SCONJ iajs-2513 33 6 : : PUNCT iajs-2513 33 7     SPACE iajs-2513 33 8 94 94 NUM iajs-2513 33 9     SPACE iajs-2513 33 10 ibn ibn PROPN iajs-2513 33 11 al al PROPN iajs-2513 33 12 - - PUNCT iajs-2513 33 13 haitham haitham PROPN iajs-2513 33 14 jour jour X iajs-2513 33 15 . . PROPN iajs-2513 34 1 for for ADP iajs-2513 34 2 pure pure ADJ iajs-2513 34 3 & & CCONJ iajs-2513 34 4 appl appl PROPN iajs-2513 34 5 . . PUNCT iajs-2513 35 1 sci sci PROPN iajs-2513 35 2 . . PROPN iajs-2513 36 1 33 33 NUM iajs-2513 36 2 ( ( PUNCT iajs-2513 36 3 4 4 NUM iajs-2513 36 4 ) ) PUNCT iajs-2513 36 5 2020 2020 NUM iajs-2513 36 6 consider consider VERB iajs-2513 36 7 the the DET iajs-2513 36 8 𝑍-module 𝑍-module PROPN iajs-2513 36 9 𝑍 𝑍 NOUN iajs-2513 36 10 , , PUNCT iajs-2513 36 11 and and CCONJ iajs-2513 36 12 the the DET iajs-2513 36 13 submodule submodule NOUN iajs-2513 36 14 4𝑍 4𝑍 PROPN iajs-2513 36 15 is be AUX iajs-2513 36 16 not not PART iajs-2513 36 17 quasi quasi ADJ iajs-2513 36 18 - - ADJ iajs-2513 36 19 prime prime ADJ iajs-2513 36 20 submodule submodule NOUN iajs-2513 36 21 of of ADP iajs-2513 36 22 𝑍 𝑍 PROPN iajs-2513 36 23 , , PUNCT iajs-2513 36 24 since since SCONJ iajs-2513 36 25 2.2.1 2.2.1 NUM iajs-2513 36 26 4 4 NUM iajs-2513 36 27 ∈ ∈ NOUN iajs-2513 36 28 4𝑍 4𝑍 NOUN iajs-2513 36 29 , , PUNCT iajs-2513 36 30 , , PUNCT iajs-2513 36 31 but but CCONJ iajs-2513 36 32 2.1 2.1 NUM iajs-2513 36 33 ∉ ∉ ADJ iajs-2513 36 34 4𝑍. 4𝑍. NUM iajs-2513 36 35 while while SCONJ iajs-2513 36 36 4𝑍 4𝑍 PROPN iajs-2513 36 37 is be AUX iajs-2513 36 38 an an DET iajs-2513 36 39 app app PROPN iajs-2513 36 40 - - PUNCT iajs-2513 36 41 qp qp NOUN iajs-2513 36 42 submodule submodule NOUN iajs-2513 36 43 of of ADP iajs-2513 36 44 the the DET iajs-2513 36 45 𝑍-module 𝑍-module PROPN iajs-2513 36 46 𝑍 𝑍 NOUN iajs-2513 36 47 , , PUNCT iajs-2513 36 48 since since SCONJ iajs-2513 36 49 for for ADP iajs-2513 36 50 all all DET iajs-2513 36 51 𝑟 𝑟 PRON iajs-2513 36 52 ∈ ∈ PROPN iajs-2513 36 53 𝑍 𝑍 NOUN iajs-2513 36 54 , , PUNCT iajs-2513 36 55 𝑞 𝑞 PROPN iajs-2513 36 56 ∈ ∈ PROPN iajs-2513 36 57 𝑍 𝑍 VERB iajs-2513 36 58 such such ADJ iajs-2513 36 59 that that DET iajs-2513 36 60 𝑟𝑞 𝑟𝑞 PROPN iajs-2513 36 61 ∈ ∈ PROPN iajs-2513 36 62 4𝑍 4𝑍 PROPN iajs-2513 36 63 , , PUNCT iajs-2513 36 64 implies imply VERB iajs-2513 36 65 that that SCONJ iajs-2513 36 66 either either CCONJ iajs-2513 36 67 𝑞 𝑞 PROPN iajs-2513 36 68 ∈ ∈ PROPN iajs-2513 36 69 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 36 70 4𝑍 4𝑍 PROPN iajs-2513 36 71 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 36 72 𝑍 𝑍 PROPN iajs-2513 36 73 〈 〈 NOUN iajs-2513 36 74 2 2 NUM iajs-2513 36 75 〉 〉 NOUN iajs-2513 36 76 0 0 NUM iajs-2513 36 77 〈 〈 NOUN iajs-2513 36 78 2 2 NUM iajs-2513 36 79 〉 〉 NOUN iajs-2513 36 80 or or CCONJ iajs-2513 36 81 𝑟 𝑟 NOUN iajs-2513 36 82 ∈ ∈ NOUN iajs-2513 36 83 4𝑍 4𝑍 NOUN iajs-2513 36 84 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 36 85 𝑍 𝑍 PROPN iajs-2513 36 86 : : PUNCT iajs-2513 36 87 𝑍 𝑍 VERB iajs-2513 36 88 √4𝑍 √4𝑍 ADV iajs-2513 36 89 2𝑍. 2𝑍. NUM iajs-2513 36 90 that that PRON iajs-2513 36 91 is be AUX iajs-2513 36 92 , , PUNCT iajs-2513 36 93 if if SCONJ iajs-2513 36 94 2.2 2.2 NUM iajs-2513 36 95 ∈ ∈ PROPN iajs-2513 36 96 4𝑍 4𝑍 NOUN iajs-2513 36 97 , , PUNCT iajs-2513 36 98 implies imply VERB iajs-2513 36 99 that that SCONJ iajs-2513 36 100 2 2 NUM iajs-2513 36 101 ∈ ∈ NOUN iajs-2513 36 102 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 36 103 4𝑍 4𝑍 PROPN iajs-2513 36 104 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 36 105 𝑍 𝑍 PROPN iajs-2513 36 106 〈 〈 NOUN iajs-2513 36 107 2 2 NUM iajs-2513 36 108 〉 〉 NOUN iajs-2513 36 109 and and CCONJ iajs-2513 36 110 2 2 NUM iajs-2513 36 111 ∈ ∈ NOUN iajs-2513 36 112 4𝑍 4𝑍 NOUN iajs-2513 36 113 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 36 114 𝑍 𝑍 PROPN iajs-2513 36 115 : : PUNCT iajs-2513 36 116 𝑍 𝑍 PROPN iajs-2513 36 117 2𝑍. 2𝑍. NUM iajs-2513 36 118 the the DET iajs-2513 36 119 following following ADJ iajs-2513 36 120 results result NOUN iajs-2513 36 121 are be AUX iajs-2513 36 122 characterizations characterization NOUN iajs-2513 36 123 of of ADP iajs-2513 36 124 app app PROPN iajs-2513 36 125 - - PUNCT iajs-2513 36 126 qp qp NOUN iajs-2513 36 127 submodules submodule NOUN iajs-2513 36 128 . . PUNCT iajs-2513 37 1 proposition proposition NOUN iajs-2513 37 2 ( ( PUNCT iajs-2513 37 3 3 3 X iajs-2513 37 4 ) ) PUNCT iajs-2513 37 5 let let VERB iajs-2513 37 6 𝑄 𝑄 PRON iajs-2513 37 7 be be AUX iajs-2513 37 8 an an DET iajs-2513 37 9 𝑅-module 𝑅-module NOUN iajs-2513 37 10 , , PUNCT iajs-2513 37 11 and and CCONJ iajs-2513 37 12 𝐸 𝐸 PROPN iajs-2513 37 13 be be VERB iajs-2513 37 14 a a DET iajs-2513 37 15 proper proper ADJ iajs-2513 37 16 submodule submodule NOUN iajs-2513 37 17 of of ADP iajs-2513 37 18 𝑄. 𝑄. PROPN iajs-2513 37 19 then then ADV iajs-2513 37 20 𝐸 𝐸 PROPN iajs-2513 37 21 is be AUX iajs-2513 37 22 an an DET iajs-2513 37 23 app app PROPN iajs-2513 37 24 - - PUNCT iajs-2513 37 25 qp qp NOUN iajs-2513 37 26 submodule submodule NOUN iajs-2513 37 27 of of ADP iajs-2513 37 28 𝑄 𝑄 PRON iajs-2513 38 1 if if SCONJ iajs-2513 39 1 and and CCONJ iajs-2513 39 2 only only ADV iajs-2513 39 3 if if SCONJ iajs-2513 39 4 𝐼𝐹 𝐼𝐹 PROPN iajs-2513 39 5 ⊆ ⊆ NUM iajs-2513 39 6 𝐸 𝐸 PROPN iajs-2513 39 7 , , PUNCT iajs-2513 39 8 for for ADP iajs-2513 39 9 𝐼 𝐼 PROPN iajs-2513 39 10 is be AUX iajs-2513 39 11 an an DET iajs-2513 39 12 ideal ideal NOUN iajs-2513 39 13 of of ADP iajs-2513 39 14 𝑅 𝑅 PROPN iajs-2513 39 15 and and CCONJ iajs-2513 39 16 𝐹 𝐹 PROPN iajs-2513 39 17 is be AUX iajs-2513 39 18 a a DET iajs-2513 39 19 submodule submodule NOUN iajs-2513 39 20 of of ADP iajs-2513 39 21 𝑄 𝑄 PROPN iajs-2513 39 22 , , PUNCT iajs-2513 39 23 implies imply VERB iajs-2513 39 24 that that SCONJ iajs-2513 39 25 either either CCONJ iajs-2513 39 26 𝐹 𝐹 PROPN iajs-2513 39 27 ⊆ ⊆ PROPN iajs-2513 39 28 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 39 29 𝐸 𝐸 PROPN iajs-2513 39 30 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 39 31 𝑄 𝑄 PROPN iajs-2513 39 32 or or CCONJ iajs-2513 39 33 𝐼 𝐼 ADP iajs-2513 39 34 𝑄 𝑄 PROPN iajs-2513 39 35 ⊆ ⊆ NUM iajs-2513 39 36 𝐸 𝐸 PROPN iajs-2513 39 37 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 39 38 𝑄 𝑄 PROPN iajs-2513 39 39 for for ADP iajs-2513 39 40 some some DET iajs-2513 39 41 𝑛 𝑛 DET iajs-2513 39 42 ∈ ∈ PROPN iajs-2513 39 43 𝑍 𝑍 NOUN iajs-2513 39 44 . . PUNCT iajs-2513 40 1 proof proof NOUN iajs-2513 40 2 suppose suppose VERB iajs-2513 40 3 𝐼𝐹 𝐼𝐹 PROPN iajs-2513 40 4 ⊆ ⊆ NUM iajs-2513 40 5 𝐸 𝐸 PROPN iajs-2513 40 6 , , PUNCT iajs-2513 40 7 for for ADP iajs-2513 40 8 𝐼 𝐼 PROPN iajs-2513 40 9 is be AUX iajs-2513 40 10 an an DET iajs-2513 40 11 ideal ideal NOUN iajs-2513 40 12 of of ADP iajs-2513 40 13 𝑅 𝑅 PROPN iajs-2513 40 14 and and CCONJ iajs-2513 40 15 𝐹 𝐹 PROPN iajs-2513 40 16 is be AUX iajs-2513 40 17 a a DET iajs-2513 40 18 submodule submodule NOUN iajs-2513 40 19 of of ADP iajs-2513 40 20 𝑄 𝑄 PRON iajs-2513 40 21 with with ADP iajs-2513 40 22 𝐹 𝐹 PRON iajs-2513 40 23 ⊈ ⊈ X iajs-2513 40 24 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 40 25 𝐸 𝐸 PROPN iajs-2513 40 26 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 40 27 𝑄 𝑄 PROPN iajs-2513 40 28 , , PUNCT iajs-2513 40 29 then then ADV iajs-2513 40 30 there there PRON iajs-2513 40 31 exists exist VERB iajs-2513 40 32 𝑘 𝑘 PRON iajs-2513 40 33 ∈ ∈ NOUN iajs-2513 40 34 𝐹 𝐹 PROPN iajs-2513 40 35 such such ADJ iajs-2513 40 36 that that SCONJ iajs-2513 40 37 𝑘 𝑘 PROPN iajs-2513 40 38 ∉ ∉ PROPN iajs-2513 40 39 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 40 40 𝐸 𝐸 PROPN iajs-2513 40 41 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 40 42 𝑄 𝑄 PROPN iajs-2513 40 43 . . PUNCT iajs-2513 41 1 now now ADV iajs-2513 41 2 we we PRON iajs-2513 41 3 have have VERB iajs-2513 41 4 𝐼𝐹 𝐼𝐹 PROPN iajs-2513 41 5 ⊆ ⊆ NUM iajs-2513 41 6 𝐸 𝐸 PROPN iajs-2513 41 7 , , PUNCT iajs-2513 41 8 then then ADV iajs-2513 41 9 for for ADP iajs-2513 41 10 any any DET iajs-2513 41 11 𝑎 𝑎 PROPN iajs-2513 41 12 ∈ ∈ PROPN iajs-2513 41 13 𝐼 𝐼 NOUN iajs-2513 41 14 , , PUNCT iajs-2513 41 15 𝑎𝑘 𝑎𝑘 PROPN iajs-2513 41 16 ∈ ∈ PROPN iajs-2513 41 17 𝐸. 𝐸. PROPN iajs-2513 41 18 since since SCONJ iajs-2513 41 19 𝐸 𝐸 PROPN iajs-2513 41 20 is be AUX iajs-2513 41 21 an an DET iajs-2513 41 22 app app PROPN iajs-2513 41 23 - - PUNCT iajs-2513 41 24 qp qp NOUN iajs-2513 41 25 submodule submodule NOUN iajs-2513 41 26 of of ADP iajs-2513 41 27 𝑄 𝑄 PROPN iajs-2513 41 28 and and CCONJ iajs-2513 41 29 𝑘 𝑘 DET iajs-2513 41 30 ∉ ∉ PROPN iajs-2513 41 31 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 41 32 𝐸 𝐸 PROPN iajs-2513 41 33 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 41 34 𝑄 𝑄 PROPN iajs-2513 41 35 , , PUNCT iajs-2513 41 36 it it PRON iajs-2513 41 37 follows follow VERB iajs-2513 41 38 that that SCONJ iajs-2513 41 39 𝑎 𝑎 PROPN iajs-2513 41 40 𝑄 𝑄 PROPN iajs-2513 41 41 ⊆ ⊆ PROPN iajs-2513 41 42 𝐸 𝐸 PROPN iajs-2513 41 43 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 41 44 𝑄 𝑄 PROPN iajs-2513 41 45 for for ADP iajs-2513 41 46 some some DET iajs-2513 41 47 𝑛 𝑛 DET iajs-2513 41 48 ∈ ∈ PROPN iajs-2513 41 49 𝑍 𝑍 NOUN iajs-2513 41 50 , , PUNCT iajs-2513 41 51 that that PRON iajs-2513 41 52 is be AUX iajs-2513 41 53 𝐼 𝐼 ADP iajs-2513 41 54 𝑄 𝑄 PROPN iajs-2513 41 55 ⊆ ⊆ NUM iajs-2513 41 56 𝐸 𝐸 PROPN iajs-2513 41 57 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 41 58 𝑄 𝑄 PROPN iajs-2513 41 59 for for ADP iajs-2513 41 60 some some PRON iajs-2513 41 61 𝑛 𝑛 DET iajs-2513 41 62 ∈ ∈ PROPN iajs-2513 41 63 𝑍 𝑍 NOUN iajs-2513 41 64 . . PUNCT iajs-2513 42 1 assume assume VERB iajs-2513 42 2 that that SCONJ iajs-2513 42 3 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 42 4 ∈ ∈ PROPN iajs-2513 42 5 𝐸 𝐸 PROPN iajs-2513 42 6 , , PUNCT iajs-2513 42 7 for for ADP iajs-2513 42 8 𝑟 𝑟 DET iajs-2513 42 9 ∈ ∈ PROPN iajs-2513 42 10 𝑅 𝑅 PROPN iajs-2513 42 11 , , PUNCT iajs-2513 42 12 𝑞 𝑞 X iajs-2513 42 13 ∈ ∈ PROPN iajs-2513 42 14 𝑄 𝑄 PROPN iajs-2513 42 15 , , PUNCT iajs-2513 42 16 then then ADV iajs-2513 42 17 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 42 18 〈 〈 PROPN iajs-2513 42 19 𝑟〉〈𝑞 𝑟〉〈𝑞 X iajs-2513 42 20 〉 〉 NOUN iajs-2513 42 21 , , PUNCT iajs-2513 42 22 that that PRON iajs-2513 42 23 is be AUX iajs-2513 42 24 𝐼𝐹 𝐼𝐹 PROPN iajs-2513 42 25 ⊆ ⊆ NUM iajs-2513 42 26 𝐸 𝐸 NOUN iajs-2513 42 27 where where SCONJ iajs-2513 42 28 𝐼 𝐼 PROPN iajs-2513 42 29 〈 〈 PROPN iajs-2513 42 30 𝑟 𝑟 X iajs-2513 42 31 〉 〉 NOUN iajs-2513 42 32 , , PUNCT iajs-2513 42 33 𝐹 𝐹 PROPN iajs-2513 42 34 〈 〈 PROPN iajs-2513 42 35 𝑞 𝑞 X iajs-2513 42 36 〉 〉 NOUN iajs-2513 42 37 , , PUNCT iajs-2513 42 38 then then ADV iajs-2513 42 39 by by ADP iajs-2513 42 40 hypothesis hypothesis NOUN iajs-2513 42 41 , , PUNCT iajs-2513 42 42 either either CCONJ iajs-2513 42 43 𝐹 𝐹 PROPN iajs-2513 42 44 ⊆ ⊆ PROPN iajs-2513 42 45 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 42 46 𝐸 𝐸 PROPN iajs-2513 42 47 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 42 48 𝑄 𝑄 PROPN iajs-2513 42 49 or or CCONJ iajs-2513 42 50 𝐼 𝐼 ADP iajs-2513 42 51 𝑄 𝑄 PROPN iajs-2513 42 52 ⊆ ⊆ NUM iajs-2513 42 53 𝐸 𝐸 PROPN iajs-2513 42 54 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 42 55 𝑄 𝑄 PROPN iajs-2513 42 56 for for ADP iajs-2513 42 57 some some DET iajs-2513 42 58 𝑛 𝑛 DET iajs-2513 42 59 ∈ ∈ PROPN iajs-2513 42 60 𝑍 𝑍 NOUN iajs-2513 42 61 . . PUNCT iajs-2513 43 1 hence hence ADV iajs-2513 43 2 either𝑞 either𝑞 NOUN iajs-2513 43 3 ∈ ∈ PROPN iajs-2513 43 4 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 43 5 𝐸 𝐸 PROPN iajs-2513 43 6 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 43 7 𝑄 𝑄 PROPN iajs-2513 43 8 or or CCONJ iajs-2513 43 9 𝑟 𝑟 PRON iajs-2513 43 10 𝑄 𝑄 PROPN iajs-2513 43 11 ⊆ ⊆ NUM iajs-2513 43 12 𝐸 𝐸 PROPN iajs-2513 43 13 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 43 14 𝑄 𝑄 PROPN iajs-2513 43 15 for for ADP iajs-2513 43 16 some some DET iajs-2513 43 17 𝑛 𝑛 DET iajs-2513 43 18 ∈ ∈ PROPN iajs-2513 43 19 𝑍 𝑍 NOUN iajs-2513 43 20 . . PUNCT iajs-2513 44 1 thus thus ADV iajs-2513 44 2 𝐸 𝐸 PROPN iajs-2513 44 3 is be AUX iajs-2513 44 4 an an DET iajs-2513 44 5 app app PROPN iajs-2513 44 6 - - PUNCT iajs-2513 44 7 qp qp NOUN iajs-2513 44 8 submodule submodule NOUN iajs-2513 44 9 of of ADP iajs-2513 44 10 𝑄. 𝑄. NOUN iajs-2513 44 11 the the DET iajs-2513 44 12 following follow VERB iajs-2513 44 13 corollary corollary NOUN iajs-2513 44 14 is be AUX iajs-2513 44 15 a a DET iajs-2513 44 16 direct direct ADJ iajs-2513 44 17 consequence consequence NOUN iajs-2513 44 18 proposition proposition NOUN iajs-2513 44 19 ( ( PUNCT iajs-2513 44 20 3 3 NUM iajs-2513 44 21 ) ) PUNCT iajs-2513 44 22 . . PUNCT iajs-2513 45 1 corollary corollary ADJ iajs-2513 45 2 ( ( PUNCT iajs-2513 45 3 4 4 X iajs-2513 45 4 ) ) PUNCT iajs-2513 45 5 let let VERB iajs-2513 45 6 𝑄 𝑄 PRON iajs-2513 45 7 be be AUX iajs-2513 45 8 an an DET iajs-2513 45 9 𝑅-module 𝑅-module NOUN iajs-2513 45 10 , , PUNCT iajs-2513 45 11 and and CCONJ iajs-2513 45 12 𝐸 𝐸 PROPN iajs-2513 45 13 be be VERB iajs-2513 45 14 a a DET iajs-2513 45 15 proper proper ADJ iajs-2513 45 16 submodule submodule NOUN iajs-2513 45 17 of of ADP iajs-2513 45 18 𝑄.then 𝑄.then PROPN iajs-2513 45 19 , , PUNCT iajs-2513 45 20 𝐸 𝐸 PROPN iajs-2513 45 21 is be AUX iajs-2513 45 22 an an DET iajs-2513 45 23 app app PROPN iajs-2513 45 24 - - PUNCT iajs-2513 45 25 qp qp NOUN iajs-2513 45 26 submodule submodule NOUN iajs-2513 45 27 of of ADP iajs-2513 45 28 𝑄 𝑄 PRON iajs-2513 45 29 if if SCONJ iajs-2513 45 30 and and CCONJ iajs-2513 45 31 only only ADV iajs-2513 45 32 if if SCONJ iajs-2513 45 33 for for ADP iajs-2513 45 34 every every DET iajs-2513 45 35 submodule submodule NOUN iajs-2513 45 36 𝐹 𝐹 PROPN iajs-2513 45 37 of of ADP iajs-2513 45 38 𝑄 𝑄 PROPN iajs-2513 45 39 and and CCONJ iajs-2513 45 40 every every DET iajs-2513 45 41 𝑟 𝑟 PRON iajs-2513 45 42 ∈ ∈ PROPN iajs-2513 45 43 𝑅 𝑅 PROPN iajs-2513 45 44 with with ADP iajs-2513 45 45 𝑟𝐹 𝑟𝐹 PROPN iajs-2513 45 46 ⊆ ⊆ NUM iajs-2513 45 47 𝐸 𝐸 PROPN iajs-2513 45 48 , , PUNCT iajs-2513 45 49 implies imply VERB iajs-2513 45 50 that that SCONJ iajs-2513 45 51 either either CCONJ iajs-2513 45 52 𝐹 𝐹 PROPN iajs-2513 45 53 ⊆ ⊆ PROPN iajs-2513 45 54 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 45 55 𝐸 𝐸 PROPN iajs-2513 45 56 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 45 57 𝑄 𝑄 PROPN iajs-2513 45 58 or or CCONJ iajs-2513 45 59 𝑟 𝑟 PRON iajs-2513 45 60 𝑄 𝑄 PROPN iajs-2513 45 61 ⊆ ⊆ NUM iajs-2513 45 62 𝐸 𝐸 PROPN iajs-2513 45 63 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 45 64 𝑄 𝑄 PROPN iajs-2513 45 65 for for ADP iajs-2513 45 66 some some DET iajs-2513 45 67 𝑛 𝑛 DET iajs-2513 45 68 ∈ ∈ PROPN iajs-2513 45 69 𝑍 𝑍 NOUN iajs-2513 45 70 . . PUNCT iajs-2513 46 1 proposition proposition NOUN iajs-2513 46 2 ( ( PUNCT iajs-2513 46 3 5 5 NUM iajs-2513 46 4 ) ) PUNCT iajs-2513 46 5 a a DET iajs-2513 46 6 zero zero NUM iajs-2513 46 7 submodule submodule NOUN iajs-2513 46 8 of of ADP iajs-2513 46 9 a a DET iajs-2513 46 10 non non ADJ iajs-2513 46 11 - - ADJ iajs-2513 46 12 zero zero ADJ iajs-2513 46 13 𝑅-module 𝑅-module PROPN iajs-2513 46 14 𝑄 𝑄 PROPN iajs-2513 46 15 is be AUX iajs-2513 46 16 an an DET iajs-2513 46 17 app app PROPN iajs-2513 46 18 - - PUNCT iajs-2513 46 19 qp qp NOUN iajs-2513 46 20 submodule submodule NOUN iajs-2513 46 21 of of ADP iajs-2513 46 22 𝑄 𝑄 PRON iajs-2513 46 23 if if SCONJ iajs-2513 47 1 and and CCONJ iajs-2513 47 2 only only ADV iajs-2513 47 3 if if SCONJ iajs-2513 47 4 𝑎𝑛𝑛 𝑎𝑛𝑛 ADP iajs-2513 47 5 𝐹 𝐹 PROPN iajs-2513 47 6 ⊆ ⊆ NUM iajs-2513 47 7 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 47 8 𝑄 𝑄 PROPN iajs-2513 47 9 : : PUNCT iajs-2513 47 10 𝑄 𝑄 PROPN iajs-2513 47 11 for for ADP iajs-2513 47 12 all all DET iajs-2513 47 13 non non ADJ iajs-2513 47 14 - - ADJ iajs-2513 47 15 zero zero NUM iajs-2513 47 16 submodule submodule NOUN iajs-2513 47 17 𝐹 𝐹 PROPN iajs-2513 47 18 of of ADP iajs-2513 47 19 𝑄 𝑄 PROPN iajs-2513 47 20 , , PUNCT iajs-2513 47 21 with with ADP iajs-2513 47 22 𝐹 𝐹 PROPN iajs-2513 47 23 ⊈ ⊈ X iajs-2513 47 24 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 47 25 0 0 NUM iajs-2513 47 26 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 47 27 𝑄 𝑄 PROPN iajs-2513 47 28 . . PUNCT iajs-2513 48 1 proof proof NOUN iajs-2513 48 2 let let VERB iajs-2513 48 3 𝐹 𝐹 PRON iajs-2513 48 4 be be AUX iajs-2513 48 5 a a DET iajs-2513 48 6 non non ADJ iajs-2513 48 7 - - ADJ iajs-2513 48 8 zero zero NUM iajs-2513 48 9 submodule submodule NOUN iajs-2513 48 10 of of ADP iajs-2513 48 11 𝑄 𝑄 PROPN iajs-2513 48 12 , , PUNCT iajs-2513 48 13 such such ADJ iajs-2513 48 14 that that SCONJ iajs-2513 48 15 𝐹 𝐹 PROPN iajs-2513 48 16 ⊈ ⊈ VERB iajs-2513 48 17 𝑟𝑎𝑑 𝑟𝑎𝑑 ADV iajs-2513 48 18 0 0 NUM iajs-2513 48 19 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 48 20 𝑄 𝑄 PROPN iajs-2513 48 21 , , PUNCT iajs-2513 48 22 and and CCONJ iajs-2513 48 23 let let VERB iajs-2513 48 24 𝑥 𝑥 X iajs-2513 48 25 ∈ ∈ VERB iajs-2513 48 26 𝑎𝑛𝑛 𝑎𝑛𝑛 X iajs-2513 48 27 𝐹 𝐹 PROPN iajs-2513 48 28 , , PUNCT iajs-2513 48 29 implies imply VERB iajs-2513 48 30 that that SCONJ iajs-2513 48 31 𝑥𝐹 𝑥𝐹 VERB iajs-2513 48 32 0 0 NUM iajs-2513 48 33 but but CCONJ iajs-2513 48 34 0 0 NUM iajs-2513 48 35 is be AUX iajs-2513 48 36 an an DET iajs-2513 48 37 app app PROPN iajs-2513 48 38 - - PUNCT iajs-2513 48 39 qp qp NOUN iajs-2513 48 40 submodule submodule NOUN iajs-2513 48 41 of of ADP iajs-2513 48 42 𝑄 𝑄 PROPN iajs-2513 48 43 and and CCONJ iajs-2513 48 44 𝐹 𝐹 PRON iajs-2513 48 45 ⊈ ⊈ VERB iajs-2513 49 1 𝑟𝑎𝑑 𝑟𝑎𝑑 ADV iajs-2513 49 2 0 0 NUM iajs-2513 49 3 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 49 4 𝑄 𝑄 PROPN iajs-2513 49 5 , , PUNCT iajs-2513 49 6 it it PRON iajs-2513 49 7 follows follow VERB iajs-2513 49 8 by by ADP iajs-2513 49 9 corollary corollary ADJ iajs-2513 49 10 ( ( PUNCT iajs-2513 49 11 4 4 NUM iajs-2513 49 12 ) ) PUNCT iajs-2513 49 13 that that PRON iajs-2513 49 14 𝑥 𝑥 VERB iajs-2513 49 15 𝑄 𝑄 NOUN iajs-2513 49 16 ⊆ ⊆ SYM iajs-2513 49 17 0 0 NUM iajs-2513 49 18 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 49 19 𝑄 𝑄 PROPN iajs-2513 49 20 for for ADP iajs-2513 49 21 some some DET iajs-2513 49 22 𝑛 𝑛 DET iajs-2513 49 23 ∈ ∈ PROPN iajs-2513 49 24 𝑍 𝑍 NOUN iajs-2513 49 25 , , PUNCT iajs-2513 49 26 that that PRON iajs-2513 49 27 is be AUX iajs-2513 49 28 𝑥 𝑥 PRON iajs-2513 49 29 ∈ ∈ NOUN iajs-2513 49 30 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 49 31 𝑄 𝑄 PROPN iajs-2513 49 32 : : PUNCT iajs-2513 49 33 𝑄 𝑄 PROPN iajs-2513 49 34 . . PUNCT iajs-2513 50 1 hence hence ADV iajs-2513 50 2 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2513 50 3 𝐹 𝐹 PROPN iajs-2513 50 4 ⊆ ⊆ NUM iajs-2513 50 5 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 50 6 𝑄 𝑄 PROPN iajs-2513 50 7 : : PUNCT iajs-2513 50 8 𝑄 𝑄 PROPN iajs-2513 50 9 . . PUNCT iajs-2513 51 1 suppose suppose VERB iajs-2513 51 2 that that SCONJ iajs-2513 51 3 𝑥𝐹 𝑥𝐹 ADP iajs-2513 51 4 ⊆ ⊆ NUM iajs-2513 51 5 0 0 NUM iajs-2513 51 6 , , PUNCT iajs-2513 51 7 for for ADP iajs-2513 51 8 𝑟 𝑟 DET iajs-2513 51 9 ∈ ∈ PROPN iajs-2513 51 10 𝑅 𝑅 PROPN iajs-2513 51 11 and and CCONJ iajs-2513 51 12 𝐹 𝐹 PROPN iajs-2513 51 13 is be AUX iajs-2513 51 14 a a DET iajs-2513 51 15 non non ADJ iajs-2513 51 16 - - ADJ iajs-2513 51 17 zero zero NUM iajs-2513 51 18 submodule submodule NOUN iajs-2513 51 19 of of ADP iajs-2513 51 20 𝑄 𝑄 PROPN iajs-2513 51 21 , , PUNCT iajs-2513 51 22 with with SCONJ iajs-2513 51 23 𝐹 𝐹 PROPN iajs-2513 51 24 ⊈ ⊈ X iajs-2513 51 25 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 51 26 0 0 NUM iajs-2513 51 27 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 51 28 𝑄 𝑄 PROPN iajs-2513 51 29 . . PUNCT iajs-2513 52 1 since since SCONJ iajs-2513 52 2 𝑥𝐹 𝑥𝐹 PROPN iajs-2513 52 3 ⊆ ⊆ NUM iajs-2513 52 4 0 0 NUM iajs-2513 52 5 it it PRON iajs-2513 52 6 follows follow VERB iajs-2513 52 7 that that SCONJ iajs-2513 52 8 𝑥 𝑥 PROPN iajs-2513 52 9 ∈ ∈ PROPN iajs-2513 52 10 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2513 52 11 𝐹 𝐹 PROPN iajs-2513 52 12 , , PUNCT iajs-2513 52 13 by by ADP iajs-2513 52 14 hypothesis hypothesis NOUN iajs-2513 52 15 𝑥 𝑥 X iajs-2513 52 16 ∈ ∈ NOUN iajs-2513 52 17 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 52 18 𝑄 𝑄 PROPN iajs-2513 52 19 : : PUNCT iajs-2513 52 20 𝑄 𝑄 PROPN iajs-2513 52 21 , , PUNCT iajs-2513 52 22 that that PRON iajs-2513 52 23 is be AUX iajs-2513 52 24 𝑥 𝑥 DET iajs-2513 52 25 ∈ ∈ NOUN iajs-2513 52 26 0 0 NUM iajs-2513 52 27 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 52 28 𝑄 𝑄 PROPN iajs-2513 52 29 : : PUNCT iajs-2513 52 30 𝑄 𝑄 PROPN iajs-2513 52 31 . . PUNCT iajs-2513 53 1 hence hence ADV iajs-2513 53 2 𝑥 𝑥 VERB iajs-2513 53 3 𝑄 𝑄 NOUN iajs-2513 53 4 ⊆ ⊆ SYM iajs-2513 53 5 0 0 NUM iajs-2513 53 6 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 53 7 𝑄 𝑄 PROPN iajs-2513 53 8 for for ADP iajs-2513 53 9 some some DET iajs-2513 53 10 𝑛 𝑛 DET iajs-2513 53 11 ∈ ∈ PROPN iajs-2513 53 12 𝑍 𝑍 NOUN iajs-2513 53 13 . . PUNCT iajs-2513 54 1 thus thus ADV iajs-2513 54 2 by by ADP iajs-2513 54 3 corollary corollary ADJ iajs-2513 54 4 ( ( PUNCT iajs-2513 54 5 4 4 NUM iajs-2513 54 6 ) ) PUNCT iajs-2513 54 7 a a DET iajs-2513 54 8 zero zero NUM iajs-2513 54 9 submodule submodule NOUN iajs-2513 54 10 of of ADP iajs-2513 54 11 an an DET iajs-2513 54 12 𝑅-module 𝑅-module PROPN iajs-2513 54 13 𝑄 𝑄 PROPN iajs-2513 54 14 is be AUX iajs-2513 54 15 an an DET iajs-2513 54 16 app app ADJ iajs-2513 54 17 - - PUNCT iajs-2513 54 18 primary primary ADJ iajs-2513 54 19 submodule submodule NOUN iajs-2513 54 20 of of ADP iajs-2513 54 21 𝑄. 𝑄. PROPN iajs-2513 54 22     SPACE iajs-2513 54 23 95 95 NUM iajs-2513 54 24     SPACE iajs-2513 54 25 ibn ibn PROPN iajs-2513 54 26 al al PROPN iajs-2513 54 27 - - PUNCT iajs-2513 54 28 haitham haitham PROPN iajs-2513 54 29 jour jour X iajs-2513 54 30 . . PROPN iajs-2513 54 31 for for ADP iajs-2513 54 32 pure pure ADJ iajs-2513 54 33 & & CCONJ iajs-2513 54 34 appl appl PROPN iajs-2513 54 35 . . PUNCT iajs-2513 55 1 sci sci PROPN iajs-2513 55 2 . . PROPN iajs-2513 56 1 33 33 NUM iajs-2513 56 2 ( ( PUNCT iajs-2513 56 3 4 4 NUM iajs-2513 56 4 ) ) PUNCT iajs-2513 56 5 2020 2020 NUM iajs-2513 56 6 proposition proposition NOUN iajs-2513 56 7 ( ( PUNCT iajs-2513 56 8 6 6 NUM iajs-2513 56 9 ) ) PUNCT iajs-2513 56 10 let let VERB iajs-2513 56 11 𝑄 𝑄 PRON iajs-2513 56 12 be be AUX iajs-2513 56 13 an an DET iajs-2513 56 14 𝑅-module 𝑅-module NOUN iajs-2513 56 15 , , PUNCT iajs-2513 56 16 and and CCONJ iajs-2513 56 17 𝐸 𝐸 PROPN iajs-2513 56 18 be be VERB iajs-2513 56 19 a a DET iajs-2513 56 20 proper proper ADJ iajs-2513 56 21 submodule submodule NOUN iajs-2513 56 22 of of ADP iajs-2513 56 23 𝑄. 𝑄. PROPN iajs-2513 56 24 then then ADV iajs-2513 56 25 , , PUNCT iajs-2513 56 26 𝐸 𝐸 PROPN iajs-2513 56 27 is be AUX iajs-2513 56 28 an an DET iajs-2513 56 29 app app PROPN iajs-2513 56 30 - - PUNCT iajs-2513 56 31 qp qp NOUN iajs-2513 56 32 submodule submodule NOUN iajs-2513 56 33 of of ADP iajs-2513 56 34 𝑄 𝑄 PRON iajs-2513 56 35 if if SCONJ iajs-2513 56 36 and and CCONJ iajs-2513 56 37 only only ADV iajs-2513 56 38 if if SCONJ iajs-2513 56 39 for for ADP iajs-2513 56 40 every every DET iajs-2513 56 41 𝑞 𝑞 PROPN iajs-2513 56 42 ∈ ∈ PROPN iajs-2513 56 43 𝑄 𝑄 PROPN iajs-2513 56 44 , , PUNCT iajs-2513 56 45 𝐸 𝐸 PROPN iajs-2513 56 46 : : PUNCT iajs-2513 56 47 𝑞 𝑞 PROPN iajs-2513 56 48 ⊆ ⊆ NUM iajs-2513 56 49 𝐸 𝐸 PROPN iajs-2513 56 50 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 56 51 𝑄 𝑄 PROPN iajs-2513 56 52 : : PUNCT iajs-2513 56 53 𝑄 𝑄 PROPN iajs-2513 56 54 with with ADP iajs-2513 56 55 𝑞 𝑞 PROPN iajs-2513 56 56 ∉ ∉ PROPN iajs-2513 56 57 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 56 58 𝐸 𝐸 PROPN iajs-2513 56 59 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 56 60 𝑄 𝑄 PROPN iajs-2513 56 61 . . PUNCT iajs-2513 57 1 proof proof NOUN iajs-2513 57 2 suppose suppose VERB iajs-2513 57 3 that that SCONJ iajs-2513 57 4 𝐸 𝐸 PROPN iajs-2513 57 5 is be AUX iajs-2513 57 6 an an DET iajs-2513 57 7 app app PROPN iajs-2513 57 8 - - PUNCT iajs-2513 57 9 qp qp NOUN iajs-2513 57 10 submodule submodule NOUN iajs-2513 57 11 of of ADP iajs-2513 57 12 𝑄 𝑄 PROPN iajs-2513 57 13 , , PUNCT iajs-2513 57 14 and and CCONJ iajs-2513 57 15 𝑟 𝑟 X iajs-2513 57 16 ∈ ∈ PROPN iajs-2513 57 17 𝐸 𝐸 PROPN iajs-2513 57 18 : : PUNCT iajs-2513 57 19 𝑞 𝑞 X iajs-2513 57 20 , , PUNCT iajs-2513 57 21 implies imply VERB iajs-2513 57 22 that that SCONJ iajs-2513 57 23 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 57 24 ∈ ∈ NOUN iajs-2513 57 25 𝐸. 𝐸. PROPN iajs-2513 57 26 since since SCONJ iajs-2513 57 27 𝐸 𝐸 PROPN iajs-2513 57 28 is be AUX iajs-2513 57 29 an an DET iajs-2513 57 30 app app PROPN iajs-2513 57 31 - - PUNCT iajs-2513 57 32 qp qp NOUN iajs-2513 57 33 submodule submodule NOUN iajs-2513 57 34 of of ADP iajs-2513 57 35 𝑄. 𝑄. PROPN iajs-2513 57 36 and and CCONJ iajs-2513 57 37 𝑞 𝑞 PROPN iajs-2513 57 38 ∉ ∉ PROPN iajs-2513 57 39 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 57 40 𝐸 𝐸 PROPN iajs-2513 57 41 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 57 42 𝑄 𝑄 PROPN iajs-2513 57 43 , , PUNCT iajs-2513 57 44 then then ADV iajs-2513 57 45 𝑟 𝑟 X iajs-2513 57 46 𝑄 𝑄 PROPN iajs-2513 57 47 ⊆ ⊆ NUM iajs-2513 57 48 𝐸 𝐸 PROPN iajs-2513 57 49 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 57 50 𝑄 𝑄 PROPN iajs-2513 57 51 for for ADP iajs-2513 57 52 some some DET iajs-2513 57 53 𝑛 𝑛 DET iajs-2513 57 54 ∈ ∈ PROPN iajs-2513 57 55 𝑍 𝑍 NOUN iajs-2513 57 56 , , PUNCT iajs-2513 57 57 that that ADV iajs-2513 57 58 is is ADV iajs-2513 57 59 , , PUNCT iajs-2513 57 60 𝑟 𝑟 X iajs-2513 57 61 ∈ ∈ NOUN iajs-2513 57 62 𝐸 𝐸 NOUN iajs-2513 57 63 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 57 64 𝑄 𝑄 PROPN iajs-2513 57 65 : : PUNCT iajs-2513 57 66 𝑄 𝑄 PROPN iajs-2513 57 67 . . PUNCT iajs-2513 58 1 thus thus ADV iajs-2513 58 2 𝐸 𝐸 ADJ iajs-2513 58 3 : : PUNCT iajs-2513 58 4 𝑞 𝑞 PROPN iajs-2513 58 5 ⊆ ⊆ NUM iajs-2513 58 6 𝐸 𝐸 PROPN iajs-2513 58 7 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 58 8 𝑄 𝑄 PROPN iajs-2513 58 9 : : PUNCT iajs-2513 58 10 𝑄 𝑄 PROPN iajs-2513 58 11 . . PUNCT iajs-2513 59 1 let let VERB iajs-2513 59 2 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 59 3 ∈ ∈ PROPN iajs-2513 59 4 𝐸 𝐸 PROPN iajs-2513 59 5 , , PUNCT iajs-2513 59 6 for for ADP iajs-2513 59 7 𝑟 𝑟 DET iajs-2513 59 8 ∈ ∈ PROPN iajs-2513 59 9 𝑅 𝑅 PROPN iajs-2513 59 10 , , PUNCT iajs-2513 59 11 𝑞 𝑞 X iajs-2513 59 12 ∈ ∈ PROPN iajs-2513 59 13 𝑄 𝑄 PROPN iajs-2513 59 14 , , PUNCT iajs-2513 59 15 and and CCONJ iajs-2513 59 16 suppose suppose VERB iajs-2513 59 17 that that SCONJ iajs-2513 59 18 𝑞 𝑞 PROPN iajs-2513 59 19 ∉ ∉ PROPN iajs-2513 59 20 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 59 21 𝐸 𝐸 PROPN iajs-2513 59 22 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 59 23 𝑄 𝑄 PROPN iajs-2513 59 24 . . PUNCT iajs-2513 60 1 since since SCONJ iajs-2513 60 2 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 60 3 ∈ ∈ PROPN iajs-2513 60 4 𝐸 𝐸 PROPN iajs-2513 60 5 it it PRON iajs-2513 60 6 follows follow VERB iajs-2513 60 7 that that SCONJ iajs-2513 60 8 𝑟 𝑟 X iajs-2513 60 9 ∈ ∈ PROPN iajs-2513 60 10 𝐸 𝐸 PROPN iajs-2513 60 11 : : PUNCT iajs-2513 60 12 𝑞 𝑞 X iajs-2513 60 13 by by ADP iajs-2513 60 14 hypothesis hypothesis NOUN iajs-2513 60 15 𝑟 𝑟 X iajs-2513 60 16 ∈ ∈ NOUN iajs-2513 60 17 𝐸 𝐸 PROPN iajs-2513 60 18 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 60 19 𝑄 𝑄 PROPN iajs-2513 60 20 : : PUNCT iajs-2513 60 21 𝑄 𝑄 PROPN iajs-2513 60 22 . . PUNCT iajs-2513 61 1 hence hence ADV iajs-2513 61 2 , , PUNCT iajs-2513 61 3 𝑟 𝑟 NOUN iajs-2513 61 4 𝑄 𝑄 PROPN iajs-2513 61 5 ⊆ ⊆ NUM iajs-2513 61 6 𝐸 𝐸 PROPN iajs-2513 61 7 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 61 8 𝑄 𝑄 PROPN iajs-2513 61 9 for for ADP iajs-2513 61 10 some some DET iajs-2513 61 11 𝑛 𝑛 DET iajs-2513 61 12 ∈ ∈ PROPN iajs-2513 61 13 𝑍 𝑍 NOUN iajs-2513 61 14 . . PUNCT iajs-2513 62 1 thus thus ADV iajs-2513 62 2 𝐸 𝐸 PROPN iajs-2513 62 3 is be AUX iajs-2513 62 4 an an DET iajs-2513 62 5 app app PROPN iajs-2513 62 6 - - PUNCT iajs-2513 62 7 qp qp NOUN iajs-2513 62 8 submodule submodule NOUN iajs-2513 62 9 of of ADP iajs-2513 62 10 𝑄. 𝑄. NOUN iajs-2513 62 11 proposition proposition NOUN iajs-2513 62 12 ( ( PUNCT iajs-2513 62 13 7 7 X iajs-2513 62 14 ) ) PUNCT iajs-2513 62 15 let let VERB iajs-2513 62 16 𝑄 𝑄 PRON iajs-2513 62 17 be be AUX iajs-2513 62 18 an an DET iajs-2513 62 19 𝑅-module 𝑅-module NOUN iajs-2513 62 20 , , PUNCT iajs-2513 62 21 and and CCONJ iajs-2513 62 22 𝐸 𝐸 PROPN iajs-2513 62 23 be be VERB iajs-2513 62 24 a a DET iajs-2513 62 25 proper proper ADJ iajs-2513 62 26 submodule submodule NOUN iajs-2513 62 27 of of ADP iajs-2513 62 28 𝑄. 𝑄. PROPN iajs-2513 62 29 then then ADV iajs-2513 62 30 , , PUNCT iajs-2513 62 31 𝐸 𝐸 PROPN iajs-2513 62 32 is be AUX iajs-2513 62 33 an an DET iajs-2513 62 34 app app PROPN iajs-2513 62 35 - - PUNCT iajs-2513 62 36 qp qp NOUN iajs-2513 62 37 submodule submodule NOUN iajs-2513 62 38 of of ADP iajs-2513 62 39 𝑄 𝑄 PRON iajs-2513 62 40 if if SCONJ iajs-2513 63 1 and and CCONJ iajs-2513 63 2 only only ADV iajs-2513 63 3 if if SCONJ iajs-2513 63 4 𝐸 𝐸 PROPN iajs-2513 63 5 : : PUNCT iajs-2513 63 6 𝑟 𝑟 NUM iajs-2513 63 7 ⊆ ⊆ NUM iajs-2513 63 8 𝐸 𝐸 PROPN iajs-2513 63 9 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 63 10 𝑄 𝑄 PROPN iajs-2513 63 11 : : PUNCT iajs-2513 63 12 𝑟 𝑟 X iajs-2513 63 13 for for ADP iajs-2513 63 14 𝑟 𝑟 DET iajs-2513 63 15 ∈ ∈ PROPN iajs-2513 63 16 𝑅 𝑅 PROPN iajs-2513 63 17 , , PUNCT iajs-2513 63 18 𝑛 𝑛 DET iajs-2513 63 19 ∈ ∈ NOUN iajs-2513 63 20 𝑍 𝑍 NOUN iajs-2513 63 21 . . PUNCT iajs-2513 64 1 proof proof NOUN iajs-2513 64 2 suppose suppose VERB iajs-2513 64 3 that that SCONJ iajs-2513 64 4 𝐸 𝐸 PROPN iajs-2513 64 5 is be AUX iajs-2513 64 6 an an DET iajs-2513 64 7 app app PROPN iajs-2513 64 8 - - PUNCT iajs-2513 64 9 qp qp NOUN iajs-2513 64 10 submodule submodule NOUN iajs-2513 64 11 of of ADP iajs-2513 64 12 𝑄 𝑄 PROPN iajs-2513 64 13 , , PUNCT iajs-2513 64 14 and and CCONJ iajs-2513 64 15 let let VERB iajs-2513 64 16 𝑞 𝑞 PROPN iajs-2513 64 17 ∈ ∈ VERB iajs-2513 64 18 𝐸 𝐸 PROPN iajs-2513 64 19 : : PUNCT iajs-2513 64 20 𝑟 𝑟 NOUN iajs-2513 64 21 , , PUNCT iajs-2513 64 22 such such ADJ iajs-2513 64 23 that that SCONJ iajs-2513 64 24 𝑞 𝑞 PROPN iajs-2513 64 25 ∉ ∉ PROPN iajs-2513 64 26 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 64 27 𝐸 𝐸 PROPN iajs-2513 64 28 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 64 29 𝑄 𝑄 PROPN iajs-2513 64 30 . . PUNCT iajs-2513 65 1 since since SCONJ iajs-2513 65 2 𝑞 𝑞 PROPN iajs-2513 65 3 ∈ ∈ PROPN iajs-2513 65 4 𝐸 𝐸 PROPN iajs-2513 65 5 : : PUNCT iajs-2513 65 6 𝑟 𝑟 NOUN iajs-2513 65 7 it it PRON iajs-2513 65 8 follows follow VERB iajs-2513 65 9 that that SCONJ iajs-2513 65 10 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 65 11 ∈ ∈ NOUN iajs-2513 65 12 𝐸. 𝐸. PROPN iajs-2513 65 13 but but CCONJ iajs-2513 65 14 𝐸 𝐸 PROPN iajs-2513 65 15 is be AUX iajs-2513 65 16 an an DET iajs-2513 65 17 app app PROPN iajs-2513 65 18 - - PUNCT iajs-2513 65 19 qp qp NOUN iajs-2513 65 20 submodule submodule NOUN iajs-2513 65 21 of of ADP iajs-2513 65 22 𝑄. 𝑄. PROPN iajs-2513 65 23 and and CCONJ iajs-2513 65 24 𝑞 𝑞 PROPN iajs-2513 65 25 ∉ ∉ PROPN iajs-2513 65 26 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 65 27 𝐸 𝐸 PROPN iajs-2513 65 28 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 65 29 𝑄 𝑄 PROPN iajs-2513 65 30 , , PUNCT iajs-2513 65 31 then then ADV iajs-2513 65 32 𝑟 𝑟 X iajs-2513 65 33 𝑄 𝑄 PROPN iajs-2513 65 34 ⊆ ⊆ NUM iajs-2513 65 35 𝐸 𝐸 PROPN iajs-2513 65 36 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 65 37 𝑄 𝑄 PROPN iajs-2513 65 38 : : PUNCT iajs-2513 65 39 𝑄 𝑄 PROPN iajs-2513 65 40 for for ADP iajs-2513 65 41 some some DET iajs-2513 65 42 𝑛 𝑛 DET iajs-2513 65 43 ∈ ∈ NOUN iajs-2513 65 44 𝑍 𝑍 NOUN iajs-2513 65 45 . . PUNCT iajs-2513 66 1 that that PRON iajs-2513 66 2 is be AUX iajs-2513 66 3 𝑟 𝑟 X iajs-2513 66 4 𝑞 𝑞 X iajs-2513 66 5 ∈ ∈ PROPN iajs-2513 66 6 𝐸 𝐸 PROPN iajs-2513 66 7 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 66 8 𝑄 𝑄 PROPN iajs-2513 66 9 for for ADP iajs-2513 66 10 all all PRON iajs-2513 66 11 𝑞 𝑞 PROPN iajs-2513 66 12 ∈ ∈ PROPN iajs-2513 66 13 𝑄 𝑄 PROPN iajs-2513 66 14 , , PUNCT iajs-2513 66 15 it it PRON iajs-2513 66 16 follows follow VERB iajs-2513 66 17 that that SCONJ iajs-2513 66 18 𝑞 𝑞 PROPN iajs-2513 66 19 ∈ ∈ PROPN iajs-2513 66 20 𝐸 𝐸 PROPN iajs-2513 66 21 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 66 22 𝑄 𝑄 PROPN iajs-2513 66 23 : : PUNCT iajs-2513 66 24 𝑟 𝑟 X iajs-2513 66 25 . . PUNCT iajs-2513 67 1 thus thus ADV iajs-2513 67 2 𝐸 𝐸 NUM iajs-2513 67 3 : : PUNCT iajs-2513 67 4 𝑟 𝑟 NUM iajs-2513 67 5 ⊆ ⊆ NUM iajs-2513 67 6 𝐸 𝐸 PROPN iajs-2513 67 7 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 67 8 𝑄 𝑄 PROPN iajs-2513 67 9 : : PUNCT iajs-2513 67 10 𝑟 𝑟 X iajs-2513 67 11 . . PUNCT iajs-2513 68 1 let let VERB iajs-2513 68 2 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 68 3 ∈ ∈ PROPN iajs-2513 68 4 𝐸 𝐸 PROPN iajs-2513 68 5 , , PUNCT iajs-2513 68 6 for for ADP iajs-2513 68 7 𝑟 𝑟 DET iajs-2513 68 8 ∈ ∈ PROPN iajs-2513 68 9 𝑅 𝑅 PROPN iajs-2513 68 10 , , PUNCT iajs-2513 68 11 𝑞 𝑞 X iajs-2513 68 12 ∈ ∈ PROPN iajs-2513 68 13 𝑄 𝑄 PROPN iajs-2513 68 14 , , PUNCT iajs-2513 68 15 and and CCONJ iajs-2513 68 16 suppose suppose VERB iajs-2513 68 17 that that SCONJ iajs-2513 68 18 𝑞 𝑞 PROPN iajs-2513 68 19 ∉ ∉ PROPN iajs-2513 68 20 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 68 21 𝐸 𝐸 PROPN iajs-2513 68 22 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 68 23 𝑄 𝑄 PROPN iajs-2513 68 24 . . PUNCT iajs-2513 69 1 since since SCONJ iajs-2513 69 2 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 69 3 ∈ ∈ PROPN iajs-2513 69 4 𝐸 𝐸 PROPN iajs-2513 69 5 it it PRON iajs-2513 69 6 follows follow VERB iajs-2513 69 7 that that SCONJ iajs-2513 69 8 𝑞 𝑞 PROPN iajs-2513 69 9 ∈ ∈ PROPN iajs-2513 69 10 𝐸 𝐸 PROPN iajs-2513 69 11 : : PUNCT iajs-2513 69 12 𝑟 𝑟 NUM iajs-2513 69 13 ⊆ ⊆ NUM iajs-2513 69 14 𝐸 𝐸 PROPN iajs-2513 69 15 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 69 16 𝑄 𝑄 PRON iajs-2513 69 17 : : PUNCT iajs-2513 69 18 𝑟 𝑟 NOUN iajs-2513 69 19 , , PUNCT iajs-2513 69 20 implies imply VERB iajs-2513 69 21 that that SCONJ iajs-2513 69 22 𝑞 𝑞 PROPN iajs-2513 69 23 ∈ ∈ PROPN iajs-2513 69 24 𝐸 𝐸 PROPN iajs-2513 69 25 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 69 26 𝑄 𝑄 PROPN iajs-2513 69 27 : : PUNCT iajs-2513 69 28 𝑟 𝑟 X iajs-2513 69 29 , , PUNCT iajs-2513 69 30 that that PRON iajs-2513 69 31 is be AUX iajs-2513 69 32 𝑟 𝑟 X iajs-2513 69 33 𝑞 𝑞 X iajs-2513 69 34 ∈ ∈ PROPN iajs-2513 69 35 𝐸 𝐸 PROPN iajs-2513 69 36 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 69 37 𝑄 𝑄 PROPN iajs-2513 69 38 for for ADP iajs-2513 69 39 all all PRON iajs-2513 69 40 𝑞 𝑞 PROPN iajs-2513 69 41 ∈ ∈ PROPN iajs-2513 69 42 𝑄 𝑄 PROPN iajs-2513 69 43 , , PUNCT iajs-2513 69 44 hence hence ADV iajs-2513 69 45 𝑟 𝑟 NUM iajs-2513 69 46 𝑄 𝑄 PROPN iajs-2513 69 47 ⊆ ⊆ NUM iajs-2513 69 48 𝐸 𝐸 PROPN iajs-2513 69 49 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 69 50 𝑄 𝑄 PROPN iajs-2513 69 51 . . PUNCT iajs-2513 70 1 thus thus ADV iajs-2513 70 2 𝐸 𝐸 PROPN iajs-2513 70 3 is be AUX iajs-2513 70 4 an an DET iajs-2513 70 5 app app PROPN iajs-2513 70 6 - - PUNCT iajs-2513 70 7 qp qp NOUN iajs-2513 70 8 submodule submodule NOUN iajs-2513 70 9 of of ADP iajs-2513 70 10 𝑄. 𝑄. PROPN iajs-2513 70 11 before before SCONJ iajs-2513 70 12 we we PRON iajs-2513 70 13 give give VERB iajs-2513 70 14 the the DET iajs-2513 70 15 next next ADJ iajs-2513 70 16 result result NOUN iajs-2513 70 17 we we PRON iajs-2513 70 18 need need VERB iajs-2513 70 19 to to PART iajs-2513 70 20 recall recall VERB iajs-2513 70 21 the the DET iajs-2513 70 22 following follow VERB iajs-2513 70 23 lemma lemma PROPN iajs-2513 70 24 . . PUNCT iajs-2513 71 1 lemma lemma PROPN iajs-2513 71 2 ( ( PUNCT iajs-2513 71 3 8) 8) NUM iajs-2513 71 4 [ [ X iajs-2513 71 5 11 11 NUM iajs-2513 71 6 , , PUNCT iajs-2513 71 7 coro coro X iajs-2513 71 8 . . PUNCT iajs-2513 72 1 ( ( PUNCT iajs-2513 72 2 9.9 9.9 NUM iajs-2513 72 3 ) ) PUNCT iajs-2513 72 4 ] ] PUNCT iajs-2513 72 5 let let VERB iajs-2513 72 6 𝐸 𝐸 PRON iajs-2513 72 7 be be AUX iajs-2513 72 8 a a DET iajs-2513 72 9 submodule submodule NOUN iajs-2513 72 10 of of ADP iajs-2513 72 11 an an DET iajs-2513 72 12 𝑅-module 𝑅-module PROPN iajs-2513 72 13 𝑄 𝑄 PROPN iajs-2513 72 14 , , PUNCT iajs-2513 72 15 then then ADV iajs-2513 72 16 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 72 17 𝐸 𝐸 PROPN iajs-2513 72 18 𝐸 𝐸 PROPN iajs-2513 72 19 ∩ ∩ NOUN iajs-2513 72 20 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 72 21 𝑄 𝑄 PROPN iajs-2513 72 22 . . PUNCT iajs-2513 73 1 proposition proposition NOUN iajs-2513 73 2 ( ( PUNCT iajs-2513 73 3 9 9 X iajs-2513 73 4 ) ) PUNCT iajs-2513 73 5 let let VERB iajs-2513 73 6 𝐸 𝐸 PRON iajs-2513 73 7 and and CCONJ iajs-2513 73 8 𝐹 𝐹 PROPN iajs-2513 73 9 are be AUX iajs-2513 73 10 proper proper ADJ iajs-2513 73 11 submodules submodule NOUN iajs-2513 73 12 of of ADP iajs-2513 73 13 an an DET iajs-2513 73 14 𝑅-module 𝑅-module PROPN iajs-2513 73 15 𝑄 𝑄 PROPN iajs-2513 73 16 with with ADP iajs-2513 73 17 𝐸 𝐸 PROPN iajs-2513 73 18 ⊂ ⊂ PUNCT iajs-2513 73 19 𝐹 𝐹 PROPN iajs-2513 73 20 and and CCONJ iajs-2513 73 21 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 73 22 𝑄 𝑄 PROPN iajs-2513 73 23 ⊆ ⊆ NUM iajs-2513 73 24 𝐹. 𝐹. NOUN iajs-2513 73 25 if if SCONJ iajs-2513 73 26 𝐸 𝐸 PROPN iajs-2513 73 27 is be AUX iajs-2513 73 28 an an DET iajs-2513 73 29 app app PROPN iajs-2513 73 30 - - PUNCT iajs-2513 73 31 qp qp NOUN iajs-2513 73 32 submodule submodule NOUN iajs-2513 73 33 of of ADP iajs-2513 73 34 𝑄 𝑄 PROPN iajs-2513 73 35 , , PUNCT iajs-2513 73 36 then then ADV iajs-2513 73 37 𝐸 𝐸 PROPN iajs-2513 73 38 is be AUX iajs-2513 73 39 an an DET iajs-2513 73 40 app app PROPN iajs-2513 73 41 - - PUNCT iajs-2513 73 42 qp qp NOUN iajs-2513 73 43 submodule submodule NOUN iajs-2513 73 44 of of ADP iajs-2513 73 45 𝐹. 𝐹. PROPN iajs-2513 73 46 proof proof NOUN iajs-2513 73 47 let let VERB iajs-2513 73 48 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 73 49 ∈ ∈ PROPN iajs-2513 73 50 𝐸 𝐸 PROPN iajs-2513 73 51 , , PUNCT iajs-2513 73 52 with with ADP iajs-2513 73 53 𝑟 𝑟 DET iajs-2513 73 54 ∈ ∈ PROPN iajs-2513 73 55 𝑅 𝑅 PROPN iajs-2513 73 56 , , PUNCT iajs-2513 73 57 𝑞 𝑞 PROPN iajs-2513 73 58 ∈ ∈ PROPN iajs-2513 73 59 𝐹 𝐹 PROPN iajs-2513 73 60 ⊆ ⊆ NUM iajs-2513 73 61 𝑄. 𝑄. NOUN iajs-2513 73 62 since since SCONJ iajs-2513 73 63 𝐸 𝐸 PROPN iajs-2513 73 64 is be AUX iajs-2513 73 65 an an DET iajs-2513 73 66 app app PROPN iajs-2513 73 67 - - PUNCT iajs-2513 73 68 qp qp NOUN iajs-2513 73 69 submodule submodule NOUN iajs-2513 73 70 of of ADP iajs-2513 73 71 𝑄 𝑄 PROPN iajs-2513 73 72 , , PUNCT iajs-2513 73 73 then then ADV iajs-2513 73 74 either either CCONJ iajs-2513 73 75 𝑞 𝑞 PROPN iajs-2513 73 76 ∈ ∈ PROPN iajs-2513 73 77 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 73 78 𝐸 𝐸 PROPN iajs-2513 73 79 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 73 80 𝑄 𝑄 PROPN iajs-2513 73 81 or or CCONJ iajs-2513 73 82 𝑟 𝑟 PRON iajs-2513 73 83 𝑄 𝑄 PROPN iajs-2513 73 84 ⊆ ⊆ NUM iajs-2513 73 85 𝐸 𝐸 PROPN iajs-2513 73 86 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 73 87 𝑄 𝑄 PROPN iajs-2513 73 88 , , PUNCT iajs-2513 73 89 for for ADP iajs-2513 73 90 some some PRON iajs-2513 73 91 𝑛 𝑛 DET iajs-2513 73 92 ∈ ∈ NOUN iajs-2513 73 93 𝑍 𝑍 NOUN iajs-2513 73 94 . . PUNCT iajs-2513 74 1 that that PRON iajs-2513 74 2 is be AUX iajs-2513 74 3 either either CCONJ iajs-2513 74 4 𝑞 𝑞 PROPN iajs-2513 74 5 ∈ ∈ PROPN iajs-2513 74 6 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 74 7 𝐸 𝐸 PROPN iajs-2513 74 8 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 74 9 𝑄 𝑄 PROPN iajs-2513 74 10 ∩ ∩ NOUN iajs-2513 74 11 𝐹 𝐹 PROPN iajs-2513 74 12 or or CCONJ iajs-2513 74 13 𝑟 𝑟 PRON iajs-2513 74 14 𝑄 𝑄 PROPN iajs-2513 74 15 ⊆ ⊆ NUM iajs-2513 74 16 𝐸 𝐸 PROPN iajs-2513 74 17 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 74 18 𝑄 𝑄 PROPN iajs-2513 74 19 ∩ ∩ ADJ iajs-2513 74 20 𝐹. 𝐹. NOUN iajs-2513 74 21 but but CCONJ iajs-2513 74 22 since since SCONJ iajs-2513 74 23 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 74 24 𝑄 𝑄 PROPN iajs-2513 74 25 ⊆ ⊆ PROPN iajs-2513 74 26 𝐹 𝐹 PROPN iajs-2513 74 27 , , PUNCT iajs-2513 74 28 then then ADV iajs-2513 74 29 by by ADP iajs-2513 74 30 modular modular ADJ iajs-2513 74 31 law law NOUN iajs-2513 74 32 we we PRON iajs-2513 74 33 have have VERB iajs-2513 74 34 either either CCONJ iajs-2513 74 35 𝑞 𝑞 PROPN iajs-2513 74 36 ∈ ∈ PROPN iajs-2513 74 37 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 74 38 𝐸 𝐸 PROPN iajs-2513 74 39 ∩ ∩ NOUN iajs-2513 74 40 𝐹 𝐹 PROPN iajs-2513 74 41 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 74 42 𝑄 𝑄 PROPN iajs-2513 74 43 ∩ ∩ NOUN iajs-2513 74 44 𝐹 𝐹 PROPN iajs-2513 74 45 or or CCONJ iajs-2513 74 46 𝑟 𝑟 PRON iajs-2513 74 47 𝑄 𝑄 PROPN iajs-2513 74 48 ⊆ ⊆ NUM iajs-2513 74 49 𝐸 𝐸 PROPN iajs-2513 74 50 ∩ ∩ NOUN iajs-2513 74 51 𝐹 𝐹 PROPN iajs-2513 74 52 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 74 53 𝑄 𝑄 PROPN iajs-2513 74 54 ∩ ∩ NOUN iajs-2513 74 55 𝐹 𝐹 PROPN iajs-2513 74 56 . . PUNCT iajs-2513 75 1 now now ADV iajs-2513 75 2 by by ADP iajs-2513 75 3 lemma lemma PROPN iajs-2513 75 4 ( ( PUNCT iajs-2513 75 5 8) 8) NUM iajs-2513 75 6 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 75 7 𝑄 𝑄 PROPN iajs-2513 75 8 ∩ ∩ NOUN iajs-2513 75 9 𝐹 𝐹 PROPN iajs-2513 75 10 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 75 11 𝐹 𝐹 PROPN iajs-2513 75 12 , , PUNCT iajs-2513 75 13 so so ADV iajs-2513 75 14 either either CCONJ iajs-2513 75 15 𝑞 𝑞 PROPN iajs-2513 75 16 ∈ ∈ PROPN iajs-2513 75 17 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 75 18 𝐸 𝐸 PROPN iajs-2513 75 19 ∩ ∩ NOUN iajs-2513 75 20 𝐹 𝐹 PROPN iajs-2513 75 21 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 75 22 𝐹 𝐹 PROPN iajs-2513 75 23 ⊆ ⊆ NUM iajs-2513 75 24 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 75 25 𝐸 𝐸 PROPN iajs-2513 75 26 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 75 27 𝐹 𝐹 PROPN iajs-2513 75 28 or or CCONJ iajs-2513 75 29 𝑟 𝑟 PRON iajs-2513 75 30 𝑄 𝑄 PROPN iajs-2513 75 31 ⊆ ⊆ NUM iajs-2513 75 32 𝐸 𝐸 PROPN iajs-2513 75 33 ∩ ∩ NOUN iajs-2513 75 34 𝐹 𝐹 PROPN iajs-2513 75 35 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 75 36 𝐹 𝐹 PROPN iajs-2513 75 37 ⊆ ⊆ NUM iajs-2513 75 38 𝐸 𝐸 PROPN iajs-2513 75 39 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 75 40 𝐹 𝐹 PROPN iajs-2513 75 41 . . PUNCT iajs-2513 76 1 hence hence ADV iajs-2513 76 2 𝐸 𝐸 PROPN iajs-2513 76 3 is be AUX iajs-2513 76 4 an an DET iajs-2513 76 5 app app PROPN iajs-2513 76 6 - - PUNCT iajs-2513 76 7 qp qp NOUN iajs-2513 76 8 submodule submodule NOUN iajs-2513 76 9 of of ADP iajs-2513 76 10 𝐹. 𝐹. PROPN iajs-2513 76 11     SPACE iajs-2513 76 12 96 96 NUM iajs-2513 76 13     SPACE iajs-2513 76 14 ibn ibn PROPN iajs-2513 76 15 al al PROPN iajs-2513 76 16 - - PUNCT iajs-2513 76 17 haitham haitham PROPN iajs-2513 76 18 jour jour X iajs-2513 76 19 . . PROPN iajs-2513 77 1 for for ADP iajs-2513 77 2 pure pure ADJ iajs-2513 77 3 & & CCONJ iajs-2513 77 4 appl appl PROPN iajs-2513 77 5 . . PUNCT iajs-2513 78 1 sci sci PROPN iajs-2513 78 2 . . PROPN iajs-2513 79 1 33 33 NUM iajs-2513 79 2 ( ( PUNCT iajs-2513 79 3 4 4 NUM iajs-2513 79 4 ) ) PUNCT iajs-2513 79 5 2020 2020 NUM iajs-2513 79 6 remark remark NOUN iajs-2513 79 7 ( ( PUNCT iajs-2513 79 8 10 10 NUM iajs-2513 79 9 ) ) PUNCT iajs-2513 79 10 if if SCONJ iajs-2513 79 11 𝐸 𝐸 PROPN iajs-2513 79 12 is be AUX iajs-2513 79 13 an an DET iajs-2513 79 14 app app PROPN iajs-2513 79 15 - - PUNCT iajs-2513 79 16 qp qp NOUN iajs-2513 79 17 submodule submodule NOUN iajs-2513 79 18 of of ADP iajs-2513 79 19 an an DET iajs-2513 79 20 𝑅-module 𝑅-module PROPN iajs-2513 79 21 𝑄 𝑄 PROPN iajs-2513 79 22 , , PUNCT iajs-2513 79 23 then then ADV iajs-2513 79 24 𝐸 𝐸 PROPN iajs-2513 79 25 : : PUNCT iajs-2513 79 26 𝑄 𝑄 PRON iajs-2513 79 27 need need VERB iajs-2513 79 28 not not PART iajs-2513 79 29 to to PART iajs-2513 79 30 be be AUX iajs-2513 79 31 an an DET iajs-2513 79 32 app app ADJ iajs-2513 79 33 - - PUNCT iajs-2513 79 34 qp qp NOUN iajs-2513 79 35 ideal ideal NOUN iajs-2513 79 36 of of ADP iajs-2513 79 37 𝑅. 𝑅. NOUN iajs-2513 79 38 the the DET iajs-2513 79 39 following follow VERB iajs-2513 79 40 example example NOUN iajs-2513 79 41 explains explain VERB iajs-2513 79 42 that that SCONJ iajs-2513 79 43 : : PUNCT iajs-2513 79 44 consider consider VERB iajs-2513 79 45 the the DET iajs-2513 79 46 𝑍-module 𝑍-module PROPN iajs-2513 79 47 𝑍 𝑍 PROPN iajs-2513 79 48 , , PUNCT iajs-2513 79 49 the the DET iajs-2513 79 50 submodule submodule NOUN iajs-2513 79 51 𝐸 𝐸 PROPN iajs-2513 79 52 〈 〈 NOUN iajs-2513 79 53 0 0 NUM iajs-2513 79 54 〉 〉 NOUN iajs-2513 79 55 is be AUX iajs-2513 79 56 an an DET iajs-2513 79 57 app app PROPN iajs-2513 79 58 - - PUNCT iajs-2513 79 59 qp qp NOUN iajs-2513 79 60 submodule submodule NOUN iajs-2513 79 61 of of ADP iajs-2513 79 62 the the DET iajs-2513 79 63 𝑍module 𝑍module PROPN iajs-2513 79 64 𝑍 𝑍 PROPN iajs-2513 79 65 [ [ X iajs-2513 79 66 see see INTJ iajs-2513 79 67 remarks remark NOUN iajs-2513 79 68 and and CCONJ iajs-2513 79 69 examples example NOUN iajs-2513 79 70 ( ( PUNCT iajs-2513 79 71 2 2 NUM iajs-2513 79 72 ) ) PUNCT iajs-2513 79 73 ( ( PUNCT iajs-2513 79 74 1 1 NUM iajs-2513 79 75 ) ) PUNCT iajs-2513 79 76 ] ] PUNCT iajs-2513 79 77 . . PUNCT iajs-2513 80 1 but but CCONJ iajs-2513 80 2 𝐸 𝐸 PROPN iajs-2513 80 3 : : PUNCT iajs-2513 80 4 𝑍 𝑍 PROPN iajs-2513 80 5 12𝑍 12𝑍 NUM iajs-2513 80 6 is be AUX iajs-2513 80 7 not not PART iajs-2513 80 8 app app NUM iajs-2513 80 9 - - PUNCT iajs-2513 80 10 qp qp NOUN iajs-2513 80 11 ideal ideal NOUN iajs-2513 80 12 of of ADP iajs-2513 80 13 𝑍 𝑍 PROPN iajs-2513 80 14 because because SCONJ iajs-2513 80 15 4.3 4.3 NUM iajs-2513 80 16 ∈ ∈ PROPN iajs-2513 80 17 12𝑍 12𝑍 NUM iajs-2513 80 18 , , PUNCT iajs-2513 80 19 for for ADP iajs-2513 80 20 4,3 4,3 NUM iajs-2513 80 21 ∈ ∈ PROPN iajs-2513 80 22 𝑍 𝑍 NOUN iajs-2513 80 23 , , PUNCT iajs-2513 80 24 but but CCONJ iajs-2513 80 25 3 3 NUM iajs-2513 80 26 ∉ ∉ NOUN iajs-2513 80 27 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 80 28 12𝑍 12𝑍 NUM iajs-2513 80 29 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 80 30 𝑍 𝑍 PROPN iajs-2513 80 31 〈 〈 PROPN iajs-2513 80 32 6 6 NUM iajs-2513 80 33 〉 〉 NOUN iajs-2513 80 34 0 0 NUM iajs-2513 80 35 〈 〈 NOUN iajs-2513 80 36 6 6 NUM iajs-2513 80 37 〉 〉 NOUN iajs-2513 80 38 and and CCONJ iajs-2513 80 39 4 4 NUM iajs-2513 80 40 ∉ ∉ X iajs-2513 80 41 12𝑍 12𝑍 NUM iajs-2513 80 42 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 80 43 𝑍 𝑍 PROPN iajs-2513 80 44 : : PUNCT iajs-2513 80 45 𝑍 𝑍 PROPN iajs-2513 80 46 √12𝑍 √12𝑍 PROPN iajs-2513 80 47 6𝑍. 6𝑍. NOUN iajs-2513 80 48 now now ADV iajs-2513 80 49 before before SCONJ iajs-2513 80 50 we we PRON iajs-2513 80 51 offer offer VERB iajs-2513 80 52 under under ADP iajs-2513 80 53 certain certain ADJ iajs-2513 80 54 condition condition NOUN iajs-2513 80 55 the the DET iajs-2513 80 56 residual residual NOUN iajs-2513 80 57 of of ADP iajs-2513 80 58 app app PROPN iajs-2513 80 59 - - PUNCT iajs-2513 80 60 qp qp NOUN iajs-2513 80 61 submodule submodule NOUN iajs-2513 80 62 is be AUX iajs-2513 80 63 an an DET iajs-2513 80 64 app app PROPN iajs-2513 80 65 - - PUNCT iajs-2513 80 66 qp qp NOUN iajs-2513 80 67 ideal ideal NOUN iajs-2513 80 68 we we PRON iajs-2513 80 69 need need VERB iajs-2513 80 70 to to PART iajs-2513 80 71 revise revise VERB iajs-2513 80 72 the the DET iajs-2513 80 73 following following ADJ iajs-2513 80 74 lemma lemma PROPN iajs-2513 80 75 : : PUNCT iajs-2513 80 76 recall recall VERB iajs-2513 80 77 that that SCONJ iajs-2513 80 78 an an DET iajs-2513 80 79 𝑅-module 𝑅-module PROPN iajs-2513 80 80 𝑄 𝑄 PROPN iajs-2513 80 81 is be AUX iajs-2513 80 82 called call VERB iajs-2513 80 83 multiplication multiplication NOUN iajs-2513 80 84 if if SCONJ iajs-2513 80 85 every every DET iajs-2513 80 86 submodule submodule NOUN iajs-2513 80 87 𝐸 𝐸 PROPN iajs-2513 80 88 of of ADP iajs-2513 80 89 𝑄 𝑄 PROPN iajs-2513 80 90 is be AUX iajs-2513 80 91 of of ADP iajs-2513 80 92 the the DET iajs-2513 80 93 form form NOUN iajs-2513 80 94 𝐸 𝐸 PROPN iajs-2513 80 95 𝐼𝑄 𝐼𝑄 PROPN iajs-2513 80 96 for for ADP iajs-2513 80 97 some some DET iajs-2513 80 98 ideal ideal ADJ iajs-2513 80 99 𝐼 𝐼 PROPN iajs-2513 80 100 of of ADP iajs-2513 80 101 𝑄 𝑄 PRON iajs-2513 81 1 [ [ X iajs-2513 81 2 12 12 NUM iajs-2513 81 3 ] ] PUNCT iajs-2513 81 4 . . PUNCT iajs-2513 82 1 lemma lemma PROPN iajs-2513 82 2 ( ( PUNCT iajs-2513 82 3 11 11 NUM iajs-2513 82 4 ) ) PUNCT iajs-2513 83 1 [ [ X iajs-2513 83 2 12 12 NUM iajs-2513 83 3 , , PUNCT iajs-2513 83 4 coro coro NOUN iajs-2513 83 5 . . PUNCT iajs-2513 84 1 14(i 14(i NUM iajs-2513 84 2 ) ) PUNCT iajs-2513 84 3 ] ] PUNCT iajs-2513 85 1 let let VERB iajs-2513 85 2 𝑄 𝑄 PRON iajs-2513 85 3 be be AUX iajs-2513 85 4 a a DET iajs-2513 85 5 faithful faithful ADJ iajs-2513 85 6 multiplication multiplication NOUN iajs-2513 85 7 𝑅-module 𝑅-module NOUN iajs-2513 85 8 , , PUNCT iajs-2513 85 9 then then ADV iajs-2513 85 10 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 85 11 𝑄 𝑄 PROPN iajs-2513 85 12 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 85 13 𝑅 𝑅 PROPN iajs-2513 85 14 𝑄. 𝑄. PROPN iajs-2513 85 15 proposition proposition NOUN iajs-2513 85 16 ( ( PUNCT iajs-2513 85 17 12 12 NUM iajs-2513 85 18 ) ) PUNCT iajs-2513 85 19 let let VERB iajs-2513 85 20 𝑄 𝑄 PRON iajs-2513 85 21 be be AUX iajs-2513 85 22 a a DET iajs-2513 85 23 faithful faithful ADJ iajs-2513 85 24 multiplication multiplication NOUN iajs-2513 85 25 𝑅-module 𝑅-module PROPN iajs-2513 85 26 and and CCONJ iajs-2513 85 27 𝐸 𝐸 PROPN iajs-2513 85 28 be be VERB iajs-2513 85 29 a a DET iajs-2513 85 30 proper proper ADJ iajs-2513 85 31 submodule submodule NOUN iajs-2513 85 32 of of ADP iajs-2513 85 33 𝑄. 𝑄. PROPN iajs-2513 85 34 then then ADV iajs-2513 85 35 𝐸 𝐸 PROPN iajs-2513 85 36 is be AUX iajs-2513 85 37 an an DET iajs-2513 85 38 app app PROPN iajs-2513 85 39 - - PUNCT iajs-2513 85 40 qp qp NOUN iajs-2513 85 41 submodule submodule NOUN iajs-2513 85 42 of of ADP iajs-2513 85 43 𝑄 𝑄 PRON iajs-2513 85 44 if if SCONJ iajs-2513 85 45 and and CCONJ iajs-2513 85 46 only only ADV iajs-2513 85 47 if if SCONJ iajs-2513 85 48 𝐸 𝐸 PROPN iajs-2513 85 49 : : PUNCT iajs-2513 85 50 𝑄 𝑄 PROPN iajs-2513 85 51 is be AUX iajs-2513 85 52 an an DET iajs-2513 85 53 app app ADJ iajs-2513 85 54 - - PUNCT iajs-2513 85 55 qp qp NOUN iajs-2513 85 56 ideal ideal NOUN iajs-2513 85 57 of of ADP iajs-2513 85 58 𝑅. 𝑅. ADJ iajs-2513 85 59 proof proof NOUN iajs-2513 85 60 ⟹ ⟹ ADV iajs-2513 85 61 let let VERB iajs-2513 85 62 𝑟𝑠 𝑟𝑠 PROPN iajs-2513 85 63 ∈ ∈ PROPN iajs-2513 85 64 𝐸 𝐸 PROPN iajs-2513 85 65 : : PUNCT iajs-2513 85 66 𝑄 𝑄 PROPN iajs-2513 85 67 , , PUNCT iajs-2513 85 68 for for ADP iajs-2513 85 69 𝑟 𝑟 NOUN iajs-2513 85 70 , , PUNCT iajs-2513 85 71 𝑠 𝑠 PROPN iajs-2513 85 72 ∈ ∈ PROPN iajs-2513 85 73 𝑅 𝑅 PROPN iajs-2513 85 74 , , PUNCT iajs-2513 85 75 so so ADV iajs-2513 85 76 𝑟𝑠𝑄 𝑟𝑠𝑄 NOUN iajs-2513 85 77 ⊆ ⊆ NUM iajs-2513 85 78 𝐸. 𝐸. NOUN iajs-2513 86 1 but but CCONJ iajs-2513 86 2 𝐸 𝐸 PROPN iajs-2513 86 3 is be AUX iajs-2513 86 4 an an DET iajs-2513 86 5 app app PROPN iajs-2513 86 6 - - PUNCT iajs-2513 86 7 qp qp NOUN iajs-2513 86 8 submodule submodule NOUN iajs-2513 86 9 of of ADP iajs-2513 86 10 𝑄 𝑄 PRON iajs-2513 86 11 then then ADV iajs-2513 86 12 by by ADP iajs-2513 86 13 corollary corollary ADJ iajs-2513 86 14 ( ( PUNCT iajs-2513 86 15 4 4 NUM iajs-2513 86 16 ) ) PUNCT iajs-2513 86 17 either either DET iajs-2513 86 18 𝑠𝑄 𝑠𝑄 NOUN iajs-2513 86 19 ⊆ ⊆ NUM iajs-2513 86 20 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 86 21 𝐸 𝐸 PROPN iajs-2513 86 22 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 86 23 𝑄 𝑄 PROPN iajs-2513 86 24 or or CCONJ iajs-2513 86 25 𝑟 𝑟 PRON iajs-2513 86 26 𝑄 𝑄 PROPN iajs-2513 86 27 ⊆ ⊆ NUM iajs-2513 86 28 𝐸 𝐸 PROPN iajs-2513 86 29 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 86 30 𝑄 𝑄 PROPN iajs-2513 86 31 , , PUNCT iajs-2513 86 32 for for ADP iajs-2513 86 33 some some DET iajs-2513 86 34 𝑛 𝑛 DET iajs-2513 86 35 ∈ ∈ PROPN iajs-2513 86 36 𝑍 𝑍 NOUN iajs-2513 86 37 . . PUNCT iajs-2513 87 1 since since SCONJ iajs-2513 87 2 𝑄 𝑄 PRON iajs-2513 87 3 is be AUX iajs-2513 87 4 multiplication multiplication NOUN iajs-2513 87 5 then then ADV iajs-2513 87 6 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 87 7 𝐸 𝐸 PROPN iajs-2513 87 8 𝐸 𝐸 PROPN iajs-2513 87 9 : : PUNCT iajs-2513 87 10 𝑄 𝑄 PROPN iajs-2513 87 11 𝑄 𝑄 PROPN iajs-2513 87 12 , , PUNCT iajs-2513 87 13 and and CCONJ iajs-2513 87 14 since since SCONJ iajs-2513 87 15 𝑄 𝑄 PRON iajs-2513 87 16 is be AUX iajs-2513 87 17 faithful faithful ADJ iajs-2513 87 18 multiplication multiplication NOUN iajs-2513 87 19 then then ADV iajs-2513 87 20 by by ADP iajs-2513 87 21 lemma lemma PROPN iajs-2513 88 1 ( ( PUNCT iajs-2513 88 2 11 11 NUM iajs-2513 88 3 ) ) PUNCT iajs-2513 88 4 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 88 5 𝑅 𝑅 PROPN iajs-2513 88 6 𝑄 𝑄 PROPN iajs-2513 88 7 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 88 8 𝑄 𝑄 PROPN iajs-2513 88 9 , , PUNCT iajs-2513 88 10 we we PRON iajs-2513 88 11 get get VERB iajs-2513 88 12 either either PRON iajs-2513 88 13 𝑠𝑄 𝑠𝑄 PROPN iajs-2513 88 14 ⊆ ⊆ NUM iajs-2513 88 15 𝐸 𝐸 PROPN iajs-2513 88 16 : : PUNCT iajs-2513 88 17 𝑄 𝑄 PROPN iajs-2513 88 18 𝑄 𝑄 PROPN iajs-2513 88 19 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 88 20 𝑅 𝑅 PROPN iajs-2513 88 21 𝑄 𝑄 PROPN iajs-2513 88 22 or or CCONJ iajs-2513 88 23 𝑟 𝑟 NOUN iajs-2513 88 24 𝑄 𝑄 PROPN iajs-2513 88 25 ⊆ ⊆ NUM iajs-2513 88 26 𝐸 𝐸 PROPN iajs-2513 88 27 : : PUNCT iajs-2513 88 28 𝑄 𝑄 PROPN iajs-2513 88 29 𝑄 𝑄 PROPN iajs-2513 88 30 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 88 31 𝑅 𝑅 PROPN iajs-2513 88 32 𝑄 𝑄 PROPN iajs-2513 88 33 , , PUNCT iajs-2513 88 34 that that PRON iajs-2513 88 35 is be AUX iajs-2513 88 36 either either CCONJ iajs-2513 88 37 𝑠 𝑠 PROPN iajs-2513 88 38 ∈ ∈ PROPN iajs-2513 88 39 𝐸 𝐸 PROPN iajs-2513 88 40 : : PUNCT iajs-2513 88 41 𝑄 𝑄 PROPN iajs-2513 88 42 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 88 43 𝑅 𝑅 PROPN iajs-2513 88 44 or or CCONJ iajs-2513 88 45 𝑟 𝑟 NUM iajs-2513 88 46 ⊆ ⊆ NUM iajs-2513 88 47 𝐸 𝐸 PROPN iajs-2513 88 48 : : PUNCT iajs-2513 88 49 𝑄 𝑄 PROPN iajs-2513 88 50 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 88 51 𝑅 𝑅 PROPN iajs-2513 88 52 ⊆ ⊆ NUM iajs-2513 88 53 𝐸 𝐸 PROPN iajs-2513 88 54 : : PUNCT iajs-2513 88 55 𝑄 𝑄 PROPN iajs-2513 88 56 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 88 57 𝑅 𝑅 PROPN iajs-2513 88 58 : : PUNCT iajs-2513 88 59 𝑅 𝑅 PROPN iajs-2513 88 60 . . PUNCT iajs-2513 89 1 hence hence ADV iajs-2513 89 2 𝐸 𝐸 PROPN iajs-2513 89 3 : : PUNCT iajs-2513 89 4 𝑄 𝑄 PROPN iajs-2513 89 5 is be AUX iajs-2513 89 6 an an DET iajs-2513 89 7 app app ADJ iajs-2513 89 8 - - PUNCT iajs-2513 89 9 qp qp NOUN iajs-2513 89 10 ideal ideal NOUN iajs-2513 89 11 of of ADP iajs-2513 89 12 𝑅. 𝑅. NOUN iajs-2513 89 13 ⟸ ⟸ NOUN iajs-2513 89 14 suppose suppose VERB iajs-2513 89 15 that that SCONJ iajs-2513 89 16 𝐸 𝐸 PROPN iajs-2513 89 17 : : PUNCT iajs-2513 89 18 𝑄 𝑄 PROPN iajs-2513 89 19 is be AUX iajs-2513 89 20 an an DET iajs-2513 89 21 app app ADJ iajs-2513 89 22 - - PUNCT iajs-2513 89 23 qp qp NOUN iajs-2513 89 24 ideal ideal NOUN iajs-2513 89 25 of of ADP iajs-2513 89 26 𝑅 𝑅 PROPN iajs-2513 89 27 , , PUNCT iajs-2513 89 28 and and CCONJ iajs-2513 89 29 𝐼𝐹 𝐼𝐹 PROPN iajs-2513 89 30 ⊆ ⊆ NUM iajs-2513 89 31 𝐸 𝐸 PROPN iajs-2513 89 32 , , PUNCT iajs-2513 89 33 for for ADP iajs-2513 89 34 𝐼 𝐼 PROPN iajs-2513 89 35 is be AUX iajs-2513 89 36 an an DET iajs-2513 89 37 ideal ideal NOUN iajs-2513 89 38 of of ADP iajs-2513 89 39 𝑅 𝑅 PROPN iajs-2513 89 40 and and CCONJ iajs-2513 89 41 𝐹 𝐹 PROPN iajs-2513 89 42 is be AUX iajs-2513 89 43 a a DET iajs-2513 89 44 submodule submodule NOUN iajs-2513 89 45 of of ADP iajs-2513 89 46 𝑄. 𝑄. NOUN iajs-2513 89 47 since since SCONJ iajs-2513 89 48 𝑄 𝑄 PRON iajs-2513 89 49 is be AUX iajs-2513 89 50 multiplication multiplication NOUN iajs-2513 89 51 then then ADV iajs-2513 89 52 𝐹 𝐹 PROPN iajs-2513 89 53 𝐽𝑄 𝐽𝑄 VERB iajs-2513 89 54 for for ADP iajs-2513 89 55 some some DET iajs-2513 89 56 ideal ideal ADJ iajs-2513 89 57 𝐽 𝐽 PROPN iajs-2513 89 58 of of ADP iajs-2513 89 59 𝑅 𝑅 PROPN iajs-2513 89 60 , , PUNCT iajs-2513 89 61 that that PRON iajs-2513 89 62 is be AUX iajs-2513 89 63 𝐼𝐽𝑄 𝐼𝐽𝑄 PROPN iajs-2513 89 64 ⊆ ⊆ PROPN iajs-2513 89 65 𝐸 𝐸 PROPN iajs-2513 89 66 , , PUNCT iajs-2513 89 67 implies imply VERB iajs-2513 89 68 that that SCONJ iajs-2513 89 69 𝐼𝐽 𝐼𝐽 PROPN iajs-2513 89 70 ⊆ ⊆ NUM iajs-2513 89 71 𝐸 𝐸 PROPN iajs-2513 89 72 : : PUNCT iajs-2513 89 73 𝑄 𝑄 PROPN iajs-2513 89 74 . . PUNCT iajs-2513 90 1 but but CCONJ iajs-2513 90 2 𝐸 𝐸 PROPN iajs-2513 90 3 : : PUNCT iajs-2513 90 4 𝑄 𝑄 PROPN iajs-2513 90 5 is be AUX iajs-2513 90 6 an an DET iajs-2513 90 7 app app ADJ iajs-2513 90 8 - - PUNCT iajs-2513 90 9 qp qp NOUN iajs-2513 90 10 ideal ideal NOUN iajs-2513 90 11 of of ADP iajs-2513 90 12 𝑅 𝑅 PROPN iajs-2513 90 13 then then ADV iajs-2513 90 14 either either CCONJ iajs-2513 90 15 𝐽 𝐽 PROPN iajs-2513 90 16 ⊆ ⊆ NUM iajs-2513 90 17 𝐸 𝐸 PROPN iajs-2513 90 18 : : PUNCT iajs-2513 90 19 𝑄 𝑄 PROPN iajs-2513 90 20 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 90 21 𝑅 𝑅 PROPN iajs-2513 90 22 or or CCONJ iajs-2513 90 23 𝐼 𝐼 PROPN iajs-2513 90 24 ⊆ ⊆ NUM iajs-2513 90 25 𝐸 𝐸 PROPN iajs-2513 90 26 : : PUNCT iajs-2513 90 27 𝑄 𝑄 PROPN iajs-2513 90 28 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 90 29 𝑅 𝑅 PROPN iajs-2513 90 30 : : PUNCT iajs-2513 90 31 𝑅 𝑅 PROPN iajs-2513 90 32 𝐸 𝐸 PROPN iajs-2513 90 33 : : PUNCT iajs-2513 90 34 𝑄 𝑄 PROPN iajs-2513 90 35 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 90 36 𝑅 𝑅 PROPN iajs-2513 90 37 for for ADP iajs-2513 90 38 some some PRON iajs-2513 90 39 𝑛 𝑛 DET iajs-2513 90 40 ∈ ∈ NOUN iajs-2513 90 41 𝑍 𝑍 NOUN iajs-2513 90 42 . . PUNCT iajs-2513 91 1 it it PRON iajs-2513 91 2 follows follow VERB iajs-2513 91 3 that that SCONJ iajs-2513 91 4 either either CCONJ iajs-2513 91 5 𝐽𝑄 𝐽𝑄 PROPN iajs-2513 91 6 ⊆ ⊆ NUM iajs-2513 91 7 𝐸 𝐸 PROPN iajs-2513 91 8 : : PUNCT iajs-2513 91 9 𝑄 𝑄 PROPN iajs-2513 91 10 𝑄 𝑄 PROPN iajs-2513 91 11 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 91 12 𝑅 𝑅 PROPN iajs-2513 91 13 𝑄 𝑄 PROPN iajs-2513 91 14 or or CCONJ iajs-2513 91 15 𝐼 𝐼 ADP iajs-2513 91 16 𝑄 𝑄 PROPN iajs-2513 91 17 ⊆ ⊆ PROPN iajs-2513 91 18 𝐸 𝐸 PROPN iajs-2513 91 19 : : PUNCT iajs-2513 91 20 𝑄 𝑄 PROPN iajs-2513 91 21 𝑄 𝑄 PROPN iajs-2513 91 22 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 91 23 𝑅 𝑅 PROPN iajs-2513 91 24 𝑄. 𝑄. PROPN iajs-2513 91 25 since since SCONJ iajs-2513 91 26 𝑄 𝑄 PRON iajs-2513 91 27 is be AUX iajs-2513 91 28 faithful faithful ADJ iajs-2513 91 29 multiplication multiplication NOUN iajs-2513 91 30 then then ADV iajs-2513 91 31 by by ADP iajs-2513 91 32 lemma lemma PROPN iajs-2513 91 33 ( ( PUNCT iajs-2513 91 34 11 11 NUM iajs-2513 91 35 ) ) PUNCT iajs-2513 91 36 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 91 37 𝑅 𝑅 PROPN iajs-2513 91 38 𝑄 𝑄 PROPN iajs-2513 91 39 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 91 40 𝑄 𝑄 PROPN iajs-2513 91 41 , , PUNCT iajs-2513 91 42 and and CCONJ iajs-2513 91 43 since since SCONJ iajs-2513 91 44 𝑄 𝑄 PRON iajs-2513 91 45 is be AUX iajs-2513 91 46 multiplication multiplication NOUN iajs-2513 91 47 then then ADV iajs-2513 91 48 𝐸 𝐸 PROPN iajs-2513 91 49 : : PUNCT iajs-2513 91 50 𝑄 𝑄 PROPN iajs-2513 91 51 𝑄 𝑄 PROPN iajs-2513 91 52 𝐸 𝐸 PROPN iajs-2513 91 53 and and CCONJ iajs-2513 91 54 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 91 55 𝐸 𝐸 PROPN iajs-2513 91 56 𝐸 𝐸 PROPN iajs-2513 91 57 : : PUNCT iajs-2513 91 58 𝑄 𝑄 PROPN iajs-2513 91 59 𝑄. 𝑄. VERB iajs-2513 91 60 hence hence ADV iajs-2513 91 61 either either CCONJ iajs-2513 91 62 𝐽𝑄 𝐽𝑄 PROPN iajs-2513 91 63 ⊆ ⊆ NUM iajs-2513 91 64 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 91 65 𝐸 𝐸 PROPN iajs-2513 91 66 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 91 67 𝑄 𝑄 PROPN iajs-2513 91 68 or or CCONJ iajs-2513 91 69 𝐼 𝐼 ADP iajs-2513 91 70 𝑄 𝑄 PROPN iajs-2513 91 71 ⊆ ⊆ NUM iajs-2513 91 72 𝐸 𝐸 PROPN iajs-2513 91 73 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 91 74 𝑄 𝑄 PROPN iajs-2513 91 75 , , PUNCT iajs-2513 91 76 that that PRON iajs-2513 91 77 is be AUX iajs-2513 91 78 either either CCONJ iajs-2513 91 79 𝐹 𝐹 PROPN iajs-2513 91 80 ⊆ ⊆ PROPN iajs-2513 91 81 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 91 82 𝐸 𝐸 PROPN iajs-2513 91 83 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 91 84 𝑄 𝑄 PROPN iajs-2513 91 85 or or CCONJ iajs-2513 91 86 𝐼 𝐼 ADP iajs-2513 91 87 𝑄 𝑄 PROPN iajs-2513 91 88 ⊆ ⊆ NUM iajs-2513 91 89 𝐸 𝐸 PROPN iajs-2513 91 90 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 91 91 𝑄 𝑄 PROPN iajs-2513 91 92 . . PUNCT iajs-2513 92 1 hence hence ADV iajs-2513 92 2 , , PUNCT iajs-2513 92 3 by by ADP iajs-2513 92 4 proposition proposition NOUN iajs-2513 92 5 ( ( PUNCT iajs-2513 92 6 3 3 X iajs-2513 92 7 ) ) PUNCT iajs-2513 92 8 𝐸 𝐸 PROPN iajs-2513 92 9 is be AUX iajs-2513 92 10 an an DET iajs-2513 92 11 app app PROPN iajs-2513 92 12 - - PUNCT iajs-2513 92 13 qp qp NOUN iajs-2513 92 14 submodule submodule NOUN iajs-2513 92 15 of of ADP iajs-2513 92 16 𝑄. 𝑄. PROPN iajs-2513 92 17 recall recall NOUN iajs-2513 92 18 that that SCONJ iajs-2513 92 19 an an DET iajs-2513 92 20 𝑅-module 𝑅-module PROPN iajs-2513 92 21 𝑄 𝑄 PROPN iajs-2513 92 22 is be AUX iajs-2513 92 23 called call VERB iajs-2513 92 24 non non ADJ iajs-2513 92 25 - - ADJ iajs-2513 92 26 singular singular ADJ iajs-2513 92 27 if if SCONJ iajs-2513 92 28 𝑍 𝑍 VERB iajs-2513 92 29 𝑄 𝑄 PROPN iajs-2513 92 30 𝑄 𝑄 PROPN iajs-2513 92 31 , , PUNCT iajs-2513 92 32 where where SCONJ iajs-2513 92 33 𝑍 𝑍 VERB iajs-2513 92 34 𝑄 𝑄 PRON iajs-2513 92 35 𝑞 𝑞 X iajs-2513 92 36 ∈ ∈ NOUN iajs-2513 92 37 𝑄 𝑄 PROPN iajs-2513 92 38 : : PUNCT iajs-2513 92 39 𝑞𝐽 𝑞𝐽 X iajs-2513 92 40 0 0 NUM iajs-2513 92 41 for for ADP iajs-2513 92 42 some some DET iajs-2513 92 43 essentail essentail NOUN iajs-2513 92 44 ideal ideal NOUN iajs-2513 92 45 𝐽 𝐽 PROPN iajs-2513 92 46 of of ADP iajs-2513 92 47 𝑅 𝑅 PROPN iajs-2513 92 48 [ [ X iajs-2513 92 49 9 9 NUM iajs-2513 92 50 ] ] PUNCT iajs-2513 92 51 . . PUNCT iajs-2513 93 1 we we PRON iajs-2513 93 2 need need VERB iajs-2513 93 3 to to PART iajs-2513 93 4 recall recall VERB iajs-2513 93 5 the the DET iajs-2513 93 6 following follow VERB iajs-2513 93 7 lemma lemma PROPN iajs-2513 93 8 : : PUNCT iajs-2513 93 9 lemma lemma PROPN iajs-2513 93 10 ( ( PUNCT iajs-2513 93 11 13 13 NUM iajs-2513 93 12 ) ) PUNCT iajs-2513 94 1 [ [ X iajs-2513 94 2 9 9 NUM iajs-2513 94 3 , , PUNCT iajs-2513 94 4 coro coro X iajs-2513 94 5 . . PUNCT iajs-2513 95 1 ( ( PUNCT iajs-2513 95 2 1.26 1.26 NUM iajs-2513 95 3 ) ) PUNCT iajs-2513 95 4 ] ] PUNCT iajs-2513 95 5 if if SCONJ iajs-2513 95 6 𝑄 𝑄 PRON iajs-2513 95 7 is be AUX iajs-2513 95 8 a a DET iajs-2513 95 9 non non ADJ iajs-2513 95 10 - - ADJ iajs-2513 95 11 singular singular ADJ iajs-2513 95 12 𝑅-module 𝑅-module NOUN iajs-2513 95 13 , , PUNCT iajs-2513 95 14 then then ADV iajs-2513 95 15 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 95 16 𝑅 𝑅 PROPN iajs-2513 95 17 𝑄 𝑄 PROPN iajs-2513 95 18 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 95 19 𝑄 𝑄 PROPN iajs-2513 95 20 . . PUNCT iajs-2513 95 21     SPACE iajs-2513 96 1 97 97 NUM iajs-2513 96 2     SPACE iajs-2513 96 3 ibn ibn PROPN iajs-2513 96 4 al al PROPN iajs-2513 96 5 - - PUNCT iajs-2513 96 6 haitham haitham PROPN iajs-2513 96 7 jour jour X iajs-2513 96 8 . . PROPN iajs-2513 96 9 for for ADP iajs-2513 96 10 pure pure ADJ iajs-2513 96 11 & & CCONJ iajs-2513 96 12 appl appl PROPN iajs-2513 96 13 . . PUNCT iajs-2513 97 1 sci sci PROPN iajs-2513 97 2 . . PROPN iajs-2513 98 1 33 33 NUM iajs-2513 98 2 ( ( PUNCT iajs-2513 98 3 4 4 NUM iajs-2513 98 4 ) ) PUNCT iajs-2513 98 5 2020 2020 NUM iajs-2513 98 6 proposition proposition NOUN iajs-2513 98 7 ( ( PUNCT iajs-2513 98 8 14 14 NUM iajs-2513 98 9 ) ) PUNCT iajs-2513 98 10 let let VERB iajs-2513 98 11 𝐸 𝐸 PRON iajs-2513 98 12 be be AUX iajs-2513 98 13 a a DET iajs-2513 98 14 propoer propoer NOUN iajs-2513 98 15 submodule submodule NOUN iajs-2513 98 16 of of ADP iajs-2513 98 17 a a DET iajs-2513 98 18 non non ADJ iajs-2513 98 19 - - ADJ iajs-2513 98 20 singular singular ADJ iajs-2513 98 21 multiplication multiplication NOUN iajs-2513 98 22 𝑅-module 𝑅-module PROPN iajs-2513 98 23 𝑇. 𝑇. PROPN iajs-2513 98 24 then then ADV iajs-2513 98 25 , , PUNCT iajs-2513 98 26 𝐸 𝐸 PROPN iajs-2513 98 27 is be AUX iajs-2513 98 28 an an DET iajs-2513 98 29 app app PROPN iajs-2513 98 30 - - PUNCT iajs-2513 98 31 qp qp NOUN iajs-2513 98 32 submodule submodule NOUN iajs-2513 98 33 of of ADP iajs-2513 98 34 𝑄 𝑄 PRON iajs-2513 98 35 if if SCONJ iajs-2513 98 36 and and CCONJ iajs-2513 98 37 only only ADV iajs-2513 98 38 if if SCONJ iajs-2513 98 39 𝐸 𝐸 PROPN iajs-2513 98 40 : : PUNCT iajs-2513 98 41 𝑄 𝑄 PROPN iajs-2513 98 42 is be AUX iajs-2513 98 43 an an DET iajs-2513 98 44 app app ADJ iajs-2513 98 45 - - PUNCT iajs-2513 98 46 qp qp NOUN iajs-2513 98 47 ideal ideal NOUN iajs-2513 98 48 of of ADP iajs-2513 98 49 𝑅. 𝑅. NOUN iajs-2513 98 50 proof proof NOUN iajs-2513 98 51 follow follow VERB iajs-2513 98 52 as as ADP iajs-2513 98 53 in in ADP iajs-2513 98 54 proposition proposition NOUN iajs-2513 98 55 ( ( PUNCT iajs-2513 98 56 12 12 NUM iajs-2513 98 57 ) ) PUNCT iajs-2513 98 58 by by ADP iajs-2513 98 59 using use VERB iajs-2513 98 60 lemma lemma PROPN iajs-2513 98 61 ( ( PUNCT iajs-2513 98 62 13 13 NUM iajs-2513 98 63 ) ) PUNCT iajs-2513 98 64 . . PUNCT iajs-2513 99 1 we we PRON iajs-2513 99 2 need need VERB iajs-2513 99 3 to to PART iajs-2513 99 4 recall recall VERB iajs-2513 99 5 the the DET iajs-2513 99 6 following follow VERB iajs-2513 99 7 lemma lemma PROPN iajs-2513 99 8 : : PUNCT iajs-2513 99 9 lemma lemma PROPN iajs-2513 99 10 ( ( PUNCT iajs-2513 99 11 15 15 NUM iajs-2513 99 12 ) ) PUNCT iajs-2513 100 1 [ [ X iajs-2513 100 2 13 13 NUM iajs-2513 100 3 , , PUNCT iajs-2513 100 4 coro coro X iajs-2513 100 5 . . PUNCT iajs-2513 100 6 of of ADP iajs-2513 100 7 theo theo PROPN iajs-2513 100 8 . . PUNCT iajs-2513 101 1 9 9 NUM iajs-2513 101 2 ] ] PUNCT iajs-2513 101 3 let let VERB iajs-2513 101 4 𝐼 𝐼 PROPN iajs-2513 101 5 and and CCONJ iajs-2513 101 6 𝐽 𝐽 PROPN iajs-2513 101 7 are be AUX iajs-2513 101 8 ideals ideal NOUN iajs-2513 101 9 of of ADP iajs-2513 101 10 a a DET iajs-2513 101 11 ring ring NOUN iajs-2513 101 12 𝑅 𝑅 PROPN iajs-2513 101 13 , , PUNCT iajs-2513 101 14 and and CCONJ iajs-2513 101 15 𝑄 𝑄 PRON iajs-2513 101 16 be be VERB iajs-2513 101 17 a a DET iajs-2513 101 18 finitely finitely ADV iajs-2513 101 19 generated generate VERB iajs-2513 101 20 multiplication multiplication NOUN iajs-2513 101 21 𝑅-module 𝑅-module PROPN iajs-2513 101 22 . . PUNCT iajs-2513 102 1 then then ADV iajs-2513 102 2 𝐼𝑄 𝐼𝑄 PROPN iajs-2513 102 3 ⊆ ⊆ NUM iajs-2513 102 4 𝐽𝑄 𝐽𝑄 PROPN iajs-2513 102 5 if if SCONJ iajs-2513 102 6 and and CCONJ iajs-2513 102 7 only only ADV iajs-2513 102 8 if if SCONJ iajs-2513 102 9 𝐼 𝐼 PROPN iajs-2513 102 10 ⊆ ⊆ PROPN iajs-2513 102 11 𝐽 𝐽 PROPN iajs-2513 102 12 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2513 102 13 𝑄 𝑄 PRON iajs-2513 102 14 . . PUNCT iajs-2513 103 1 proposition proposition NOUN iajs-2513 103 2 ( ( PUNCT iajs-2513 103 3 16 16 NUM iajs-2513 103 4 ) ) PUNCT iajs-2513 103 5 let let VERB iajs-2513 103 6 𝑄 𝑄 PRON iajs-2513 103 7 be be AUX iajs-2513 103 8 a a DET iajs-2513 103 9 faithful faithful ADJ iajs-2513 103 10 finitely finitely ADV iajs-2513 103 11 generated generate VERB iajs-2513 103 12 multiplication multiplication NOUN iajs-2513 103 13 𝑅-module 𝑅-module PROPN iajs-2513 103 14 and and CCONJ iajs-2513 103 15 𝐼 𝐼 PROPN iajs-2513 103 16 is be AUX iajs-2513 103 17 an an DET iajs-2513 103 18 app app ADJ iajs-2513 103 19 - - PUNCT iajs-2513 103 20 qp qp NOUN iajs-2513 103 21 ideal ideal NOUN iajs-2513 103 22 of of ADP iajs-2513 103 23 𝑅. 𝑅. NOUN iajs-2513 103 24 then then ADV iajs-2513 103 25 𝐼𝑄 𝐼𝑄 PROPN iajs-2513 103 26 is be AUX iajs-2513 103 27 an an DET iajs-2513 103 28 app app PROPN iajs-2513 103 29 - - PUNCT iajs-2513 103 30 qp qp NOUN iajs-2513 103 31 submodule submodule NOUN iajs-2513 103 32 of of ADP iajs-2513 103 33 𝑄. 𝑄. PROPN iajs-2513 103 34 proof proof NOUN iajs-2513 103 35 let let VERB iajs-2513 103 36 𝑟𝐹 𝑟𝐹 PROPN iajs-2513 103 37 ⊆ ⊆ NUM iajs-2513 103 38 𝐼𝑄 𝐼𝑄 PROPN iajs-2513 103 39 for for ADP iajs-2513 103 40 𝑟 𝑟 DET iajs-2513 103 41 ∈ ∈ PROPN iajs-2513 103 42 𝑅 𝑅 PROPN iajs-2513 103 43 , , PUNCT iajs-2513 103 44 and and CCONJ iajs-2513 103 45 𝐹 𝐹 PROPN iajs-2513 103 46 is be AUX iajs-2513 103 47 a a DET iajs-2513 103 48 submodule submodule NOUN iajs-2513 103 49 of of ADP iajs-2513 103 50 𝑄 𝑄 PRON iajs-2513 103 51 with with ADP iajs-2513 103 52 𝑟 𝑟 DET iajs-2513 103 53 𝑄 𝑄 PROPN iajs-2513 103 54 ⊈ ⊈ PROPN iajs-2513 103 55 𝐼𝑄 𝐼𝑄 X iajs-2513 103 56 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 103 57 𝑄 𝑄 PROPN iajs-2513 103 58 for for ADP iajs-2513 103 59 some some DET iajs-2513 103 60 𝑛 𝑛 DET iajs-2513 103 61 ∈ ∈ PROPN iajs-2513 103 62 𝑍 𝑍 NOUN iajs-2513 103 63 . . PUNCT iajs-2513 104 1 since since SCONJ iajs-2513 104 2 𝑄 𝑄 PRON iajs-2513 104 3 is be AUX iajs-2513 104 4 faithful faithful ADJ iajs-2513 104 5 multiplication multiplication NOUN iajs-2513 104 6 then then ADV iajs-2513 104 7 by by ADP iajs-2513 104 8 lemma lemma PROPN iajs-2513 104 9 ( ( PUNCT iajs-2513 104 10 11 11 NUM iajs-2513 104 11 ) ) PUNCT iajs-2513 104 12 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 104 13 𝑄 𝑄 PROPN iajs-2513 104 14 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 104 15 𝑅 𝑅 PROPN iajs-2513 104 16 𝑄 𝑄 PROPN iajs-2513 104 17 , , PUNCT iajs-2513 104 18 that that PRON iajs-2513 104 19 is be AUX iajs-2513 104 20 𝑟 𝑟 PRON iajs-2513 104 21 𝑄 𝑄 PROPN iajs-2513 104 22 ⊈ ⊈ PROPN iajs-2513 104 23 𝐼𝑄 𝐼𝑄 X iajs-2513 104 24 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 104 25 𝑅 𝑅 PROPN iajs-2513 104 26 𝑄 𝑄 PROPN iajs-2513 104 27 for for ADP iajs-2513 104 28 some some PRON iajs-2513 104 29 𝑛 𝑛 DET iajs-2513 104 30 ∈ ∈ PROPN iajs-2513 104 31 𝑍 𝑍 NOUN iajs-2513 104 32 , , PUNCT iajs-2513 104 33 it it PRON iajs-2513 104 34 follows follow VERB iajs-2513 104 35 that that SCONJ iajs-2513 104 36 𝑟 𝑟 PRON iajs-2513 104 37 ∉ ∉ PROPN iajs-2513 104 38 𝐼 𝐼 PROPN iajs-2513 104 39 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 104 40 𝑅 𝑅 PROPN iajs-2513 104 41 𝐼 𝐼 PROPN iajs-2513 104 42 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 104 43 𝑅 𝑅 PROPN iajs-2513 104 44 : : PUNCT iajs-2513 104 45 𝑅 𝑅 PROPN iajs-2513 104 46 implies imply VERB iajs-2513 104 47 that that SCONJ iajs-2513 104 48 𝑟 𝑟 X iajs-2513 104 49 𝑅 𝑅 PROPN iajs-2513 104 50 ⊈ ⊈ PROPN iajs-2513 104 51 𝐼 𝐼 PROPN iajs-2513 104 52 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 104 53 𝑅 𝑅 PROPN iajs-2513 104 54 , , PUNCT iajs-2513 104 55 now now ADV iajs-2513 104 56 , , PUNCT iajs-2513 104 57 since since SCONJ iajs-2513 104 58 𝑟𝐹 𝑟𝐹 PROPN iajs-2513 104 59 ⊆ ⊆ NUM iajs-2513 104 60 𝐼𝑄 𝐼𝑄 PROPN iajs-2513 104 61 and and CCONJ iajs-2513 104 62 𝑄 𝑄 PROPN iajs-2513 104 63 is be AUX iajs-2513 104 64 a a DET iajs-2513 104 65 multiplication multiplication NOUN iajs-2513 104 66 then then ADV iajs-2513 104 67 𝐹 𝐹 PROPN iajs-2513 104 68 𝐽𝑄 𝐽𝑄 VERB iajs-2513 104 69 for for ADP iajs-2513 104 70 some some DET iajs-2513 104 71 ideal ideal ADJ iajs-2513 104 72 𝐽 𝐽 PROPN iajs-2513 104 73 of of ADP iajs-2513 104 74 𝑅 𝑅 PROPN iajs-2513 104 75 , , PUNCT iajs-2513 104 76 thus thus ADV iajs-2513 104 77 𝑟𝐽𝑄 𝑟𝐽𝑄 ADP iajs-2513 104 78 ⊆ ⊆ NUM iajs-2513 104 79 𝐼𝑄. 𝐼𝑄. NOUN iajs-2513 104 80 hence hence ADV iajs-2513 104 81 by by ADP iajs-2513 104 82 lemma lemma PROPN iajs-2513 104 83 ( ( PUNCT iajs-2513 104 84 15 15 NUM iajs-2513 104 85 ) ) PUNCT iajs-2513 104 86 𝑟𝐽 𝑟𝐽 NOUN iajs-2513 104 87 ⊆ ⊆ NUM iajs-2513 104 88 𝐼 𝐼 PROPN iajs-2513 104 89 𝑎𝑛𝑛 𝑎𝑛𝑛 VERB iajs-2513 104 90 𝑄 𝑄 PRON iajs-2513 104 91 , , PUNCT iajs-2513 104 92 but but CCONJ iajs-2513 104 93 𝑄 𝑄 PRON iajs-2513 104 94 is be AUX iajs-2513 104 95 a a DET iajs-2513 104 96 faithful faithful ADJ iajs-2513 104 97 , , PUNCT iajs-2513 104 98 then then ADV iajs-2513 104 99 𝑟𝐽 𝑟𝐽 X iajs-2513 104 100 ⊆ ⊆ NUM iajs-2513 104 101 𝐼 𝐼 PROPN iajs-2513 104 102 0 0 X iajs-2513 104 103 𝐼. 𝐼. NOUN iajs-2513 104 104 since since SCONJ iajs-2513 104 105 𝐼 𝐼 PROPN iajs-2513 104 106 is be AUX iajs-2513 104 107 an an DET iajs-2513 104 108 app app ADJ iajs-2513 104 109 - - PUNCT iajs-2513 104 110 qp qp NOUN iajs-2513 104 111 ideal ideal NOUN iajs-2513 104 112 of of ADP iajs-2513 104 113 𝑅 𝑅 PROPN iajs-2513 104 114 and and CCONJ iajs-2513 104 115 𝑟 𝑟 NOUN iajs-2513 104 116 𝑅 𝑅 PROPN iajs-2513 104 117 ⊈ ⊈ PROPN iajs-2513 104 118 𝐼 𝐼 PROPN iajs-2513 104 119 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 104 120 𝑅 𝑅 PROPN iajs-2513 104 121 then then ADV iajs-2513 104 122 by by ADP iajs-2513 104 123 corollary corollary ADJ iajs-2513 104 124 ( ( PUNCT iajs-2513 104 125 4 4 NUM iajs-2513 104 126 ) ) PUNCT iajs-2513 104 127 either either CCONJ iajs-2513 104 128 ⊆ ⊆ NUM iajs-2513 104 129 √𝐼 √𝐼 X iajs-2513 104 130 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 104 131 𝑅 𝑅 PROPN iajs-2513 104 132 , , PUNCT iajs-2513 104 133 hence hence ADV iajs-2513 104 134 𝐽𝑄 𝐽𝑄 PROPN iajs-2513 104 135 ⊆ ⊆ NUM iajs-2513 104 136 √𝐼𝑄 √𝐼𝑄 ADP iajs-2513 104 137 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 104 138 𝑅 𝑅 PROPN iajs-2513 104 139 𝑄. 𝑄. PROPN iajs-2513 104 140 it it PRON iajs-2513 104 141 follows follow VERB iajs-2513 104 142 by by ADP iajs-2513 104 143 lemma lemma PROPN iajs-2513 104 144 ( ( PUNCT iajs-2513 104 145 11 11 NUM iajs-2513 104 146 ) ) PUNCT iajs-2513 104 147 𝐽𝑄 𝐽𝑄 PROPN iajs-2513 104 148 ⊆ ⊆ NUM iajs-2513 104 149 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 104 150 𝐼𝑄 𝐼𝑄 X iajs-2513 104 151 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 104 152 𝑄 𝑄 PROPN iajs-2513 104 153 . . PUNCT iajs-2513 105 1 that that PRON iajs-2513 105 2 is be AUX iajs-2513 105 3 𝐹 𝐹 PROPN iajs-2513 105 4 ⊆ ⊆ NUM iajs-2513 105 5 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 105 6 𝐼𝑄 𝐼𝑄 PROPN iajs-2513 105 7 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 105 8 𝑄 𝑄 PROPN iajs-2513 105 9 . . PUNCT iajs-2513 106 1 hence hence ADV iajs-2513 106 2 by by ADP iajs-2513 106 3 corollary corollary ADJ iajs-2513 106 4 ( ( PUNCT iajs-2513 106 5 4 4 NUM iajs-2513 106 6 ) ) PUNCT iajs-2513 106 7 𝐼𝑄 𝐼𝑄 PROPN iajs-2513 106 8 is be AUX iajs-2513 106 9 an an DET iajs-2513 106 10 app app PROPN iajs-2513 106 11 - - PUNCT iajs-2513 106 12 qp qp NOUN iajs-2513 106 13 submodule submodule NOUN iajs-2513 106 14 of of ADP iajs-2513 106 15 𝑄. 𝑄. PROPN iajs-2513 106 16 proposition proposition NOUN iajs-2513 106 17 ( ( PUNCT iajs-2513 106 18 17 17 NUM iajs-2513 106 19 ) ) PUNCT iajs-2513 106 20 let let VERB iajs-2513 106 21 𝑄 𝑄 PRON iajs-2513 106 22 be be AUX iajs-2513 106 23 a a DET iajs-2513 106 24 finitely finitely ADV iajs-2513 106 25 generated generate VERB iajs-2513 106 26 multiplication multiplication NOUN iajs-2513 106 27 non non ADJ iajs-2513 106 28 - - ADJ iajs-2513 106 29 singular singular ADJ iajs-2513 106 30 𝑅-module 𝑅-module PROPN iajs-2513 106 31 and and CCONJ iajs-2513 106 32 𝐼 𝐼 PROPN iajs-2513 106 33 is be AUX iajs-2513 106 34 an an DET iajs-2513 106 35 app app ADJ iajs-2513 106 36 - - PUNCT iajs-2513 106 37 qp qp NOUN iajs-2513 106 38 ideal ideal NOUN iajs-2513 106 39 of of ADP iajs-2513 106 40 𝑅 𝑅 PROPN iajs-2513 106 41 with with ADP iajs-2513 106 42 𝑎𝑛𝑛 𝑎𝑛𝑛 NOUN iajs-2513 106 43 𝑄 𝑄 PROPN iajs-2513 106 44 ⊆ ⊆ X iajs-2513 106 45 𝐼. 𝐼. PROPN iajs-2513 106 46 then then ADV iajs-2513 106 47 𝐼𝑄 𝐼𝑄 PROPN iajs-2513 106 48 is be AUX iajs-2513 106 49 an an DET iajs-2513 106 50 app app PROPN iajs-2513 106 51 - - PUNCT iajs-2513 106 52 qp qp NOUN iajs-2513 106 53 submodule submodule NOUN iajs-2513 106 54 of of ADP iajs-2513 106 55 𝑄. 𝑄. PROPN iajs-2513 106 56 proof proof NOUN iajs-2513 106 57 follows follow VERB iajs-2513 106 58 similar similar ADJ iajs-2513 106 59 as as ADP iajs-2513 106 60 in in ADP iajs-2513 106 61 proposition proposition NOUN iajs-2513 106 62 ( ( PUNCT iajs-2513 106 63 16 16 NUM iajs-2513 106 64 ) ) PUNCT iajs-2513 106 65 and and CCONJ iajs-2513 106 66 using use VERB iajs-2513 106 67 lemma lemma PROPN iajs-2513 106 68 ( ( PUNCT iajs-2513 106 69 13 13 NUM iajs-2513 106 70 ) ) PUNCT iajs-2513 106 71 . . PUNCT iajs-2513 107 1 proposition proposition NOUN iajs-2513 107 2 ( ( PUNCT iajs-2513 107 3 18 18 NUM iajs-2513 107 4 ) ) PUNCT iajs-2513 107 5 let let VERB iajs-2513 107 6 𝑄 𝑄 PRON iajs-2513 107 7 be be AUX iajs-2513 107 8 a a DET iajs-2513 107 9 faithful faithful ADJ iajs-2513 107 10 finitely finitely ADV iajs-2513 107 11 generated generate VERB iajs-2513 107 12 multiplication multiplication NOUN iajs-2513 107 13 𝑅-module 𝑅-module PROPN iajs-2513 107 14 and and CCONJ iajs-2513 107 15 𝐸 𝐸 PROPN iajs-2513 107 16 be be VERB iajs-2513 107 17 a a DET iajs-2513 107 18 proper proper ADJ iajs-2513 107 19 submodule submodule NOUN iajs-2513 107 20 of of ADP iajs-2513 107 21 𝑄. 𝑄. PROPN iajs-2513 107 22 then then ADV iajs-2513 107 23 the the DET iajs-2513 107 24 following follow VERB iajs-2513 107 25 statements statement NOUN iajs-2513 107 26 are be AUX iajs-2513 107 27 equivalent equivalent ADJ iajs-2513 107 28 . . PUNCT iajs-2513 108 1 1 1 X iajs-2513 108 2 ) ) PUNCT iajs-2513 108 3 𝐸 𝐸 PROPN iajs-2513 108 4 is be AUX iajs-2513 108 5 an an DET iajs-2513 108 6 app app PROPN iajs-2513 108 7 - - PUNCT iajs-2513 108 8 qp qp NOUN iajs-2513 108 9 submodule submodule NOUN iajs-2513 108 10 of of ADP iajs-2513 108 11 𝑄. 𝑄. PROPN iajs-2513 108 12 2 2 NUM iajs-2513 108 13 ) ) PUNCT iajs-2513 108 14 𝐸 𝐸 PROPN iajs-2513 108 15 : : PUNCT iajs-2513 108 16 𝑄 𝑄 PROPN iajs-2513 108 17 is be AUX iajs-2513 108 18 an an DET iajs-2513 108 19 app app ADJ iajs-2513 108 20 - - PUNCT iajs-2513 108 21 qp qp NOUN iajs-2513 108 22 ideal ideal NOUN iajs-2513 108 23 of of ADP iajs-2513 108 24 𝑅. 𝑅. NOUN iajs-2513 108 25 3 3 NUM iajs-2513 108 26 ) ) PUNCT iajs-2513 108 27 𝐸 𝐸 PROPN iajs-2513 108 28 𝐼𝑄 𝐼𝑄 PROPN iajs-2513 108 29 for for ADP iajs-2513 108 30 some some DET iajs-2513 108 31 an an DET iajs-2513 108 32 app app PROPN iajs-2513 108 33 - - PUNCT iajs-2513 108 34 qp qp NOUN iajs-2513 108 35 ideal ideal NOUN iajs-2513 108 36 𝐼 𝐼 ADP iajs-2513 108 37 of of ADP iajs-2513 108 38 𝑅. 𝑅. ADJ iajs-2513 108 39 proof proof NOUN iajs-2513 108 40 ( ( PUNCT iajs-2513 108 41 1 1 NUM iajs-2513 108 42 ) ) PUNCT iajs-2513 108 43 2 2 NUM iajs-2513 108 44 ) ) PUNCT iajs-2513 108 45 it it PRON iajs-2513 108 46 follows follow VERB iajs-2513 108 47 by by ADP iajs-2513 108 48 proposition proposition NOUN iajs-2513 108 49 ( ( PUNCT iajs-2513 108 50 12 12 NUM iajs-2513 108 51 ) ) PUNCT iajs-2513 108 52 . . PUNCT iajs-2513 109 1 ( ( PUNCT iajs-2513 109 2 2 2 X iajs-2513 109 3 ) ) PUNCT iajs-2513 109 4 ( ( PUNCT iajs-2513 109 5 3 3 X iajs-2513 109 6 ) ) PUNCT iajs-2513 109 7 it it PRON iajs-2513 109 8 is be AUX iajs-2513 109 9 clear clear ADJ iajs-2513 109 10 . . PUNCT iajs-2513 110 1 ( ( PUNCT iajs-2513 110 2 3 3 X iajs-2513 110 3 ) ) PUNCT iajs-2513 110 4 ( ( PUNCT iajs-2513 110 5 2 2 X iajs-2513 110 6 ) ) PUNCT iajs-2513 110 7 suppose suppose VERB iajs-2513 110 8 that that SCONJ iajs-2513 110 9 𝐸 𝐸 PROPN iajs-2513 110 10 𝐼𝑄 𝐼𝑄 PROPN iajs-2513 110 11 for for ADP iajs-2513 110 12 some some DET iajs-2513 110 13 app app PROPN iajs-2513 110 14 - - PUNCT iajs-2513 110 15 qp qp NOUN iajs-2513 110 16 ideal ideal NOUN iajs-2513 110 17 𝐼 𝐼 PROPN iajs-2513 110 18 of of ADP iajs-2513 110 19 𝑅. 𝑅. NOUN iajs-2513 110 20 since since SCONJ iajs-2513 110 21 𝑄 𝑄 PRON iajs-2513 110 22 is be AUX iajs-2513 110 23 a a DET iajs-2513 110 24 multiplication multiplication NOUN iajs-2513 110 25 , , PUNCT iajs-2513 110 26 then then ADV iajs-2513 110 27 𝐸 𝐸 PROPN iajs-2513 110 28 𝐸 𝐸 PROPN iajs-2513 110 29 : : PUNCT iajs-2513 110 30 𝑄 𝑄 PROPN iajs-2513 110 31 𝑄 𝑄 PROPN iajs-2513 110 32 𝐼𝑄. 𝐼𝑄. NOUN iajs-2513 110 33 but but CCONJ iajs-2513 110 34 𝑄 𝑄 PRON iajs-2513 110 35 is be AUX iajs-2513 110 36 faithful faithful ADJ iajs-2513 110 37 finitely finitely ADV iajs-2513 110 38 generated generate VERB iajs-2513 110 39 multiplication multiplication NOUN iajs-2513 110 40 , , PUNCT iajs-2513 110 41 then then ADV iajs-2513 110 42 𝐼 𝐼 PROPN iajs-2513 110 43 𝐸 𝐸 PROPN iajs-2513 110 44 : : PUNCT iajs-2513 110 45 𝑄 𝑄 NOUN iajs-2513 110 46 , , PUNCT iajs-2513 110 47 it it PRON iajs-2513 110 48 follows follow VERB iajs-2513 110 49 that that SCONJ iajs-2513 110 50 𝐸 𝐸 PROPN iajs-2513 110 51 : : PUNCT iajs-2513 110 52 𝑄 𝑄 PROPN iajs-2513 110 53 an an DET iajs-2513 110 54 app app NOUN iajs-2513 110 55 - - PUNCT iajs-2513 110 56 qp qp NOUN iajs-2513 110 57 ideal ideal NOUN iajs-2513 110 58 of of ADP iajs-2513 110 59 𝑅. 𝑅. NOUN iajs-2513 110 60     SPACE iajs-2513 110 61 98 98 NUM iajs-2513 110 62     SPACE iajs-2513 110 63 ibn ibn PROPN iajs-2513 110 64 al al PROPN iajs-2513 110 65 - - PUNCT iajs-2513 110 66 haitham haitham PROPN iajs-2513 110 67 jour jour X iajs-2513 110 68 . . PROPN iajs-2513 110 69 for for ADP iajs-2513 110 70 pure pure ADJ iajs-2513 110 71 & & CCONJ iajs-2513 110 72 appl appl PROPN iajs-2513 110 73 . . PUNCT iajs-2513 111 1 sci sci PROPN iajs-2513 111 2 . . PROPN iajs-2513 112 1 33 33 NUM iajs-2513 112 2 ( ( PUNCT iajs-2513 112 3 4 4 NUM iajs-2513 112 4 ) ) PUNCT iajs-2513 112 5 2020 2020 NUM iajs-2513 112 6 proposition proposition NOUN iajs-2513 112 7 ( ( PUNCT iajs-2513 112 8 19 19 NUM iajs-2513 112 9 ) ) PUNCT iajs-2513 112 10 let let VERB iajs-2513 112 11 𝑄 𝑄 PRON iajs-2513 112 12 be be AUX iajs-2513 112 13 a a DET iajs-2513 112 14 finitely finitely ADV iajs-2513 112 15 generated generate VERB iajs-2513 112 16 multiplication multiplication NOUN iajs-2513 112 17 non non ADJ iajs-2513 112 18 - - ADJ iajs-2513 112 19 singular singular ADJ iajs-2513 112 20 𝑅-module 𝑅-module PROPN iajs-2513 112 21 and and CCONJ iajs-2513 112 22 𝐸 𝐸 PROPN iajs-2513 112 23 be be VERB iajs-2513 112 24 a a DET iajs-2513 112 25 proper proper ADJ iajs-2513 112 26 submodule submodule NOUN iajs-2513 112 27 of of ADP iajs-2513 112 28 𝑄. 𝑄. PROPN iajs-2513 112 29 then then ADV iajs-2513 112 30 the the DET iajs-2513 112 31 following follow VERB iajs-2513 112 32 statements statement NOUN iajs-2513 112 33 are be AUX iajs-2513 112 34 equivalent equivalent ADJ iajs-2513 112 35 . . PUNCT iajs-2513 113 1 1 1 X iajs-2513 113 2 ) ) PUNCT iajs-2513 113 3 𝐸 𝐸 PROPN iajs-2513 113 4 is be AUX iajs-2513 113 5 an an DET iajs-2513 113 6 app app PROPN iajs-2513 113 7 - - PUNCT iajs-2513 113 8 qp qp NOUN iajs-2513 113 9 submodule submodule NOUN iajs-2513 113 10 of of ADP iajs-2513 113 11 𝑄. 𝑄. PROPN iajs-2513 113 12 2 2 NUM iajs-2513 113 13 ) ) PUNCT iajs-2513 113 14 𝐸 𝐸 PROPN iajs-2513 113 15 : : PUNCT iajs-2513 113 16 𝑄 𝑄 PROPN iajs-2513 113 17 is be AUX iajs-2513 113 18 an an DET iajs-2513 113 19 app app ADJ iajs-2513 113 20 - - PUNCT iajs-2513 113 21 qp qp NOUN iajs-2513 113 22 ideal ideal NOUN iajs-2513 113 23 of of ADP iajs-2513 113 24 𝑅. 𝑅. NOUN iajs-2513 113 25 3 3 NUM iajs-2513 113 26 ) ) PUNCT iajs-2513 113 27 𝐸 𝐸 PROPN iajs-2513 113 28 𝐽𝑄 𝐽𝑄 NOUN iajs-2513 113 29 for for ADP iajs-2513 113 30 some some PRON iajs-2513 113 31 an an DET iajs-2513 113 32 app app PROPN iajs-2513 113 33 - - PUNCT iajs-2513 113 34 qp qp NOUN iajs-2513 113 35 ideal ideal PROPN iajs-2513 113 36 𝐽 𝐽 PROPN iajs-2513 113 37 of of ADP iajs-2513 113 38 𝑅 𝑅 PROPN iajs-2513 113 39 with with ADP iajs-2513 113 40 𝑎𝑛𝑛 𝑎𝑛𝑛 NOUN iajs-2513 113 41 𝑄 𝑄 PROPN iajs-2513 113 42 ⊆ ⊆ NUM iajs-2513 113 43 𝐽. 𝐽. ADJ iajs-2513 113 44 proof proof NOUN iajs-2513 113 45 it it PRON iajs-2513 113 46 follows follow VERB iajs-2513 113 47 similar similar ADJ iajs-2513 113 48 as as ADP iajs-2513 113 49 proposition proposition NOUN iajs-2513 113 50 ( ( PUNCT iajs-2513 113 51 18 18 NUM iajs-2513 113 52 ) ) PUNCT iajs-2513 113 53 by by ADP iajs-2513 113 54 using use VERB iajs-2513 113 55 proposition proposition NOUN iajs-2513 113 56 ( ( PUNCT iajs-2513 113 57 14 14 NUM iajs-2513 113 58 ) ) PUNCT iajs-2513 113 59 and and CCONJ iajs-2513 113 60 lemma lemma PROPN iajs-2513 113 61 ( ( PUNCT iajs-2513 113 62 15 15 NUM iajs-2513 113 63 ) ) PUNCT iajs-2513 113 64 . . PUNCT iajs-2513 114 1 we we PRON iajs-2513 114 2 need need VERB iajs-2513 114 3 the the DET iajs-2513 114 4 following follow VERB iajs-2513 114 5 lemma lemma PROPN iajs-2513 114 6 . . PUNCT iajs-2513 115 1 lemma lemma PROPN iajs-2513 115 2 ( ( PUNCT iajs-2513 115 3 20 20 NUM iajs-2513 115 4 ) ) PUNCT iajs-2513 116 1 [ [ X iajs-2513 116 2 14 14 NUM iajs-2513 116 3 . . PUNCT iajs-2513 116 4 coro coro X iajs-2513 116 5 . . PUNCT iajs-2513 117 1 ( ( PUNCT iajs-2513 117 2 1.3 1.3 NUM iajs-2513 117 3 ) ) PUNCT iajs-2513 117 4 ] ] PUNCT iajs-2513 117 5 let let VERB iajs-2513 117 6 𝑓 𝑓 X iajs-2513 117 7 : : PUNCT iajs-2513 117 8 𝑄 𝑄 PROPN iajs-2513 117 9 ⟶ ⟶ NOUN iajs-2513 117 10 𝑄 𝑄 PRON iajs-2513 117 11 be be VERB iajs-2513 117 12 an an DET iajs-2513 117 13 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2513 117 14 and and CCONJ iajs-2513 117 15 𝐸 𝐸 PROPN iajs-2513 117 16 is be AUX iajs-2513 117 17 a a DET iajs-2513 117 18 submodule submodule NOUN iajs-2513 117 19 of of ADP iajs-2513 117 20 𝑄 𝑄 PRON iajs-2513 117 21 with with ADP iajs-2513 117 22 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-2513 117 23 𝑓 𝑓 PRON iajs-2513 117 24 ⊆ ⊆ NUM iajs-2513 117 25 𝐸 𝐸 PROPN iajs-2513 117 26 , , PUNCT iajs-2513 117 27 then then ADV iajs-2513 117 28 𝑓 𝑓 DET iajs-2513 117 29 𝑟𝑎𝑑 𝑟𝑎𝑑 NOUN iajs-2513 117 30 𝐸 𝐸 PROPN iajs-2513 117 31 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 117 32 𝑓 𝑓 DET iajs-2513 117 33 𝐸 𝐸 PROPN iajs-2513 117 34 . . PUNCT iajs-2513 118 1 proposition proposition NOUN iajs-2513 118 2 ( ( PUNCT iajs-2513 118 3 21 21 NUM iajs-2513 118 4 ) ) PUNCT iajs-2513 118 5 let let VERB iajs-2513 118 6 𝑓 𝑓 X iajs-2513 118 7 : : PUNCT iajs-2513 118 8 𝑄 𝑄 PROPN iajs-2513 118 9 ⟶ ⟶ NOUN iajs-2513 118 10 𝑄 𝑄 PRON iajs-2513 118 11 be be VERB iajs-2513 118 12 an an DET iajs-2513 118 13 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2513 118 14 and and CCONJ iajs-2513 118 15 𝐸 𝐸 PROPN iajs-2513 118 16 is be AUX iajs-2513 118 17 an an DET iajs-2513 118 18 app app PROPN iajs-2513 118 19 - - PUNCT iajs-2513 118 20 qp qp NOUN iajs-2513 118 21 submodule submodule NOUN iajs-2513 118 22 of of ADP iajs-2513 118 23 𝑄 𝑄 PROPN iajs-2513 118 24 . . PUNCT iajs-2513 119 1 then then ADV iajs-2513 119 2 𝑓 𝑓 X iajs-2513 119 3 𝐸 𝐸 PROPN iajs-2513 119 4 is be AUX iajs-2513 119 5 an an DET iajs-2513 119 6 app app PROPN iajs-2513 119 7 - - PUNCT iajs-2513 119 8 qp qp NOUN iajs-2513 119 9 submodule submodule NOUN iajs-2513 119 10 of of ADP iajs-2513 119 11 𝑄. 𝑄. PROPN iajs-2513 119 12 proof proof NOUN iajs-2513 119 13 it it PRON iajs-2513 119 14 is be AUX iajs-2513 119 15 clear clear ADJ iajs-2513 119 16 that that SCONJ iajs-2513 119 17 𝑓 𝑓 DET iajs-2513 119 18 𝐸 𝐸 PROPN iajs-2513 119 19 is be AUX iajs-2513 119 20 a a DET iajs-2513 119 21 proper proper ADJ iajs-2513 119 22 submodule submodule NOUN iajs-2513 119 23 of of ADP iajs-2513 119 24 𝑄. 𝑄. PROPN iajs-2513 119 25 now now ADV iajs-2513 119 26 , , PUNCT iajs-2513 119 27 suppose suppose VERB iajs-2513 119 28 that that SCONJ iajs-2513 119 29 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 119 30 ∈ ∈ PROPN iajs-2513 119 31 𝑓 𝑓 DET iajs-2513 119 32 𝐸 𝐸 PROPN iajs-2513 119 33 , , PUNCT iajs-2513 119 34 for for ADP iajs-2513 119 35 𝑟 𝑟 DET iajs-2513 119 36 ∈ ∈ PROPN iajs-2513 119 37 𝑅 𝑅 PROPN iajs-2513 119 38 , , PUNCT iajs-2513 119 39 𝑞 𝑞 X iajs-2513 119 40 ∈ ∈ PROPN iajs-2513 119 41 𝑄 𝑄 PROPN iajs-2513 119 42 , , PUNCT iajs-2513 119 43 implies imply VERB iajs-2513 119 44 that that SCONJ iajs-2513 119 45 𝑟𝑓 𝑟𝑓 X iajs-2513 119 46 𝑞 𝑞 X iajs-2513 119 47 ∈ ∈ PROPN iajs-2513 119 48 𝐸 𝐸 PROPN iajs-2513 119 49 . . PUNCT iajs-2513 120 1 but but CCONJ iajs-2513 120 2 𝐸 𝐸 PROPN iajs-2513 120 3 is be AUX iajs-2513 120 4 an an DET iajs-2513 120 5 app app PROPN iajs-2513 120 6 - - PUNCT iajs-2513 120 7 qp qp NOUN iajs-2513 120 8 submodule submodule NOUN iajs-2513 120 9 of of ADP iajs-2513 120 10 𝑄 𝑄 PROPN iajs-2513 120 11 , , PUNCT iajs-2513 120 12 it it PRON iajs-2513 120 13 follows follow VERB iajs-2513 120 14 that that SCONJ iajs-2513 120 15 either either CCONJ iajs-2513 120 16 𝑓 𝑓 PRON iajs-2513 120 17 𝑞 𝑞 X iajs-2513 120 18 ∈ ∈ PROPN iajs-2513 120 19 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 120 20 𝐸 𝐸 PROPN iajs-2513 120 21 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 120 22 𝑄 𝑄 PROPN iajs-2513 120 23 or or CCONJ iajs-2513 120 24 𝑟 𝑟 PRON iajs-2513 120 25 𝑄 𝑄 PROPN iajs-2513 120 26 ⊆ ⊆ NUM iajs-2513 120 27 𝐸 𝐸 PROPN iajs-2513 120 28 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 120 29 𝑄 𝑄 PROPN iajs-2513 120 30 for for ADP iajs-2513 120 31 some some DET iajs-2513 120 32 𝑛 𝑛 DET iajs-2513 120 33 ∈ ∈ NOUN iajs-2513 120 34 𝑍 𝑍 NOUN iajs-2513 120 35 . . PUNCT iajs-2513 121 1 it it PRON iajs-2513 121 2 follows follow VERB iajs-2513 121 3 that that SCONJ iajs-2513 121 4 by by ADP iajs-2513 121 5 lemma lemma PROPN iajs-2513 121 6 ( ( PUNCT iajs-2513 121 7 20 20 NUM iajs-2513 121 8 ) ) PUNCT iajs-2513 121 9 , , PUNCT iajs-2513 121 10 either either CCONJ iajs-2513 121 11 𝑞 𝑞 PROPN iajs-2513 121 12 ∈ ∈ PROPN iajs-2513 121 13 𝑓 𝑓 PRON iajs-2513 121 14 𝑟𝑎𝑑 𝑟𝑎𝑑 NOUN iajs-2513 121 15 𝐸 𝐸 PROPN iajs-2513 121 16 𝑓 𝑓 DET iajs-2513 121 17 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 121 18 𝑄 𝑄 PROPN iajs-2513 121 19 ⊆ ⊆ PROPN iajs-2513 121 20 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 121 21 𝑓 𝑓 DET iajs-2513 121 22 𝐸 𝐸 NOUN iajs-2513 121 23 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 121 24 𝑄 𝑄 PROPN iajs-2513 121 25 or or CCONJ iajs-2513 121 26 𝑟 𝑟 PRON iajs-2513 121 27 𝑓 𝑓 PRON iajs-2513 121 28 𝑓 𝑓 DET iajs-2513 121 29 𝑄 𝑄 PROPN iajs-2513 121 30 ⊆ ⊆ PROPN iajs-2513 121 31 𝑓 𝑓 PRON iajs-2513 121 32 𝐸 𝐸 ADJ iajs-2513 122 1 𝑓 𝑓 PRON iajs-2513 122 2 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 122 3 𝑄 𝑄 PROPN iajs-2513 122 4 ⊆ ⊆ PROPN iajs-2513 122 5 𝑓 𝑓 DET iajs-2513 122 6 𝐸 𝐸 NOUN iajs-2513 122 7 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 122 8 𝑄 𝑄 PROPN iajs-2513 122 9 .that .that PRON iajs-2513 122 10 is be AUX iajs-2513 122 11 either either CCONJ iajs-2513 122 12 𝑞 𝑞 PROPN iajs-2513 122 13 ∈ ∈ PROPN iajs-2513 122 14 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 122 15 𝑓 𝑓 DET iajs-2513 122 16 𝐸 𝐸 ADJ iajs-2513 122 17 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 122 18 𝑄 𝑄 PROPN iajs-2513 122 19 or or CCONJ iajs-2513 122 20 𝑟 𝑟 PRON iajs-2513 122 21 𝑄 𝑄 PROPN iajs-2513 122 22 ⊆ ⊆ PROPN iajs-2513 122 23 𝑓 𝑓 DET iajs-2513 122 24 𝐸 𝐸 ADJ iajs-2513 122 25 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 122 26 𝑄 𝑄 PROPN iajs-2513 122 27 . . PUNCT iajs-2513 123 1 hence hence ADV iajs-2513 123 2 𝑓 𝑓 DET iajs-2513 123 3 𝐸 𝐸 PROPN iajs-2513 123 4 be be VERB iajs-2513 123 5 an an DET iajs-2513 123 6 app app PROPN iajs-2513 123 7 - - PUNCT iajs-2513 123 8 qp qp NOUN iajs-2513 123 9 submodule submodule NOUN iajs-2513 123 10 of of ADP iajs-2513 123 11 𝑄. 𝑄. NOUN iajs-2513 123 12 proposition proposition NOUN iajs-2513 123 13 ( ( PUNCT iajs-2513 123 14 22 22 NUM iajs-2513 123 15 ) ) PUNCT iajs-2513 123 16 let let VERB iajs-2513 123 17 𝑓 𝑓 X iajs-2513 123 18 : : PUNCT iajs-2513 123 19 𝑄 𝑄 PROPN iajs-2513 123 20 ⟶ ⟶ NOUN iajs-2513 123 21 𝑄 𝑄 PRON iajs-2513 123 22 be be VERB iajs-2513 123 23 an an DET iajs-2513 123 24 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-2513 123 25 and and CCONJ iajs-2513 123 26 𝐸 𝐸 PROPN iajs-2513 123 27 is be AUX iajs-2513 123 28 an an DET iajs-2513 123 29 app app PROPN iajs-2513 123 30 - - PUNCT iajs-2513 123 31 qp qp NOUN iajs-2513 123 32 submodule submodule NOUN iajs-2513 123 33 of of ADP iajs-2513 123 34 𝑄 𝑄 PROPN iajs-2513 123 35 with with ADP iajs-2513 123 36 ker ker PROPN iajs-2513 123 37 𝑓 𝑓 DET iajs-2513 123 38 ⊆ ⊆ NUM iajs-2513 123 39 𝐸 𝐸 PROPN iajs-2513 123 40 . . PUNCT iajs-2513 124 1 then then ADV iajs-2513 124 2 𝑓 𝑓 X iajs-2513 124 3 𝐸 𝐸 PROPN iajs-2513 124 4 is be AUX iajs-2513 124 5 an an DET iajs-2513 124 6 app app PROPN iajs-2513 124 7 - - PUNCT iajs-2513 124 8 qp qp NOUN iajs-2513 124 9 submodule submodule NOUN iajs-2513 124 10 of of ADP iajs-2513 124 11 𝑄 𝑄 PROPN iajs-2513 124 12 . . PUNCT iajs-2513 125 1 proof proof NOUN iajs-2513 125 2 𝑓 𝑓 DET iajs-2513 125 3 𝐸 𝐸 PROPN iajs-2513 125 4 is be AUX iajs-2513 125 5 a a DET iajs-2513 125 6 proper proper ADJ iajs-2513 125 7 submodule submodule NOUN iajs-2513 125 8 of of ADP iajs-2513 125 9 𝑄 𝑄 PROPN iajs-2513 125 10 . . PUNCT iajs-2513 126 1 if if SCONJ iajs-2513 126 2 not not PART iajs-2513 126 3 , , PUNCT iajs-2513 126 4 that that PRON iajs-2513 126 5 is be AUX iajs-2513 126 6 𝑓 𝑓 DET iajs-2513 126 7 𝐸 𝐸 ADJ iajs-2513 126 8 𝑄 𝑄 PROPN iajs-2513 126 9 . . PUNCT iajs-2513 127 1 let let VERB iajs-2513 127 2 𝑞 𝑞 PRON iajs-2513 127 3 ∈ ∈ VERB iajs-2513 127 4 𝑄 𝑄 PROPN iajs-2513 127 5 , , PUNCT iajs-2513 127 6 then then ADV iajs-2513 127 7 𝑓 𝑓 PRON iajs-2513 127 8 𝑞 𝑞 X iajs-2513 127 9 ∈ ∈ PROPN iajs-2513 127 10 𝑄 𝑄 PROPN iajs-2513 127 11 𝑓 𝑓 DET iajs-2513 127 12 𝐸 𝐸 PROPN iajs-2513 127 13 , , PUNCT iajs-2513 127 14 so so CCONJ iajs-2513 127 15 there there PRON iajs-2513 127 16 exists exist VERB iajs-2513 127 17 𝑥 𝑥 DET iajs-2513 127 18 ∈ ∈ PROPN iajs-2513 127 19 𝐸 𝐸 PROPN iajs-2513 127 20 such such ADJ iajs-2513 127 21 that that SCONJ iajs-2513 127 22 𝑓 𝑓 DET iajs-2513 127 23 𝑞 𝑞 X iajs-2513 127 24 𝑓 𝑓 PRON iajs-2513 127 25 𝑥 𝑥 PROPN iajs-2513 127 26 , , PUNCT iajs-2513 127 27 implies imply VERB iajs-2513 127 28 that that SCONJ iajs-2513 127 29 𝑓 𝑓 DET iajs-2513 127 30 𝑞 𝑞 X iajs-2513 127 31 𝑥 𝑥 PROPN iajs-2513 127 32 0 0 NUM iajs-2513 127 33 , , PUNCT iajs-2513 127 34 that that PRON iajs-2513 127 35 is be AUX iajs-2513 127 36 𝑞 𝑞 X iajs-2513 127 37 𝑥 𝑥 X iajs-2513 127 38 ∈ ∈ PROPN iajs-2513 128 1 𝐹𝑒𝑟 𝐹𝑒𝑟 PROPN iajs-2513 128 2 𝑓 𝑓 PROPN iajs-2513 128 3 ⊆ ⊆ NUM iajs-2513 128 4 𝐸 𝐸 PROPN iajs-2513 128 5 , , PUNCT iajs-2513 128 6 it it PRON iajs-2513 128 7 follows follow VERB iajs-2513 128 8 that that SCONJ iajs-2513 128 9 𝑞 𝑞 PROPN iajs-2513 128 10 ∈ ∈ PROPN iajs-2513 128 11 𝐸. 𝐸. PROPN iajs-2513 128 12 thus thus ADV iajs-2513 128 13 , , PUNCT iajs-2513 128 14     SPACE iajs-2513 128 15 𝐸 𝐸 NOUN iajs-2513 128 16 𝑄 𝑄 PROPN iajs-2513 128 17 contradiction contradiction NOUN iajs-2513 128 18 . . PUNCT iajs-2513 129 1 now now ADV iajs-2513 129 2 suppose suppose VERB iajs-2513 129 3 that that SCONJ iajs-2513 129 4 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 129 5 ∈ ∈ PROPN iajs-2513 129 6 𝑓 𝑓 DET iajs-2513 129 7 𝐸 𝐸 PROPN iajs-2513 129 8 , , PUNCT iajs-2513 129 9 for for ADP iajs-2513 129 10 𝑟 𝑟 DET iajs-2513 129 11 ∈ ∈ PROPN iajs-2513 129 12 𝑅 𝑅 PROPN iajs-2513 129 13 , , PUNCT iajs-2513 129 14 𝑞 𝑞 X iajs-2513 129 15 ∈ ∈ PROPN iajs-2513 129 16 𝑄 𝑄 PROPN iajs-2513 129 17 , , PUNCT iajs-2513 129 18 𝑓 𝑓 PRON iajs-2513 129 19 𝑞 𝑞 X iajs-2513 129 20 𝑞 𝑞 NOUN iajs-2513 129 21 for for ADP iajs-2513 129 22 some some DET iajs-2513 129 23 𝑞 𝑞 X iajs-2513 129 24 ∈ ∈ PROPN iajs-2513 129 25 𝑄 𝑄 PROPN iajs-2513 129 26 ( ( PUNCT iajs-2513 129 27 since since SCONJ iajs-2513 129 28 𝑓 𝑓 PRON iajs-2513 129 29 is be AUX iajs-2513 129 30 onto onto ADP iajs-2513 129 31 ) ) PUNCT iajs-2513 129 32 , , PUNCT iajs-2513 129 33 that that PRON iajs-2513 129 34 is be AUX iajs-2513 129 35 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 129 36 𝑟𝑓 𝑟𝑓 X iajs-2513 129 37 𝑞 𝑞 X iajs-2513 129 38 𝑓 𝑓 DET iajs-2513 129 39 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 129 40 ∈ ∈ PROPN iajs-2513 129 41 𝑓 𝑓 DET iajs-2513 129 42 𝐸 𝐸 PROPN iajs-2513 129 43 , , PUNCT iajs-2513 129 44 it it PRON iajs-2513 129 45 follows follow VERB iajs-2513 129 46 that that SCONJ iajs-2513 129 47 there there PRON iajs-2513 129 48 exists exist VERB iajs-2513 129 49 𝑒 𝑒 PROPN iajs-2513 129 50 ∈ ∈ PROPN iajs-2513 129 51 𝐸 𝐸 PROPN iajs-2513 129 52 such such ADJ iajs-2513 129 53 that that SCONJ iajs-2513 129 54 𝑓 𝑓 PRON iajs-2513 129 55 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 129 56 𝑓 𝑓 PRON iajs-2513 129 57 𝑒 𝑒 PROPN iajs-2513 129 58 , , PUNCT iajs-2513 129 59 that that PRON iajs-2513 129 60 is be AUX iajs-2513 129 61 𝑓 𝑓 PRON iajs-2513 129 62 𝑒 𝑒 NOUN iajs-2513 129 63 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 129 64 0 0 NUM iajs-2513 129 65 , , PUNCT iajs-2513 129 66 so so ADV iajs-2513 129 67 𝑒 𝑒 ADJ iajs-2513 129 68 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 129 69 ∈ ∈ PROPN iajs-2513 129 70 𝑘𝑒𝑟 𝑘𝑒𝑟 NOUN iajs-2513 129 71 𝑓 𝑓 DET iajs-2513 129 72 ⊆ ⊆ NUM iajs-2513 129 73 𝐸 𝐸 PROPN iajs-2513 129 74 , , PUNCT iajs-2513 129 75 implies imply VERB iajs-2513 129 76 that that SCONJ iajs-2513 129 77 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 129 78 ∈ ∈ NOUN iajs-2513 129 79 𝐸. 𝐸. PROPN iajs-2513 129 80 but but CCONJ iajs-2513 129 81 𝐸 𝐸 PROPN iajs-2513 129 82 is be AUX iajs-2513 129 83 an an DET iajs-2513 129 84 app app PROPN iajs-2513 129 85 - - PUNCT iajs-2513 129 86 qp qp NOUN iajs-2513 129 87 submodule submodule NOUN iajs-2513 129 88 of of ADP iajs-2513 129 89 𝑄 𝑄 PROPN iajs-2513 129 90 , , PUNCT iajs-2513 129 91 then then ADV iajs-2513 129 92 either either CCONJ iajs-2513 129 93 𝑞 𝑞 PROPN iajs-2513 129 94 ∈ ∈ PROPN iajs-2513 129 95 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 129 96 𝐸 𝐸 PROPN iajs-2513 129 97 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 129 98 𝑄 𝑄 PROPN iajs-2513 129 99 or or CCONJ iajs-2513 129 100 𝑟 𝑟 PRON iajs-2513 129 101 𝑄 𝑄 PROPN iajs-2513 129 102 ⊆ ⊆ NUM iajs-2513 129 103 𝐸 𝐸 PROPN iajs-2513 129 104 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 129 105 𝑄 𝑄 PROPN iajs-2513 129 106 for for ADP iajs-2513 129 107 some some DET iajs-2513 129 108 𝑛 𝑛 DET iajs-2513 129 109 ∈ ∈ PROPN iajs-2513 129 110 𝑍 𝑍 NOUN iajs-2513 129 111 . . PUNCT iajs-2513 130 1 hence hence ADV iajs-2513 130 2 , , PUNCT iajs-2513 130 3 by by ADP iajs-2513 130 4 using use VERB iajs-2513 130 5 lemma lemma PROPN iajs-2513 130 6 ( ( PUNCT iajs-2513 130 7 20 20 NUM iajs-2513 130 8 ) ) PUNCT iajs-2513 130 9 either either CCONJ iajs-2513 130 10 𝑞 𝑞 X iajs-2513 130 11 𝑓 𝑓 PROPN iajs-2513 130 12 𝑞 𝑞 X iajs-2513 130 13 ∈ ∈ PROPN iajs-2513 130 14 𝑓 𝑓 PRON iajs-2513 130 15 𝑟𝑎𝑑 𝑟𝑎𝑑 NOUN iajs-2513 130 16 𝐸 𝐸 PROPN iajs-2513 130 17 𝑓 𝑓 DET iajs-2513 130 18 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 130 19 𝑄 𝑄 PROPN iajs-2513 130 20 ⊆ ⊆ PROPN iajs-2513 130 21 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 130 22 𝑓 𝑓 DET iajs-2513 130 23 𝐸 𝐸 NOUN iajs-2513 130 24 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 130 25 𝑄 𝑄 PROPN iajs-2513 130 26 or or CCONJ iajs-2513 130 27 𝑟 𝑟 PRON iajs-2513 130 28 𝑄 𝑄 ADP iajs-2513 130 29 𝑟 𝑟 NOUN iajs-2513 130 30 𝑓 𝑓 DET iajs-2513 130 31 𝑄 𝑄 PROPN iajs-2513 130 32 ⊆ ⊆ PROPN iajs-2513 130 33 𝑓 𝑓 PRON iajs-2513 130 34 𝐸 𝐸 ADJ iajs-2513 131 1 𝑓 𝑓 PRON iajs-2513 131 2 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 131 3 𝑄 𝑄 PROPN iajs-2513 131 4 ⊆ ⊆ PROPN iajs-2513 131 5 𝑓 𝑓 DET iajs-2513 131 6 𝐸 𝐸 ADJ iajs-2513 131 7 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 131 8 𝑄 𝑄 PROPN iajs-2513 131 9 . . PUNCT iajs-2513 132 1 thus thus ADV iajs-2513 132 2 𝑓 𝑓 DET iajs-2513 132 3 𝐸 𝐸 PROPN iajs-2513 132 4 is be AUX iajs-2513 132 5 an an DET iajs-2513 132 6 app app PROPN iajs-2513 132 7 - - PUNCT iajs-2513 132 8 qp qp NOUN iajs-2513 132 9 submodule submodule NOUN iajs-2513 132 10 of of ADP iajs-2513 132 11 𝑄 𝑄 PROPN iajs-2513 132 12 . . PUNCT iajs-2513 133 1 remark remark NOUN iajs-2513 133 2 ( ( PUNCT iajs-2513 133 3 23 23 NUM iajs-2513 133 4 ) ) PUNCT iajs-2513 133 5 the the DET iajs-2513 133 6 intersection intersection NOUN iajs-2513 133 7 of of ADP iajs-2513 133 8 two two NUM iajs-2513 133 9 app app ADJ iajs-2513 133 10 - - PUNCT iajs-2513 133 11 qp qp NOUN iajs-2513 133 12 submodules submodule NOUN iajs-2513 133 13 of of ADP iajs-2513 133 14 an an DET iajs-2513 133 15 𝑅-module 𝑅-module PROPN iajs-2513 133 16 𝑄 𝑄 PRON iajs-2513 133 17 need need VERB iajs-2513 133 18 not not PART iajs-2513 133 19 to to PART iajs-2513 133 20 be be AUX iajs-2513 133 21 an an DET iajs-2513 133 22 app app ADJ iajs-2513 133 23 - - PUNCT iajs-2513 133 24 qp qp NOUN iajs-2513 133 25 submodule submodule NOUN iajs-2513 133 26 of of ADP iajs-2513 133 27 𝑄.the 𝑄.the DET iajs-2513 133 28 following following ADJ iajs-2513 133 29 example example NOUN iajs-2513 133 30 explains explain VERB iajs-2513 133 31 that that SCONJ iajs-2513 133 32 : : PUNCT iajs-2513 133 33     SPACE iajs-2513 133 34 99 99 NUM iajs-2513 133 35     SPACE iajs-2513 133 36 ibn ibn PROPN iajs-2513 133 37 al al PROPN iajs-2513 133 38 - - PUNCT iajs-2513 133 39 haitham haitham PROPN iajs-2513 133 40 jour jour X iajs-2513 133 41 . . PROPN iajs-2513 134 1 for for ADP iajs-2513 134 2 pure pure ADJ iajs-2513 134 3 & & CCONJ iajs-2513 134 4 appl appl PROPN iajs-2513 134 5 . . PUNCT iajs-2513 135 1 sci sci PROPN iajs-2513 135 2 . . PROPN iajs-2513 136 1 33 33 NUM iajs-2513 136 2 ( ( PUNCT iajs-2513 136 3 4 4 NUM iajs-2513 136 4 ) ) PUNCT iajs-2513 136 5 2020 2020 NUM iajs-2513 136 6 consider consider VERB iajs-2513 136 7 the the DET iajs-2513 136 8 𝑍-module 𝑍-module PROPN iajs-2513 136 9 𝑍 𝑍 NOUN iajs-2513 136 10 and and CCONJ iajs-2513 136 11 the the DET iajs-2513 136 12 submodules submodule NOUN iajs-2513 136 13 2𝑍 2𝑍 NOUN iajs-2513 136 14 , , PUNCT iajs-2513 136 15 3𝑍 3𝑍 PROPN iajs-2513 136 16 are be AUX iajs-2513 136 17 app app ADJ iajs-2513 136 18 - - PUNCT iajs-2513 136 19 qp qp NOUN iajs-2513 136 20 submodules submodule NOUN iajs-2513 136 21 of of ADP iajs-2513 136 22 𝑍-modules 𝑍-modules PROPN iajs-2513 136 23 𝑍 𝑍 PROPN iajs-2513 136 24 ( ( PUNCT iajs-2513 136 25 because because SCONJ iajs-2513 136 26 they they PRON iajs-2513 136 27 are be AUX iajs-2513 136 28 prime prime ADJ iajs-2513 136 29 ) ) PUNCT iajs-2513 137 1 but but CCONJ iajs-2513 137 2 2𝑍 2𝑍 PROPN iajs-2513 137 3 ∩ ∩ PROPN iajs-2513 137 4 3𝑍 3𝑍 PROPN iajs-2513 137 5 6𝑍 6𝑍 NOUN iajs-2513 137 6 is be AUX iajs-2513 137 7 not not PART iajs-2513 137 8 app app ADJ iajs-2513 137 9 - - PUNCT iajs-2513 137 10 qp qp NOUN iajs-2513 137 11 submodule submodule NOUN iajs-2513 137 12 of of ADP iajs-2513 137 13 𝑍-module 𝑍-module PROPN iajs-2513 137 14 𝑍 𝑍 NOUN iajs-2513 137 15 , , PUNCT iajs-2513 137 16 since since SCONJ iajs-2513 137 17 2.3 2.3 NUM iajs-2513 137 18 ∈ ∈ NOUN iajs-2513 137 19 6𝑍 6𝑍 NOUN iajs-2513 137 20 , , PUNCT iajs-2513 137 21 but but CCONJ iajs-2513 137 22 3 3 NUM iajs-2513 137 23 ∉ ∉ NOUN iajs-2513 137 24 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 137 25 6𝑍 6𝑍 NOUN iajs-2513 137 26 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 137 27 𝑍 𝑍 PROPN iajs-2513 137 28 6𝑍 6𝑍 NOUN iajs-2513 137 29 0 0 NUM iajs-2513 137 30 6𝑍 6𝑍 NOUN iajs-2513 137 31 and and CCONJ iajs-2513 137 32 2 2 NUM iajs-2513 137 33 ∉ ∉ ADJ iajs-2513 137 34 6𝑍 6𝑍 NOUN iajs-2513 137 35 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 137 36 𝑍 𝑍 PROPN iajs-2513 137 37 : : PUNCT iajs-2513 137 38 𝑍 𝑍 PROPN iajs-2513 137 39 6𝑍 6𝑍 NOUN iajs-2513 137 40 : : PUNCT iajs-2513 137 41 𝑍 𝑍 VERB iajs-2513 137 42 √6𝑍 √6𝑍 X iajs-2513 137 43 6𝑍. 6𝑍. NOUN iajs-2513 137 44 we we PRON iajs-2513 137 45 need need VERB iajs-2513 137 46 the the DET iajs-2513 137 47 following follow VERB iajs-2513 137 48 lemma lemma PROPN iajs-2513 137 49 : : PUNCT iajs-2513 137 50 lemma lemma PROPN iajs-2513 137 51 ( ( PUNCT iajs-2513 137 52 24 24 NUM iajs-2513 137 53 ) ) PUNCT iajs-2513 138 1 [ [ X iajs-2513 138 2 15 15 NUM iajs-2513 138 3 , , PUNCT iajs-2513 138 4 theo theo PROPN iajs-2513 138 5 . . PUNCT iajs-2513 138 6 15(3 15(3 NUM iajs-2513 138 7 ) ) PUNCT iajs-2513 138 8 ] ] PUNCT iajs-2513 138 9 let let VERB iajs-2513 138 10 𝑄 𝑄 PRON iajs-2513 138 11 be be AUX iajs-2513 138 12 a a DET iajs-2513 138 13 multiplication multiplication NOUN iajs-2513 138 14 𝑅-module 𝑅-module PROPN iajs-2513 138 15 and and CCONJ iajs-2513 138 16 𝐸 𝐸 PROPN iajs-2513 138 17 , , PUNCT iajs-2513 138 18 𝐹 𝐹 PRON iajs-2513 138 19 be be VERB iajs-2513 138 20 a a DET iajs-2513 138 21 submodules submodule NOUN iajs-2513 138 22 of of ADP iajs-2513 138 23 𝑄. 𝑄. NOUN iajs-2513 138 24 then then ADV iajs-2513 138 25 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 138 26 𝐸 𝐸 PROPN iajs-2513 138 27 ∩ ∩ NOUN iajs-2513 138 28 𝐹 𝐹 PROPN iajs-2513 138 29 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 138 30 𝐸 𝐸 PROPN iajs-2513 138 31 ∩ ∩ NOUN iajs-2513 138 32 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 138 33 𝐹 𝐹 PROPN iajs-2513 138 34 . . PUNCT iajs-2513 139 1 proposition proposition NOUN iajs-2513 139 2 ( ( PUNCT iajs-2513 139 3 25 25 NUM iajs-2513 139 4 ) ) PUNCT iajs-2513 139 5 let let VERB iajs-2513 139 6 𝐸 𝐸 PRON iajs-2513 139 7 and and CCONJ iajs-2513 139 8 𝐹 𝐹 PRON iajs-2513 139 9 be be VERB iajs-2513 139 10 a a DET iajs-2513 139 11 proper proper ADJ iajs-2513 139 12 submodules submodule NOUN iajs-2513 139 13 of of ADP iajs-2513 139 14 multiplication multiplication NOUN iajs-2513 139 15 𝑅-module 𝑅-module ADP iajs-2513 139 16 𝑄 𝑄 PRON iajs-2513 139 17 with with ADP iajs-2513 139 18 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 139 19 𝑄 𝑄 PROPN iajs-2513 139 20 ⊆ ⊆ PROPN iajs-2513 139 21 𝐸 𝐸 PROPN iajs-2513 139 22 or or CCONJ iajs-2513 139 23 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 139 24 𝑄 𝑄 PROPN iajs-2513 139 25 ⊆ ⊆ NUM iajs-2513 139 26 𝐹. 𝐹. NOUN iajs-2513 139 27 if if SCONJ iajs-2513 139 28 𝐸 𝐸 PROPN iajs-2513 139 29 and and CCONJ iajs-2513 139 30 𝐹 𝐹 PROPN iajs-2513 139 31 are be AUX iajs-2513 139 32 app app ADJ iajs-2513 139 33 - - PUNCT iajs-2513 139 34 qp qp NOUN iajs-2513 139 35 submodules submodule NOUN iajs-2513 139 36 of of ADP iajs-2513 139 37 𝑄 𝑄 PROPN iajs-2513 139 38 , , PUNCT iajs-2513 139 39 then then ADV iajs-2513 139 40 𝐸 𝐸 PROPN iajs-2513 139 41 ∩ ∩ NOUN iajs-2513 139 42 𝐹 𝐹 PRON iajs-2513 139 43 is be AUX iajs-2513 139 44 an an DET iajs-2513 139 45 app app PROPN iajs-2513 139 46 - - PUNCT iajs-2513 139 47 qp qp NOUN iajs-2513 139 48 submodule submodule NOUN iajs-2513 139 49 of of ADP iajs-2513 139 50 𝑄. 𝑄. PROPN iajs-2513 139 51 proof proof NOUN iajs-2513 139 52 suppose suppose VERB iajs-2513 139 53 𝑟𝑞 𝑟𝑞 SCONJ iajs-2513 139 54 ∈ ∈ PROPN iajs-2513 139 55 𝐸 𝐸 PROPN iajs-2513 139 56 ∩ ∩ NOUN iajs-2513 139 57 𝐹 𝐹 PROPN iajs-2513 139 58 for for ADP iajs-2513 139 59 𝑟,∈ 𝑟,∈ PROPN iajs-2513 139 60 𝑅 𝑅 PROPN iajs-2513 139 61 , , PUNCT iajs-2513 139 62 𝑞 𝑞 X iajs-2513 139 63 ∈ ∈ PROPN iajs-2513 139 64 𝑄 𝑄 PROPN iajs-2513 139 65 , , PUNCT iajs-2513 139 66 then then ADV iajs-2513 139 67 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 139 68 ∈ ∈ PROPN iajs-2513 139 69 𝐸 𝐸 PROPN iajs-2513 139 70 and and CCONJ iajs-2513 139 71 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 139 72 ∈ ∈ PROPN iajs-2513 139 73 𝐹. 𝐹. PROPN iajs-2513 139 74 but but CCONJ iajs-2513 139 75 both both DET iajs-2513 139 76 𝐸 𝐸 PROPN iajs-2513 139 77 and and CCONJ iajs-2513 139 78 𝐹 𝐹 PROPN iajs-2513 139 79 are be AUX iajs-2513 139 80 app app ADJ iajs-2513 139 81 - - PUNCT iajs-2513 139 82 qp qp NOUN iajs-2513 139 83 submodules submodule NOUN iajs-2513 139 84 of of ADP iajs-2513 139 85 𝑄 𝑄 PROPN iajs-2513 139 86 , , PUNCT iajs-2513 139 87 then then ADV iajs-2513 139 88 either either CCONJ iajs-2513 139 89 𝑞 𝑞 PROPN iajs-2513 139 90 ∈ ∈ PROPN iajs-2513 139 91 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 139 92 𝐸 𝐸 PROPN iajs-2513 139 93 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 139 94 𝑄 𝑄 PROPN iajs-2513 139 95 or or CCONJ iajs-2513 139 96 𝑟 𝑟 PRON iajs-2513 139 97 𝑄 𝑄 PROPN iajs-2513 139 98 ⊆ ⊆ NUM iajs-2513 139 99 𝐸 𝐸 PROPN iajs-2513 139 100 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 139 101 𝑄 𝑄 PROPN iajs-2513 139 102 and and CCONJ iajs-2513 139 103 either either CCONJ iajs-2513 139 104 𝑞 𝑞 PROPN iajs-2513 139 105 ∈ ∈ PROPN iajs-2513 139 106 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 139 107 𝐹 𝐹 PROPN iajs-2513 139 108 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 139 109 𝑄 𝑄 PROPN iajs-2513 139 110 or or CCONJ iajs-2513 139 111 𝑟 𝑟 NOUN iajs-2513 139 112 𝑄 𝑄 PROPN iajs-2513 139 113 ⊆ ⊆ NUM iajs-2513 139 114 𝐹 𝐹 PROPN iajs-2513 139 115 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 139 116 𝑄 𝑄 PROPN iajs-2513 139 117 for for ADP iajs-2513 139 118 some some DET iajs-2513 139 119 𝑛 𝑛 DET iajs-2513 139 120 ∈ ∈ PROPN iajs-2513 139 121 𝑍 𝑍 NOUN iajs-2513 139 122 . . PUNCT iajs-2513 140 1 hence hence ADV iajs-2513 140 2 either either CCONJ iajs-2513 140 3 𝑞 𝑞 PROPN iajs-2513 140 4 ∈ ∈ PROPN iajs-2513 140 5 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 140 6 𝐸 𝐸 PROPN iajs-2513 140 7 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 140 8 𝑄 𝑄 PROPN iajs-2513 140 9 ∩ ∩ NOUN iajs-2513 140 10 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 140 11 𝐹 𝐹 PROPN iajs-2513 140 12 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 140 13 𝑄 𝑄 PROPN iajs-2513 140 14 or or CCONJ iajs-2513 140 15 𝑟 𝑟 PRON iajs-2513 140 16 𝑄 𝑄 PROPN iajs-2513 140 17 ⊆ ⊆ NUM iajs-2513 140 18 𝐸 𝐸 PROPN iajs-2513 140 19 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 140 20 𝑄 𝑄 PROPN iajs-2513 140 21 ∩ ∩ NOUN iajs-2513 140 22 𝐹 𝐹 PROPN iajs-2513 140 23 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 140 24 𝑄 𝑄 PROPN iajs-2513 140 25 . . PUNCT iajs-2513 141 1 if if SCONJ iajs-2513 141 2 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 141 3 𝑄 𝑄 PROPN iajs-2513 141 4 ⊆ ⊆ NUM iajs-2513 141 5 𝐹 𝐹 PROPN iajs-2513 141 6 ⊆ ⊆ NUM iajs-2513 141 7 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 141 8 𝐸 𝐸 PROPN iajs-2513 141 9 , , PUNCT iajs-2513 141 10 then then ADV iajs-2513 141 11 𝐹 𝐹 PROPN iajs-2513 141 12 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 141 13 𝑄 𝑄 PROPN iajs-2513 141 14 𝐹 𝐹 PROPN iajs-2513 141 15 and and CCONJ iajs-2513 141 16 𝑟𝑎𝑑 𝑟𝑎𝑑 INTJ iajs-2513 141 17 𝐹 𝐹 PROPN iajs-2513 141 18 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 141 19 𝑄 𝑄 PROPN iajs-2513 141 20 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 141 21 𝐹 𝐹 PROPN iajs-2513 141 22 . . PUNCT iajs-2513 142 1 thus thus ADV iajs-2513 142 2 either either CCONJ iajs-2513 142 3 𝑞 𝑞 PROPN iajs-2513 142 4 ∈ ∈ PROPN iajs-2513 142 5 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 142 6 𝐸 𝐸 PROPN iajs-2513 142 7 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 142 8 𝑇 𝑇 PROPN iajs-2513 142 9 ∩ ∩ PROPN iajs-2513 142 10 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 142 11 𝐹 𝐹 PROPN iajs-2513 142 12 or or CCONJ iajs-2513 142 13 𝑟 𝑟 PRON iajs-2513 142 14 𝑄 𝑄 PROPN iajs-2513 142 15 ⊆ ⊆ NUM iajs-2513 142 16 𝐸 𝐸 PROPN iajs-2513 142 17 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 142 18 𝑄 𝑄 PROPN iajs-2513 142 19 ∩ ∩ NOUN iajs-2513 142 20 𝐹. 𝐹. PROPN iajs-2513 142 21 it it PRON iajs-2513 142 22 follows follow VERB iajs-2513 142 23 that that SCONJ iajs-2513 142 24 by by ADP iajs-2513 142 25 modular modular ADJ iajs-2513 142 26 law law NOUN iajs-2513 142 27 either either CCONJ iajs-2513 142 28 𝑞 𝑞 PROPN iajs-2513 142 29 ∈ ∈ PROPN iajs-2513 142 30 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 142 31 𝐸 𝐸 PROPN iajs-2513 142 32 ∩ ∩ NOUN iajs-2513 142 33 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 142 34 𝐹 𝐹 PROPN iajs-2513 142 35 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 142 36 𝑄 𝑄 PROPN iajs-2513 142 37 or or CCONJ iajs-2513 142 38 𝑟 𝑟 PRON iajs-2513 142 39 𝑄 𝑄 PROPN iajs-2513 142 40 ⊆ ⊆ NUM iajs-2513 142 41 𝐸 𝐸 PROPN iajs-2513 142 42 ∩ ∩ NOUN iajs-2513 142 43 𝐹 𝐹 PROPN iajs-2513 142 44 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 142 45 𝑄 𝑄 PROPN iajs-2513 142 46 . . PUNCT iajs-2513 143 1 hence hence ADV iajs-2513 143 2 by by ADP iajs-2513 143 3 lemma lemma PROPN iajs-2513 143 4 ( ( PUNCT iajs-2513 143 5 24 24 NUM iajs-2513 143 6 ) ) PUNCT iajs-2513 143 7 either either CCONJ iajs-2513 143 8 𝑞 𝑞 PROPN iajs-2513 143 9 ∈ ∈ PROPN iajs-2513 143 10 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 143 11 𝐸 𝐸 PROPN iajs-2513 143 12 ∩ ∩ NOUN iajs-2513 143 13 𝐹 𝐹 PROPN iajs-2513 143 14 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 143 15 𝑄 𝑄 PROPN iajs-2513 143 16 or or CCONJ iajs-2513 143 17 𝑟 𝑟 PRON iajs-2513 143 18 𝑄 𝑄 PROPN iajs-2513 143 19 ⊆ ⊆ NUM iajs-2513 143 20 𝐸 𝐸 PROPN iajs-2513 143 21 ∩ ∩ NOUN iajs-2513 143 22 𝐹 𝐹 PROPN iajs-2513 143 23 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 143 24 𝑄 𝑄 PROPN iajs-2513 143 25 for for ADP iajs-2513 143 26 some some DET iajs-2513 143 27 𝑛 𝑛 DET iajs-2513 143 28 ∈ ∈ PROPN iajs-2513 143 29 𝑍 𝑍 NOUN iajs-2513 143 30 . . PUNCT iajs-2513 144 1 thus thus ADV iajs-2513 144 2 𝐸 𝐸 PROPN iajs-2513 144 3 ∩ ∩ NOUN iajs-2513 144 4 𝐹 𝐹 PRON iajs-2513 144 5 is be AUX iajs-2513 144 6 an an DET iajs-2513 144 7 app app PROPN iajs-2513 144 8 - - PUNCT iajs-2513 144 9 qp qp NOUN iajs-2513 144 10 submodule submodule NOUN iajs-2513 144 11 of of ADP iajs-2513 144 12 𝑄. 𝑄. NOUN iajs-2513 144 13 similarly similarly ADV iajs-2513 144 14 if if SCONJ iajs-2513 144 15 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 144 16 𝑄 𝑄 PROPN iajs-2513 144 17 ⊆ ⊆ PROPN iajs-2513 144 18 𝐸 𝐸 PROPN iajs-2513 144 19 , , PUNCT iajs-2513 144 20 we we PRON iajs-2513 144 21 got get VERB iajs-2513 144 22 𝐸 𝐸 PROPN iajs-2513 144 23 ∩ ∩ NOUN iajs-2513 144 24 𝐹 𝐹 PRON iajs-2513 144 25 is be AUX iajs-2513 144 26 an an DET iajs-2513 144 27 app app PROPN iajs-2513 144 28 - - PUNCT iajs-2513 144 29 qp qp NOUN iajs-2513 144 30 submodule submodule NOUN iajs-2513 144 31 of of ADP iajs-2513 144 32 𝑄. 𝑄. NOUN iajs-2513 144 33 proposition proposition NOUN iajs-2513 144 34 ( ( PUNCT iajs-2513 144 35 26 26 NUM iajs-2513 144 36 ) ) PUNCT iajs-2513 144 37 let let VERB iajs-2513 144 38 𝑄 𝑄 PRON iajs-2513 144 39 𝑄 𝑄 PROPN iajs-2513 144 40 ⊕ ⊕ PROPN iajs-2513 144 41 𝑄 𝑄 PRON iajs-2513 144 42 be be VERB iajs-2513 144 43 an an DET iajs-2513 144 44 𝑅-module 𝑅-module PROPN iajs-2513 144 45 , , PUNCT iajs-2513 144 46 where where SCONJ iajs-2513 144 47 𝑄 𝑄 PRON iajs-2513 144 48 , , PUNCT iajs-2513 144 49 𝑄 𝑄 PRON iajs-2513 144 50 are be AUX iajs-2513 144 51 𝑅-modules 𝑅-modules PROPN iajs-2513 144 52 , , PUNCT iajs-2513 144 53 and and CCONJ iajs-2513 144 54 𝐸 𝐸 PROPN iajs-2513 144 55 𝐸 𝐸 PROPN iajs-2513 144 56 ⊕ ⊕ PROPN iajs-2513 144 57 𝐸 𝐸 PROPN iajs-2513 144 58 be be VERB iajs-2513 144 59 a a DET iajs-2513 144 60 submodule submodule NOUN iajs-2513 144 61 of of ADP iajs-2513 144 62 𝑄 𝑄 PROPN iajs-2513 144 63 , , PUNCT iajs-2513 144 64 with with ADP iajs-2513 144 65 𝐸 𝐸 PROPN iajs-2513 144 66 , , PUNCT iajs-2513 144 67 𝐸 𝐸 PROPN iajs-2513 144 68 are be AUX iajs-2513 144 69 submodules submodule NOUN iajs-2513 144 70 of of ADP iajs-2513 144 71 𝑄 𝑄 PRON iajs-2513 144 72 , , PUNCT iajs-2513 144 73 𝑄 𝑄 PROPN iajs-2513 144 74 respectively respectively ADV iajs-2513 144 75 with with ADP iajs-2513 144 76 𝑟𝑎𝑑 𝑟𝑎𝑑 DET iajs-2513 144 77 𝐸 𝐸 PROPN iajs-2513 144 78 ⊆ ⊆ NUM iajs-2513 144 79 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 144 80 𝑄 𝑄 PROPN iajs-2513 144 81 . . PUNCT iajs-2513 145 1 if if SCONJ iajs-2513 145 2 𝐸 𝐸 PROPN iajs-2513 145 3 is be AUX iajs-2513 145 4 an an DET iajs-2513 145 5 app app PROPN iajs-2513 145 6 - - PUNCT iajs-2513 145 7 qp qp NOUN iajs-2513 145 8 submodule submodule NOUN iajs-2513 145 9 of of ADP iajs-2513 145 10 𝑄 𝑄 PROPN iajs-2513 145 11 , , PUNCT iajs-2513 145 12 then then ADV iajs-2513 145 13 𝐸 𝐸 PROPN iajs-2513 145 14 is be AUX iajs-2513 145 15 an an DET iajs-2513 145 16 app app PROPN iajs-2513 145 17 - - PUNCT iajs-2513 145 18 qp qp NOUN iajs-2513 145 19 submodule submodule NOUN iajs-2513 145 20 of of ADP iajs-2513 145 21 𝑄 𝑄 PROPN iajs-2513 145 22 and and CCONJ iajs-2513 145 23 𝐸 𝐸 PROPN iajs-2513 145 24 is be AUX iajs-2513 145 25 an an DET iajs-2513 145 26 appqp appqp ADJ iajs-2513 145 27 submodule submodule NOUN iajs-2513 145 28 of of ADP iajs-2513 145 29 𝑄 𝑄 PROPN iajs-2513 145 30 . . PUNCT iajs-2513 146 1 proof proof NOUN iajs-2513 146 2 let let VERB iajs-2513 146 3 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 146 4 ∈ ∈ PROPN iajs-2513 146 5 𝐸 𝐸 PROPN iajs-2513 146 6 , , PUNCT iajs-2513 146 7 for for ADP iajs-2513 146 8 𝑟 𝑟 DET iajs-2513 146 9 ∈ ∈ PROPN iajs-2513 146 10 𝑅 𝑅 PROPN iajs-2513 146 11 , , PUNCT iajs-2513 146 12 𝑞 𝑞 X iajs-2513 146 13 ∈ ∈ PROPN iajs-2513 146 14 𝑄 𝑄 PROPN iajs-2513 146 15 , , PUNCT iajs-2513 147 1 then then ADV iajs-2513 147 2 𝑟 𝑟 DET iajs-2513 147 3 𝑞 𝑞 X iajs-2513 147 4 , , PUNCT iajs-2513 147 5 0 0 NUM iajs-2513 147 6 ∈ ∈ NOUN iajs-2513 147 7 𝐸. 𝐸. NOUN iajs-2513 147 8 since since SCONJ iajs-2513 147 9 𝐸 𝐸 PROPN iajs-2513 147 10 is be AUX iajs-2513 147 11 an an DET iajs-2513 147 12 app app PROPN iajs-2513 147 13 - - PUNCT iajs-2513 147 14 qp qp NOUN iajs-2513 147 15 submodule submodule NOUN iajs-2513 147 16 of of ADP iajs-2513 147 17 𝑄 𝑄 PROPN iajs-2513 147 18 , , PUNCT iajs-2513 147 19 then then ADV iajs-2513 147 20 𝑞 𝑞 X iajs-2513 147 21 , , PUNCT iajs-2513 147 22 0 0 NUM iajs-2513 147 23 ∈ ∈ PROPN iajs-2513 147 24 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 147 25 𝐸 𝐸 PROPN iajs-2513 147 26 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 147 27 𝑄 𝑄 PROPN iajs-2513 147 28 or or CCONJ iajs-2513 147 29 𝑟 𝑟 PRON iajs-2513 147 30 𝑄 𝑄 PROPN iajs-2513 147 31 ⊆ ⊆ NUM iajs-2513 147 32 𝐸 𝐸 PROPN iajs-2513 147 33 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 147 34 𝑄 𝑄 PROPN iajs-2513 147 35 for for ADP iajs-2513 147 36 some some DET iajs-2513 147 37 𝑛 𝑛 DET iajs-2513 147 38 ∈ ∈ PROPN iajs-2513 147 39 𝑍 𝑍 NOUN iajs-2513 147 40 . . PUNCT iajs-2513 148 1 but but CCONJ iajs-2513 148 2 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 148 3 𝐸 𝐸 PROPN iajs-2513 148 4 ⊆ ⊆ NUM iajs-2513 148 5 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 148 6 𝑄 𝑄 PROPN iajs-2513 148 7 , , PUNCT iajs-2513 148 8 implies imply VERB iajs-2513 148 9 that that SCONJ iajs-2513 148 10 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 148 11 𝐸 𝐸 PROPN iajs-2513 148 12 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 148 13 𝑄 𝑄 PRON iajs-2513 148 14 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 148 15 𝑄 𝑄 PROPN iajs-2513 148 16 , , PUNCT iajs-2513 148 17 and and CCONJ iajs-2513 148 18 𝐸 𝐸 PROPN iajs-2513 148 19 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 148 20 𝑄 𝑄 PRON iajs-2513 148 21 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 148 22 𝑄 𝑄 PROPN iajs-2513 149 1 [ [ X iajs-2513 149 2 since since SCONJ iajs-2513 149 3 𝐸 𝐸 PROPN iajs-2513 149 4 ⊆ ⊆ NUM iajs-2513 149 5 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 149 6 𝐸 𝐸 PROPN iajs-2513 149 7 ⊆ ⊆ NUM iajs-2513 149 8 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 149 9 𝑄 𝑄 PROPN iajs-2513 149 10 . . PUNCT iajs-2513 150 1 it it PRON iajs-2513 150 2 follows follow VERB iajs-2513 150 3 that that SCONJ iajs-2513 150 4 either either CCONJ iajs-2513 150 5 𝑞 𝑞 X iajs-2513 150 6 , , PUNCT iajs-2513 150 7 0 0 NUM iajs-2513 150 8 ∈ ∈ NOUN iajs-2513 150 9 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 150 10 𝑄 𝑄 PRON iajs-2513 150 11 𝑠𝑜𝑐 𝑠𝑜𝑐 AUX iajs-2513 150 12 𝑄 𝑄 PRON iajs-2513 150 13 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 150 14 𝑄 𝑄 PROPN iajs-2513 150 15 ⊕ ⊕ VERB iajs-2513 150 16 𝑄 𝑄 PROPN iajs-2513 150 17 or or CCONJ iajs-2513 150 18 𝑟 𝑟 NOUN iajs-2513 150 19 𝑄 𝑄 PROPN iajs-2513 150 20 ⊕ ⊕ VERB iajs-2513 150 21 𝑄 𝑄 PRON iajs-2513 150 22 ⊆ ⊆ NUM iajs-2513 150 23 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 150 24 𝑄 𝑄 PROPN iajs-2513 150 25 𝑠𝑜𝑐 𝑠𝑜𝑐 AUX iajs-2513 150 26 𝑄 𝑄 PROPN iajs-2513 150 27 ⊕ ⊕ PROPN iajs-2513 150 28 𝑄 𝑄 PROPN iajs-2513 150 29 , , PUNCT iajs-2513 150 30 that that PRON iajs-2513 150 31 is be AUX iajs-2513 150 32 either either CCONJ iajs-2513 150 33 𝑞 𝑞 X iajs-2513 150 34 , , PUNCT iajs-2513 150 35 0 0 NUM iajs-2513 150 36 ∈ ∈ NOUN iajs-2513 150 37 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 150 38 𝑄 𝑄 PROPN iajs-2513 150 39 ⊕ ⊕ PROPN iajs-2513 150 40 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 150 41 𝑄 𝑄 PROPN iajs-2513 150 42 or or CCONJ iajs-2513 150 43 𝑟 𝑟 NOUN iajs-2513 150 44 𝑄 𝑄 PROPN iajs-2513 150 45 ⊕ ⊕ VERB iajs-2513 150 46 𝑄 𝑄 PRON iajs-2513 150 47 ⊆ ⊆ NUM iajs-2513 150 48 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 150 49 𝑄 𝑄 PROPN iajs-2513 150 50 ⊕ ⊕ PROPN iajs-2513 150 51 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 150 52 𝑄 𝑄 PROPN iajs-2513 150 53 , , PUNCT iajs-2513 150 54 hence hence ADV iajs-2513 150 55 either either CCONJ iajs-2513 150 56 𝑞 𝑞 PROPN iajs-2513 150 57 ∈ ∈ PROPN iajs-2513 150 58 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 150 59 𝑄 𝑄 PROPN iajs-2513 150 60 ⊆ ⊆ PROPN iajs-2513 150 61 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 150 62 𝐸 𝐸 PROPN iajs-2513 150 63 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 150 64 𝑄 𝑄 PROPN iajs-2513 150 65 or or CCONJ iajs-2513 150 66 𝑟 𝑟 NOUN iajs-2513 150 67 𝑄 𝑄 PROPN iajs-2513 150 68 ⊆ ⊆ NUM iajs-2513 150 69 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 150 70 𝑄 𝑄 PROPN iajs-2513 150 71 ⊆ ⊆ NUM iajs-2513 150 72 𝐸 𝐸 PROPN iajs-2513 150 73 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 150 74 𝑄 𝑄 PROPN iajs-2513 150 75 . . PUNCT iajs-2513 151 1 thus thus ADV iajs-2513 151 2 𝐸 𝐸 PROPN iajs-2513 151 3 is be AUX iajs-2513 151 4 an an DET iajs-2513 151 5 app app PROPN iajs-2513 151 6 - - PUNCT iajs-2513 151 7 qp qp NOUN iajs-2513 151 8 submodule submodule NOUN iajs-2513 151 9 of of ADP iajs-2513 151 10 𝑄 𝑄 PROPN iajs-2513 151 11 .similarly .similarly SCONJ iajs-2513 151 12 we we PRON iajs-2513 151 13 can can AUX iajs-2513 151 14 prove prove VERB iajs-2513 151 15 that that SCONJ iajs-2513 151 16 𝐸 𝐸 PROPN iajs-2513 151 17 is be AUX iajs-2513 151 18 an an DET iajs-2513 151 19 app app PROPN iajs-2513 151 20 - - PUNCT iajs-2513 151 21 qp qp NOUN iajs-2513 151 22 submodule submodule NOUN iajs-2513 151 23 of of ADP iajs-2513 151 24 𝑄 𝑄 PROPN iajs-2513 151 25 . . PUNCT iajs-2513 151 26     SPACE iajs-2513 152 1 100 100 NUM iajs-2513 152 2     SPACE iajs-2513 152 3 ibn ibn PROPN iajs-2513 152 4 al al PROPN iajs-2513 152 5 - - PUNCT iajs-2513 152 6 haitham haitham PROPN iajs-2513 152 7 jour jour X iajs-2513 152 8 . . PROPN iajs-2513 152 9 for for ADP iajs-2513 152 10 pure pure ADJ iajs-2513 152 11 & & CCONJ iajs-2513 152 12 appl appl PROPN iajs-2513 152 13 . . PUNCT iajs-2513 153 1 sci sci PROPN iajs-2513 153 2 . . PROPN iajs-2513 154 1 33 33 NUM iajs-2513 154 2 ( ( PUNCT iajs-2513 154 3 4 4 NUM iajs-2513 154 4 ) ) PUNCT iajs-2513 154 5 2020 2020 NUM iajs-2513 154 6 proposition proposition NOUN iajs-2513 154 7 ( ( PUNCT iajs-2513 154 8 27 27 NUM iajs-2513 154 9 ) ) PUNCT iajs-2513 154 10 let let VERB iajs-2513 154 11 𝑄 𝑄 PRON iajs-2513 154 12 𝑄 𝑄 PROPN iajs-2513 154 13 ⊕ ⊕ PROPN iajs-2513 154 14 𝑄 𝑄 PRON iajs-2513 154 15 be be VERB iajs-2513 154 16 an an DET iajs-2513 154 17 𝑅-module 𝑅-module PROPN iajs-2513 154 18 , , PUNCT iajs-2513 154 19 where where SCONJ iajs-2513 154 20 𝑄 𝑄 PRON iajs-2513 154 21 and and CCONJ iajs-2513 154 22 𝑄 𝑄 PRON iajs-2513 154 23 are be AUX iajs-2513 154 24 𝑅-modules 𝑅-modules PROPN iajs-2513 154 25 . . PUNCT iajs-2513 155 1 then then ADV iajs-2513 155 2 , , PUNCT iajs-2513 155 3 the the DET iajs-2513 155 4 following follow VERB iajs-2513 155 5 are be AUX iajs-2513 155 6 held hold VERB iajs-2513 155 7 : : PUNCT iajs-2513 155 8     SPACE iajs-2513 155 9 1 1 X iajs-2513 155 10 ) ) PUNCT iajs-2513 155 11 𝐸 𝐸 PROPN iajs-2513 155 12 is be AUX iajs-2513 155 13 an an DET iajs-2513 155 14 app app PROPN iajs-2513 155 15 - - PUNCT iajs-2513 155 16 qp qp NOUN iajs-2513 155 17 submodule submodule NOUN iajs-2513 155 18 of of ADP iajs-2513 155 19 𝑄 𝑄 PRON iajs-2513 155 20 such such ADJ iajs-2513 155 21 that that SCONJ iajs-2513 155 22 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 155 23 𝐸 𝐸 PROPN iajs-2513 155 24 ⊆ ⊆ NUM iajs-2513 155 25 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 155 26 𝑄 𝑄 PROPN iajs-2513 155 27 and and CCONJ iajs-2513 155 28 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 155 29 𝑄 𝑄 PROPN iajs-2513 155 30 𝑄 𝑄 PROPN iajs-2513 155 31 if if SCONJ iajs-2513 156 1 and and CCONJ iajs-2513 156 2 only only ADV iajs-2513 156 3 if if SCONJ iajs-2513 156 4 𝐸 𝐸 PROPN iajs-2513 156 5 ⊕ ⊕ NOUN iajs-2513 156 6 𝑄 𝑄 PROPN iajs-2513 156 7 is be AUX iajs-2513 156 8 an an DET iajs-2513 156 9 app app PROPN iajs-2513 156 10 - - PUNCT iajs-2513 156 11 qp qp NOUN iajs-2513 156 12 submodule submodule NOUN iajs-2513 156 13 of of ADP iajs-2513 156 14 𝑄. 𝑄. PROPN iajs-2513 156 15 2 2 NUM iajs-2513 156 16 ) ) PUNCT iajs-2513 156 17 𝐸 𝐸 PROPN iajs-2513 156 18 is be AUX iajs-2513 156 19 an an DET iajs-2513 156 20 app app PROPN iajs-2513 156 21 - - PUNCT iajs-2513 156 22 qp qp NOUN iajs-2513 156 23 submodule submodule NOUN iajs-2513 156 24 of of ADP iajs-2513 156 25 𝑄 𝑄 PRON iajs-2513 156 26 such such ADJ iajs-2513 156 27 that that SCONJ iajs-2513 156 28 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 156 29 𝐸 𝐸 PROPN iajs-2513 156 30 ⊆ ⊆ NUM iajs-2513 156 31 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 156 32 2 2 NUM iajs-2513 156 33 and and CCONJ iajs-2513 156 34 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 156 35 𝑄 𝑄 PROPN iajs-2513 156 36 𝑄 𝑄 PROPN iajs-2513 156 37 if if SCONJ iajs-2513 157 1 and and CCONJ iajs-2513 157 2 only only ADV iajs-2513 157 3 if if SCONJ iajs-2513 157 4 𝑄 𝑄 PROPN iajs-2513 157 5 ⊕ ⊕ PROPN iajs-2513 157 6 𝐸 𝐸 PROPN iajs-2513 157 7 is be AUX iajs-2513 157 8 an an DET iajs-2513 157 9 app app PROPN iajs-2513 157 10 - - PUNCT iajs-2513 157 11 qp qp NOUN iajs-2513 157 12 submodule submodule NOUN iajs-2513 157 13 of of ADP iajs-2513 157 14 𝑄. 𝑄. PROPN iajs-2513 157 15 proof proof NOUN iajs-2513 157 16 1 1 NUM iajs-2513 157 17 ) ) PUNCT iajs-2513 157 18 ⟹ ⟹ VERB iajs-2513 158 1 let let VERB iajs-2513 158 2 𝑟 𝑟 PRON iajs-2513 158 3 𝑞 𝑞 PROPN iajs-2513 158 4 , , PUNCT iajs-2513 158 5 𝑞 𝑞 PROPN iajs-2513 158 6 ∈ ∈ PROPN iajs-2513 158 7 𝐸 𝐸 PROPN iajs-2513 158 8 ⊕ ⊕ PROPN iajs-2513 158 9 𝑄 𝑄 PROPN iajs-2513 158 10 , , PUNCT iajs-2513 158 11 for for ADP iajs-2513 158 12 𝑟 𝑟 DET iajs-2513 158 13 ∈ ∈ PROPN iajs-2513 158 14 𝑅 𝑅 PROPN iajs-2513 158 15 , , PUNCT iajs-2513 158 16 𝑞 𝑞 X iajs-2513 158 17 , , PUNCT iajs-2513 158 18 𝑞 𝑞 PROPN iajs-2513 158 19 ∈ ∈ PROPN iajs-2513 158 20 𝑄 𝑄 PROPN iajs-2513 158 21 , , PUNCT iajs-2513 158 22 then then ADV iajs-2513 158 23 𝑟𝑞 𝑟𝑞 NOUN iajs-2513 158 24 ∈ ∈ PROPN iajs-2513 158 25 𝐸 𝐸 PROPN iajs-2513 158 26 . . PUNCT iajs-2513 159 1 but but CCONJ iajs-2513 159 2 𝐸 𝐸 PROPN iajs-2513 159 3 is be AUX iajs-2513 159 4 an an DET iajs-2513 159 5 app app PROPN iajs-2513 159 6 - - PUNCT iajs-2513 159 7 qp qp NOUN iajs-2513 159 8 submodule submodule NOUN iajs-2513 159 9 of of ADP iajs-2513 159 10 𝑄 𝑄 PROPN iajs-2513 159 11 and and CCONJ iajs-2513 159 12 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 159 13 𝐸 𝐸 ADJ iajs-2513 159 14 ⊆ ⊆ NUM iajs-2513 159 15 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 159 16 𝑄 𝑄 PROPN iajs-2513 159 17 , , PUNCT iajs-2513 159 18 then then ADV iajs-2513 159 19 either either CCONJ iajs-2513 159 20 𝑞 𝑞 PROPN iajs-2513 159 21 ∈ ∈ PROPN iajs-2513 159 22 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 159 23 𝐸 𝐸 PROPN iajs-2513 159 24 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 159 25 𝑄 𝑄 PRON iajs-2513 159 26 𝑠𝑜𝑐 𝑠𝑜𝑐 AUX iajs-2513 159 27 𝑄 𝑄 PROPN iajs-2513 159 28 or or CCONJ iajs-2513 159 29 𝑟 𝑟 PRON iajs-2513 159 30 𝑄 𝑄 PROPN iajs-2513 159 31 ⊆ ⊆ NUM iajs-2513 159 32 𝐸 𝐸 PROPN iajs-2513 159 33 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 159 34 𝑄 𝑄 PRON iajs-2513 159 35 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 159 36 𝑄 𝑄 PROPN iajs-2513 159 37 for for ADP iajs-2513 159 38 some some DET iajs-2513 159 39 𝑛 𝑛 DET iajs-2513 159 40 ∈ ∈ PROPN iajs-2513 159 41 𝑍 𝑍 NOUN iajs-2513 159 42 . . PUNCT iajs-2513 160 1 since since SCONJ iajs-2513 160 2 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 160 3 𝑄 𝑄 PROPN iajs-2513 160 4 𝑄 𝑄 PROPN iajs-2513 160 5 , , PUNCT iajs-2513 160 6 then then ADV iajs-2513 160 7 either either CCONJ iajs-2513 160 8 𝑞 𝑞 X iajs-2513 160 9 , , PUNCT iajs-2513 160 10 𝑞 𝑞 X iajs-2513 160 11 ∈ ∈ PROPN iajs-2513 160 12 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 160 13 𝑄 𝑄 PROPN iajs-2513 160 14 ⊕ ⊕ PROPN iajs-2513 160 15 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 160 16 𝑄 𝑄 PRON iajs-2513 160 17 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 160 18 𝑄 𝑄 PROPN iajs-2513 160 19 ⊕ ⊕ VERB iajs-2513 160 20 𝑄 𝑄 PRON iajs-2513 160 21 ⊆ ⊆ PROPN iajs-2513 160 22 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 160 23 𝐸 𝐸 PROPN iajs-2513 160 24 ⊕ ⊕ NOUN iajs-2513 160 25 𝑄 𝑄 PROPN iajs-2513 160 26 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 160 27 𝑄 𝑄 PROPN iajs-2513 160 28 ⊕ ⊕ VERB iajs-2513 160 29 𝑄 𝑄 PROPN iajs-2513 160 30 or or CCONJ iajs-2513 160 31 𝑟 𝑟 NOUN iajs-2513 160 32 𝑄 𝑄 PROPN iajs-2513 160 33 ⊕ ⊕ VERB iajs-2513 160 34 𝑄 𝑄 PRON iajs-2513 160 35 ⊆ ⊆ NUM iajs-2513 160 36 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 160 37 𝑄 𝑄 PROPN iajs-2513 160 38 ⊕ ⊕ PROPN iajs-2513 160 39 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 160 40 𝑄 𝑄 PRON iajs-2513 160 41 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 160 42 𝑄 𝑄 PROPN iajs-2513 160 43 ⊕ ⊕ VERB iajs-2513 160 44 𝑄 𝑄 PRON iajs-2513 160 45 ⊆ ⊆ NUM iajs-2513 160 46 𝐸 𝐸 PROPN iajs-2513 160 47 ⊕ ⊕ NOUN iajs-2513 160 48 𝑄 𝑄 PROPN iajs-2513 160 49 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 160 50 𝑄 𝑄 PROPN iajs-2513 160 51 ⊕ ⊕ PROPN iajs-2513 160 52 𝑄 𝑄 PROPN iajs-2513 160 53 . . PUNCT iajs-2513 161 1 thus thus ADV iajs-2513 161 2 𝐸 𝐸 PROPN iajs-2513 161 3 ⊕ ⊕ PROPN iajs-2513 161 4 𝑄 𝑄 PROPN iajs-2513 161 5 is be AUX iajs-2513 161 6 an an DET iajs-2513 161 7 app app PROPN iajs-2513 161 8 - - PUNCT iajs-2513 161 9 qp qp NOUN iajs-2513 161 10 submodule submodule NOUN iajs-2513 161 11 of of ADP iajs-2513 161 12 𝑄. 𝑄. PROPN iajs-2513 161 13 ⟸ ⟸ NOUN iajs-2513 161 14 suppose suppose VERB iajs-2513 161 15 𝑟𝑞 𝑟𝑞 ADP iajs-2513 161 16 ∈ ∈ PROPN iajs-2513 161 17 𝐸 𝐸 PROPN iajs-2513 161 18 , , PUNCT iajs-2513 161 19 for for ADP iajs-2513 161 20 𝑟 𝑟 DET iajs-2513 161 21 ∈ ∈ PROPN iajs-2513 161 22 𝑅 𝑅 PROPN iajs-2513 161 23 , , PUNCT iajs-2513 161 24 𝑞 𝑞 X iajs-2513 161 25 ∈ ∈ PROPN iajs-2513 161 26 𝑄 𝑄 PROPN iajs-2513 161 27 . . PUNCT iajs-2513 162 1 then then ADV iajs-2513 162 2 for for ADP iajs-2513 162 3 each each DET iajs-2513 162 4 𝑞 𝑞 PROPN iajs-2513 162 5 ∈ ∈ PROPN iajs-2513 162 6 𝑄 𝑄 PROPN iajs-2513 162 7 , , PUNCT iajs-2513 162 8 𝑞 𝑞 X iajs-2513 162 9 , , PUNCT iajs-2513 162 10 𝑞 𝑞 PROPN iajs-2513 162 11 ∈ ∈ PROPN iajs-2513 162 12 𝐸 𝐸 PROPN iajs-2513 162 13 ⊕ ⊕ PROPN iajs-2513 162 14 𝑄 𝑄 PROPN iajs-2513 162 15 , , PUNCT iajs-2513 162 16 but but CCONJ iajs-2513 162 17 𝐸 𝐸 PROPN iajs-2513 162 18 ⊕ ⊕ PROPN iajs-2513 162 19 𝑄 𝑄 PROPN iajs-2513 162 20 is be AUX iajs-2513 162 21 an an DET iajs-2513 162 22 app app PROPN iajs-2513 162 23 - - PUNCT iajs-2513 162 24 qp qp NOUN iajs-2513 162 25 submodule submodule NOUN iajs-2513 162 26 of of ADP iajs-2513 162 27 𝑄 𝑄 PROPN iajs-2513 162 28 , , PUNCT iajs-2513 162 29 implies imply VERB iajs-2513 162 30 that that SCONJ iajs-2513 162 31 either either CCONJ iajs-2513 162 32 𝑞 𝑞 X iajs-2513 162 33 , , PUNCT iajs-2513 162 34 𝑞 𝑞 PROPN iajs-2513 162 35 ∈ ∈ PROPN iajs-2513 162 36 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 162 37 𝐸 𝐸 PROPN iajs-2513 162 38 ⊕ ⊕ NOUN iajs-2513 162 39 𝑄 𝑄 PROPN iajs-2513 163 1 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 163 2 𝑄 𝑄 PROPN iajs-2513 163 3 or or CCONJ iajs-2513 163 4 𝑟 𝑟 PRON iajs-2513 163 5 𝑄 𝑄 PROPN iajs-2513 163 6 ⊆ ⊆ NUM iajs-2513 163 7 𝐸 𝐸 NOUN iajs-2513 163 8 ⊕ ⊕ NOUN iajs-2513 163 9 𝑄 𝑄 PROPN iajs-2513 163 10 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 163 11 𝑄 𝑄 PROPN iajs-2513 163 12 for for ADP iajs-2513 163 13 some some PRON iajs-2513 163 14 𝑛 𝑛 DET iajs-2513 163 15 ∈ ∈ PROPN iajs-2513 163 16 𝑍 𝑍 PROPN iajs-2513 163 17 .it .it PUNCT iajs-2513 163 18 follows follow VERB iajs-2513 163 19 that that SCONJ iajs-2513 163 20 either either CCONJ iajs-2513 163 21 𝑞 𝑞 X iajs-2513 163 22 , , PUNCT iajs-2513 163 23 𝑞 𝑞 PROPN iajs-2513 163 24 ∈ ∈ PROPN iajs-2513 163 25 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 163 26 𝐸 𝐸 PROPN iajs-2513 163 27 ⊕ ⊕ PROPN iajs-2513 163 28 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 163 29 𝑄 𝑄 PRON iajs-2513 163 30 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 163 31 𝑄 𝑄 PROPN iajs-2513 163 32 ⊕ ⊕ VERB iajs-2513 163 33 𝑄 𝑄 PROPN iajs-2513 163 34 or or CCONJ iajs-2513 163 35 𝑟 𝑟 NOUN iajs-2513 163 36 ( ( PUNCT iajs-2513 163 37 𝑄 𝑄 PROPN iajs-2513 163 38 ⊕ ⊕ VERB iajs-2513 163 39 𝑄 𝑄 PRON iajs-2513 163 40 ⊆ ⊆ NUM iajs-2513 163 41 𝐸 𝐸 PROPN iajs-2513 163 42 ⊕ ⊕ NOUN iajs-2513 163 43 𝑄 𝑄 PROPN iajs-2513 163 44 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 163 45 𝑄 𝑄 PROPN iajs-2513 163 46 ⊕ ⊕ PROPN iajs-2513 163 47 𝑄 𝑄 PROPN iajs-2513 163 48 , , PUNCT iajs-2513 163 49 that that PRON iajs-2513 163 50 is be AUX iajs-2513 163 51 either either CCONJ iajs-2513 163 52 𝑞 𝑞 PROPN iajs-2513 163 53 , , PUNCT iajs-2513 163 54 𝑞 𝑞 PROPN iajs-2513 163 55 ∈ ∈ PROPN iajs-2513 163 56 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 163 57 𝐸 𝐸 PROPN iajs-2513 163 58 ⊕ ⊕ PROPN iajs-2513 163 59 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 163 60 𝑄 𝑄 PRON iajs-2513 163 61 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 163 62 𝑄 𝑄 PROPN iajs-2513 163 63 ⊕ ⊕ PROPN iajs-2513 163 64 𝑠𝑜𝑐 𝑠𝑜𝑐 VERB iajs-2513 163 65 𝑄 𝑄 PROPN iajs-2513 163 66 or or CCONJ iajs-2513 163 67 𝑟 𝑟 NOUN iajs-2513 163 68 ( ( PUNCT iajs-2513 163 69 𝑄 𝑄 PROPN iajs-2513 163 70 ⊕ ⊕ VERB iajs-2513 163 71 𝑄 𝑄 PRON iajs-2513 163 72 ⊆ ⊆ NUM iajs-2513 163 73 𝐸 𝐸 PROPN iajs-2513 163 74 ⊕ ⊕ NOUN iajs-2513 164 1 𝑄 𝑄 PROPN iajs-2513 164 2 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 164 3 𝑄 𝑄 PROPN iajs-2513 164 4 ⊕ ⊕ PROPN iajs-2513 164 5 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 164 6 𝑄 𝑄 PROPN iajs-2513 164 7 . . PUNCT iajs-2513 165 1 since since SCONJ iajs-2513 165 2 𝑠𝑜𝑐 𝑠𝑜𝑐 PROPN iajs-2513 165 3 𝑄 𝑄 PROPN iajs-2513 165 4 𝑄 𝑄 PROPN iajs-2513 165 5 implies imply VERB iajs-2513 165 6 that that SCONJ iajs-2513 165 7 either either CCONJ iajs-2513 165 8 𝑞 𝑞 X iajs-2513 165 9 , , PUNCT iajs-2513 165 10 𝑞 𝑞 PROPN iajs-2513 165 11 ∈ ∈ PROPN iajs-2513 165 12 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 165 13 𝐸 𝐸 PROPN iajs-2513 165 14 𝑠𝑜𝑐 𝑠𝑜𝑐 NOUN iajs-2513 165 15 𝑄 𝑄 PROPN iajs-2513 165 16 ⊕ ⊕ PROPN iajs-2513 165 17 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 165 18 𝑄 𝑄 PROPN iajs-2513 165 19 𝑄 𝑄 PROPN iajs-2513 165 20 or or CCONJ iajs-2513 165 21 𝑟 𝑟 NOUN iajs-2513 165 22 ( ( PUNCT iajs-2513 165 23 𝑄 𝑄 PROPN iajs-2513 165 24 ⊕ ⊕ VERB iajs-2513 165 25 𝑄 𝑄 PRON iajs-2513 165 26 ⊆ ⊆ NUM iajs-2513 165 27 𝐸 𝐸 PROPN iajs-2513 165 28 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 165 29 𝑄 𝑄 PROPN iajs-2513 165 30 ⊕ ⊕ PROPN iajs-2513 165 31 𝑄 𝑄 PROPN iajs-2513 165 32 , , PUNCT iajs-2513 165 33 that that PRON iajs-2513 165 34 is be AUX iajs-2513 165 35 either either CCONJ iajs-2513 165 36 𝑞 𝑞 PROPN iajs-2513 165 37 ∈ ∈ PROPN iajs-2513 165 38 𝑟𝑎𝑑 𝑟𝑎𝑑 ADP iajs-2513 165 39 𝐸 𝐸 PROPN iajs-2513 165 40 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 165 41 𝑄 𝑄 PROPN iajs-2513 165 42 or or CCONJ iajs-2513 165 43 𝑟 𝑟 PRON iajs-2513 165 44 𝑄 𝑄 PROPN iajs-2513 165 45 ⊆ ⊆ NUM iajs-2513 165 46 𝐸 𝐸 PROPN iajs-2513 165 47 𝑠𝑜𝑐 𝑠𝑜𝑐 X iajs-2513 165 48 𝑄 𝑄 PROPN iajs-2513 165 49 for for ADP iajs-2513 165 50 some some DET iajs-2513 165 51 𝑛 𝑛 DET iajs-2513 165 52 ∈ ∈ PROPN iajs-2513 165 53 𝑍 𝑍 NOUN iajs-2513 165 54 . . PUNCT iajs-2513 166 1 hence hence ADV iajs-2513 166 2 𝐸 𝐸 PROPN iajs-2513 166 3 is be AUX iajs-2513 166 4 an an DET iajs-2513 166 5 app app PROPN iajs-2513 166 6 - - PUNCT iajs-2513 166 7 qp qp NOUN iajs-2513 166 8 submodule submodule NOUN iajs-2513 166 9 of of ADP iajs-2513 166 10 𝑄 𝑄 PROPN iajs-2513 166 11 . . PUNCT iajs-2513 167 1 2 2 X iajs-2513 167 2 ) ) PUNCT iajs-2513 167 3 its its PRON iajs-2513 167 4 follows follow VERB iajs-2513 167 5 as as ADP iajs-2513 167 6 in in ADP iajs-2513 167 7 part part NOUN iajs-2513 167 8 ( ( PUNCT iajs-2513 167 9 1 1 NUM iajs-2513 167 10 ) ) PUNCT iajs-2513 167 11 . . PUNCT iajs-2513 168 1 3 3 X iajs-2513 168 2 . . X iajs-2513 168 3 conclusion conclusion NOUN iajs-2513 168 4 in in ADP iajs-2513 168 5 this this DET iajs-2513 168 6 paper paper NOUN iajs-2513 168 7 , , PUNCT iajs-2513 168 8 we we PRON iajs-2513 168 9 introduce introduce VERB iajs-2513 168 10 a a DET iajs-2513 168 11 new new ADJ iajs-2513 168 12 generalization generalization NOUN iajs-2513 168 13 of of ADP iajs-2513 168 14 prime prime ADJ iajs-2513 168 15 and and CCONJ iajs-2513 168 16 primary primary ADJ iajs-2513 168 17 submodules submodule NOUN iajs-2513 168 18 called call VERB iajs-2513 168 19 an an DET iajs-2513 168 20 approximaitly approximaitly ADV iajs-2513 168 21 quasi quasi ADJ iajs-2513 168 22 - - ADJ iajs-2513 168 23 primary primary ADJ iajs-2513 168 24 submodule submodule NOUN iajs-2513 168 25 . . PUNCT iajs-2513 169 1 many many ADJ iajs-2513 169 2 characterizations characterization NOUN iajs-2513 169 3 of of ADP iajs-2513 169 4 this this DET iajs-2513 169 5 generalization generalization NOUN iajs-2513 169 6 are be AUX iajs-2513 169 7 introduced introduce VERB iajs-2513 169 8 . . PUNCT iajs-2513 170 1 relationships relationship NOUN iajs-2513 170 2 of of ADP iajs-2513 170 3 this this DET iajs-2513 170 4 generalization generalization NOUN iajs-2513 170 5 with with ADP iajs-2513 170 6 other other ADJ iajs-2513 170 7 classes class NOUN iajs-2513 170 8 of of ADP iajs-2513 170 9 modules module NOUN iajs-2513 170 10 are be AUX iajs-2513 170 11 given give VERB iajs-2513 170 12 . . PUNCT iajs-2513 171 1 references reference NOUN iajs-2513 171 2 1 1 NUM iajs-2513 171 3 . . PUNCT iajs-2513 171 4 dauns daun NOUN iajs-2513 171 5 , , PUNCT iajs-2513 171 6 j. j. PROPN iajs-2513 171 7 prime prime PROPN iajs-2513 171 8 modules modules PROPN iajs-2513 171 9 , , PUNCT iajs-2513 171 10 j. j. PROPN iajs-2513 171 11 reine reine PROPN iajs-2513 171 12 angew angew PROPN iajs-2513 171 13 , , PUNCT iajs-2513 171 14 math math NOUN iajs-2513 171 15 . . PUNCT iajs-2513 172 1 1978 1978 NUM iajs-2513 172 2 , , PUNCT iajs-2513 172 3 2 2 NUM iajs-2513 172 4 , , PUNCT iajs-2513 172 5 156 156 NUM iajs-2513 172 6 - - SYM iajs-2513 172 7 181 181 NUM iajs-2513 172 8 . . PUNCT iajs-2513 173 1 2 2 NUM iajs-2513 173 2 . . X iajs-2513 173 3 haibat haibat PROPN iajs-2513 173 4 , , PUNCT iajs-2513 173 5 k.m k.m PROPN iajs-2513 173 6 . . PROPN iajs-2513 173 7 ; ; PUNCT iajs-2513 174 1 omar omar PROPN iajs-2513 174 2 , , PUNCT iajs-2513 174 3 a.a a.a PROPN iajs-2513 174 4 . . PROPN iajs-2513 174 5 pseudo-2 pseudo-2 PROPN iajs-2513 174 6 - - PUNCT iajs-2513 174 7 absorbing absorbing ADJ iajs-2513 174 8 and and CCONJ iajs-2513 174 9 pseudo pseudo NOUN iajs-2513 174 10 semi-2 semi-2 NOUN iajs-2513 174 11 - - PUNCT iajs-2513 174 12 absorbing absorbing ADJ iajs-2513 174 13 submodules submodule NOUN iajs-2513 174 14 , , PUNCT iajs-2513 174 15 aip aip PROPN iajs-2513 174 16 conference conference NOUN iajs-2513 174 17 proceeding proceeding NOUN iajs-2513 174 18 , , PUNCT iajs-2513 174 19 2069 2069 NUM iajs-2513 174 20 , , PUNCT iajs-2513 174 21 020006(2019 020006(2019 NUM iajs-2513 174 22 ) ) PUNCT iajs-2513 174 23 , , PUNCT iajs-2513 174 24 1 1 NUM iajs-2513 174 25 - - SYM iajs-2513 174 26 9 9 NUM iajs-2513 174 27 . . PUNCT iajs-2513 174 28 scopus scopus PROPN iajs-2513 174 29 . . PUNCT iajs-2513 175 1 3 3 X iajs-2513 175 2 . . X iajs-2513 175 3 haibat haibat PROPN iajs-2513 175 4 k.m k.m PROPN iajs-2513 175 5 . . PROPN iajs-2513 175 6 ; ; PUNCT iajs-2513 175 7 omer omer PROPN iajs-2513 175 8 a.a a.a PROPN iajs-2513 175 9 . . PROPN iajs-2513 175 10 pseudo pseudo NOUN iajs-2513 175 11 quasi-2 quasi-2 NOUN iajs-2513 175 12 - - PUNCT iajs-2513 175 13 absorbing absorbing ADJ iajs-2513 175 14 submodules submodule NOUN iajs-2513 175 15 and and CCONJ iajs-2513 175 16 some some DET iajs-2513 175 17 related related ADJ iajs-2513 175 18 concepts concept NOUN iajs-2513 175 19 , , PUNCT iajs-2513 175 20 ibn ibn PROPN iajs-2513 175 21 al al PROPN iajs-2513 175 22 - - PUNCT iajs-2513 175 23 haitham haitham PROPN iajs-2513 175 24 journal journal PROPN iajs-2513 175 25 for for ADP iajs-2513 175 26 pure pure ADJ iajs-2513 175 27 and and CCONJ iajs-2513 175 28 applied apply VERB iajs-2513 175 29 sci sci PROPN iajs-2513 175 30 . . PROPN iajs-2513 175 31 2019 2019 NUM iajs-2513 175 32 , , PUNCT iajs-2513 175 33 32 32 NUM iajs-2513 175 34 , , PUNCT iajs-2513 175 35 2 2 NUM iajs-2513 175 36 , , PUNCT iajs-2513 175 37 114 114 NUM iajs-2513 175 38 - - SYM iajs-2513 175 39 122 122 NUM iajs-2513 175 40 . . PUNCT iajs-2513 176 1 4 4 X iajs-2513 176 2 . . X iajs-2513 176 3 haibat haibat PROPN iajs-2513 176 4 k.m k.m PROPN iajs-2513 176 5 . . PROPN iajs-2513 176 6 ; ; PUNCT iajs-2513 176 7 omer omer PROPN iajs-2513 176 8 a.a a.a PROPN iajs-2513 176 9 . . PROPN iajs-2513 176 10 pseudo pseudo PROPN iajs-2513 176 11 primary-2 primary-2 NOUN iajs-2513 176 12 - - PUNCT iajs-2513 176 13 absorbing absorbing ADJ iajs-2513 176 14 submodules submodule NOUN iajs-2513 176 15 and and CCONJ iajs-2513 176 16 some some DET iajs-2513 176 17 related related ADJ iajs-2513 176 18 concepts concept NOUN iajs-2513 176 19 , , PUNCT iajs-2513 176 20 ibn ibn PROPN iajs-2513 176 21 al al PROPN iajs-2513 176 22 - - PUNCT iajs-2513 176 23 haitham haitham PROPN iajs-2513 176 24 journal journal PROPN iajs-2513 176 25 for for ADP iajs-2513 176 26 pure pure ADJ iajs-2513 176 27 and and CCONJ iajs-2513 176 28 applied apply VERB iajs-2513 176 29 sci sci PROPN iajs-2513 176 30 . . PROPN iajs-2513 176 31 2019 2019 NUM iajs-2513 176 32 , , PUNCT iajs-2513 176 33 32 32 NUM iajs-2513 176 34 , , PUNCT iajs-2513 176 35 3 3 NUM iajs-2513 176 36 , , PUNCT iajs-2513 176 37 129 129 NUM iajs-2513 176 38 - - SYM iajs-2513 176 39 139 139 NUM iajs-2513 176 40 . . PUNCT iajs-2513 177 1 5 5 X iajs-2513 177 2 . . X iajs-2513 177 3 haibat haibat PROPN iajs-2513 177 4 , , PUNCT iajs-2513 177 5 k.m k.m PROPN iajs-2513 177 6 . . PROPN iajs-2513 177 7 ; ; PUNCT iajs-2513 178 1 akram akram PROPN iajs-2513 178 2 , , PUNCT iajs-2513 178 3 s.m s.m PROPN iajs-2513 178 4 . . PROPN iajs-2513 178 5 nearly nearly ADV iajs-2513 178 6 semi semi ADJ iajs-2513 178 7 -2 -2 ADV iajs-2513 178 8 - - PUNCT iajs-2513 178 9 absorbing absorb VERB iajs-2513 178 10 submodules submodule NOUN iajs-2513 178 11 and and CCONJ iajs-2513 178 12 related related ADJ iajs-2513 178 13 concepts concept NOUN iajs-2513 178 14 , , PUNCT iajs-2513 178 15 italian italian ADJ iajs-2513 178 16 journal journal NOUN iajs-2513 178 17 of of ADP iajs-2513 178 18 pure pure ADJ iajs-2513 178 19 and and CCONJ iajs-2513 178 20 applied applied ADJ iajs-2513 178 21 mathematics mathematic NOUN iajs-2513 178 22 . . 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