id sid tid token lemma pos iajs-3013 1 1 ihjpas ihjpas PROPN iajs-3013 1 2 . . PUNCT iajs-3013 2 1 36(1)2023 36(1)2023 NUM iajs-3013 2 2 300 300 NUM iajs-3013 2 3 this this DET iajs-3013 2 4 work work NOUN iajs-3013 2 5 is be AUX iajs-3013 2 6 licensed license VERB iajs-3013 2 7 under under ADP iajs-3013 2 8 a a DET iajs-3013 2 9 creative creative ADJ iajs-3013 2 10 commons common NOUN iajs-3013 2 11 attribution attribution NOUN iajs-3013 2 12 4.0 4.0 NUM iajs-3013 2 13 international international ADJ iajs-3013 2 14 license license NOUN iajs-3013 2 15 almost almost ADV iajs-3013 2 16 and and CCONJ iajs-3013 2 17 strongly strongly ADV iajs-3013 2 18 almost almost ADV iajs-3013 2 19 approximately approximately ADV iajs-3013 2 20 nearly nearly ADV iajs-3013 2 21 quasi quasi ADJ iajs-3013 2 22 compactly compactly ADV iajs-3013 2 23 packed pack VERB iajs-3013 2 24 modules module NOUN iajs-3013 2 25 abstract abstract ADJ iajs-3013 2 26 in in ADP iajs-3013 2 27 this this DET iajs-3013 2 28 paper paper NOUN iajs-3013 2 29 we we PRON iajs-3013 2 30 present present VERB iajs-3013 2 31 the the DET iajs-3013 2 32 almost almost ADV iajs-3013 2 33 approximately approximately ADV iajs-3013 2 34 nearly nearly ADV iajs-3013 2 35 quasi quasi ADJ iajs-3013 2 36 compactly compactly ADV iajs-3013 2 37 packed pack VERB iajs-3013 2 38 ( ( PUNCT iajs-3013 2 39 submodules submodule NOUN iajs-3013 2 40 ) ) PUNCT iajs-3013 2 41 modules module NOUN iajs-3013 2 42 as as ADP iajs-3013 2 43 an an DET iajs-3013 2 44 application application NOUN iajs-3013 2 45 of of ADP iajs-3013 2 46 almost almost ADV iajs-3013 2 47 approximately approximately ADV iajs-3013 2 48 nearly nearly ADV iajs-3013 2 49 quasiprime quasiprime ADJ iajs-3013 2 50 submodule submodule NOUN iajs-3013 2 51 . . PUNCT iajs-3013 3 1 we we PRON iajs-3013 3 2 give give VERB iajs-3013 3 3 some some DET iajs-3013 3 4 examples example NOUN iajs-3013 3 5 , , PUNCT iajs-3013 3 6 remarks remark VERB iajs-3013 3 7 , , PUNCT iajs-3013 3 8 and and CCONJ iajs-3013 3 9 properties property NOUN iajs-3013 3 10 of of ADP iajs-3013 3 11 this this DET iajs-3013 3 12 concept concept NOUN iajs-3013 3 13 . . PUNCT iajs-3013 4 1 also also ADV iajs-3013 4 2 , , PUNCT iajs-3013 4 3 as as ADP iajs-3013 4 4 the the DET iajs-3013 4 5 strong strong ADJ iajs-3013 4 6 form form NOUN iajs-3013 4 7 of of ADP iajs-3013 4 8 this this DET iajs-3013 4 9 concept concept NOUN iajs-3013 4 10 , , PUNCT iajs-3013 4 11 we we PRON iajs-3013 4 12 introduce introduce VERB iajs-3013 4 13 the the PRON iajs-3013 4 14 strongly strongly ADV iajs-3013 4 15 , , PUNCT iajs-3013 4 16 almost almost ADV iajs-3013 4 17 approximately approximately ADV iajs-3013 4 18 nearly nearly ADV iajs-3013 4 19 quasi quasi ADJ iajs-3013 4 20 compactly compactly ADV iajs-3013 4 21 packed pack VERB iajs-3013 4 22 ( ( PUNCT iajs-3013 4 23 submodules submodule NOUN iajs-3013 4 24 ) ) PUNCT iajs-3013 4 25 modules module NOUN iajs-3013 4 26 . . PUNCT iajs-3013 5 1 moreover moreover ADV iajs-3013 5 2 , , PUNCT iajs-3013 5 3 we we PRON iajs-3013 5 4 present present VERB iajs-3013 5 5 the the DET iajs-3013 5 6 definitions definition NOUN iajs-3013 5 7 of of ADP iajs-3013 5 8 almost almost ADV iajs-3013 5 9 approximately approximately ADV iajs-3013 5 10 nearly nearly ADV iajs-3013 5 11 quasiprime quasiprime ADJ iajs-3013 5 12 radical radical ADJ iajs-3013 5 13 submodules submodule NOUN iajs-3013 5 14 and and CCONJ iajs-3013 5 15 almost almost ADV iajs-3013 5 16 approximately approximately ADV iajs-3013 5 17 nearly nearly ADV iajs-3013 5 18 quasiprime quasiprime ADJ iajs-3013 5 19 radical radical ADJ iajs-3013 5 20 submodules submodule NOUN iajs-3013 5 21 and and CCONJ iajs-3013 5 22 give give VERB iajs-3013 5 23 some some DET iajs-3013 5 24 basic basic ADJ iajs-3013 5 25 properties property NOUN iajs-3013 5 26 of of ADP iajs-3013 5 27 these these DET iajs-3013 5 28 concepts concept NOUN iajs-3013 5 29 that that PRON iajs-3013 5 30 will will AUX iajs-3013 5 31 be be AUX iajs-3013 5 32 needed need VERB iajs-3013 5 33 in in ADP iajs-3013 5 34 section section NOUN iajs-3013 5 35 four four NUM iajs-3013 5 36 of of ADP iajs-3013 5 37 this this DET iajs-3013 5 38 research research NOUN iajs-3013 5 39 . . PUNCT iajs-3013 6 1 we we PRON iajs-3013 6 2 study study VERB iajs-3013 6 3 these these DET iajs-3013 6 4 two two NUM iajs-3013 6 5 concepts concept NOUN iajs-3013 6 6 extensively extensively ADV iajs-3013 6 7 . . PUNCT iajs-3013 7 1 keywords keyword NOUN iajs-3013 7 2 : : PUNCT iajs-3013 7 3 alappnq alappnq ADJ iajs-3013 7 4 - - PUNCT iajs-3013 7 5 prime prime NOUN iajs-3013 7 6 submodules submodule NOUN iajs-3013 7 7 , , PUNCT iajs-3013 7 8 alappnq alappnq PROPN iajs-3013 7 9 compactly compactly ADV iajs-3013 7 10 packed pack VERB iajs-3013 7 11 , , PUNCT iajs-3013 7 12 strongly strongly ADV iajs-3013 7 13 alappnq alappnq PROPN iajs-3013 7 14 compactly compactly ADV iajs-3013 7 15 packed pack VERB iajs-3013 7 16 , , PUNCT iajs-3013 7 17 alappnq alappnq ADJ iajs-3013 7 18 - - PUNCT iajs-3013 7 19 prime prime NOUN iajs-3013 7 20 radical radical NOUN iajs-3013 7 21 of of ADP iajs-3013 7 22 submodule submodule NOUN iajs-3013 7 23 , , PUNCT iajs-3013 7 24 alappnq alappnq NOUN iajs-3013 7 25 - - PUNCT iajs-3013 7 26 prime prime ADJ iajs-3013 7 27 radical radical ADJ iajs-3013 7 28 submodule submodule NOUN iajs-3013 7 29 . . PUNCT iajs-3013 8 1 1 1 X iajs-3013 8 2 . . X iajs-3013 8 3 introduction introduction NOUN iajs-3013 8 4 the the DET iajs-3013 8 5 concept concept NOUN iajs-3013 8 6 of of ADP iajs-3013 8 7 almost almost ADV iajs-3013 8 8 approximately approximately ADV iajs-3013 8 9 nearly nearly ADV iajs-3013 8 10 quasiprime quasiprime ADJ iajs-3013 8 11 was be AUX iajs-3013 8 12 recently recently ADV iajs-3013 8 13 introduced introduce VERB iajs-3013 8 14 by by ADP iajs-3013 8 15 [ [ X iajs-3013 8 16 1 1 NUM iajs-3013 8 17 ] ] PUNCT iajs-3013 8 18 as as ADP iajs-3013 8 19 a a DET iajs-3013 8 20 generation generation NOUN iajs-3013 8 21 of of ADP iajs-3013 8 22 β€œ " PUNCT iajs-3013 8 23 quasiprime quasiprime ADJ iajs-3013 8 24 , , PUNCT iajs-3013 8 25 nearly nearly ADV iajs-3013 8 26 quasiprime quasiprime ADJ iajs-3013 8 27 and and CCONJ iajs-3013 8 28 approximately approximately ADV iajs-3013 8 29 quasiprime quasiprime ADJ iajs-3013 8 30 ” " PUNCT iajs-3013 8 31 submodules submodule NOUN iajs-3013 8 32 see see VERB iajs-3013 8 33 [ [ X iajs-3013 8 34 24 24 NUM iajs-3013 8 35 ] ] PUNCT iajs-3013 8 36 . . PUNCT iajs-3013 9 1 the the DET iajs-3013 9 2 submodule submodule NOUN iajs-3013 9 3 𝐹 𝐹 PROPN iajs-3013 9 4 of of ADP iajs-3013 9 5 𝑄 𝑄 PROPN iajs-3013 9 6 is be AUX iajs-3013 9 7 called call VERB iajs-3013 9 8 almost almost ADV iajs-3013 9 9 approximately approximately ADV iajs-3013 9 10 nearly nearly ADV iajs-3013 9 11 quasiprime quasiprime ADJ iajs-3013 9 12 ( ( PUNCT iajs-3013 9 13 simply simply ADV iajs-3013 9 14 alappnqprime alappnqprime NOUN iajs-3013 9 15 ) ) PUNCT iajs-3013 9 16 submodule submodule NOUN iajs-3013 9 17 , , PUNCT iajs-3013 9 18 if if SCONJ iajs-3013 9 19 for for ADP iajs-3013 9 20 any any DET iajs-3013 9 21 π‘Ÿπ‘ π‘ž π‘Ÿπ‘ π‘ž NOUN iajs-3013 9 22 ∈ ∈ PROPN iajs-3013 9 23 𝐹 𝐹 PROPN iajs-3013 9 24 , , PUNCT iajs-3013 9 25 for for ADP iajs-3013 9 26 π‘Ÿ π‘Ÿ NOUN iajs-3013 9 27 , , PUNCT iajs-3013 9 28 𝑠 𝑠 PROPN iajs-3013 9 29 ∈ ∈ PROPN iajs-3013 9 30 𝑅 𝑅 PROPN iajs-3013 9 31 , , PUNCT iajs-3013 9 32 π‘ž π‘ž X iajs-3013 9 33 ∈ ∈ PROPN iajs-3013 9 34 𝑄 𝑄 PROPN iajs-3013 9 35 , , PUNCT iajs-3013 9 36 implying imply VERB iajs-3013 9 37 that that SCONJ iajs-3013 9 38 either either CCONJ iajs-3013 9 39 π‘Ÿπ‘ž π‘Ÿπ‘ž NOUN iajs-3013 9 40 ∈ ∈ PROPN iajs-3013 9 41 𝐹 𝐹 PROPN iajs-3013 9 42 + + CCONJ iajs-3013 9 43 ( ( PUNCT iajs-3013 9 44 π‘ π‘œπ‘(𝑄 π‘ π‘œπ‘(𝑄 PROPN iajs-3013 9 45 ) ) PUNCT iajs-3013 9 46 + + NUM iajs-3013 9 47 𝐽(𝑄 𝐽(𝑄 NOUN iajs-3013 9 48 ) ) PUNCT iajs-3013 9 49 ) ) PUNCT iajs-3013 9 50 or or CCONJ iajs-3013 9 51 π‘ π‘ž π‘ π‘ž ADP iajs-3013 9 52 ∈ ∈ PROPN iajs-3013 9 53 𝐹 𝐹 PROPN iajs-3013 9 54 + + CCONJ iajs-3013 9 55 ( ( PUNCT iajs-3013 9 56 π‘ π‘œπ‘(𝑄 π‘ π‘œπ‘(𝑄 PROPN iajs-3013 9 57 ) ) PUNCT iajs-3013 9 58 + + NUM iajs-3013 9 59 𝐽(𝑄 𝐽(𝑄 NOUN iajs-3013 9 60 ) ) PUNCT iajs-3013 9 61 ) ) PUNCT iajs-3013 9 62 . . PUNCT iajs-3013 10 1 as as ADP iajs-3013 10 2 an an DET iajs-3013 10 3 application application NOUN iajs-3013 10 4 of of ADP iajs-3013 10 5 alappnq alappnq ADJ iajs-3013 10 6 - - PUNCT iajs-3013 10 7 prime prime NOUN iajs-3013 10 8 submodule submodule NOUN iajs-3013 10 9 , , PUNCT iajs-3013 10 10 we we PRON iajs-3013 10 11 introduce introduce VERB iajs-3013 10 12 the the DET iajs-3013 10 13 concepts concept NOUN iajs-3013 10 14 of of ADP iajs-3013 10 15 [ [ X iajs-3013 10 16 almost almost ADV iajs-3013 10 17 approximately approximately ADV iajs-3013 10 18 nearly nearly ADV iajs-3013 10 19 quasi quasi ADJ iajs-3013 10 20 compactly compactly ADV iajs-3013 10 21 packed pack VERB iajs-3013 10 22 ( ( PUNCT iajs-3013 10 23 submodules submodule NOUN iajs-3013 10 24 ) ) PUNCT iajs-3013 10 25 modules module NOUN iajs-3013 10 26 , , PUNCT iajs-3013 10 27 strongly strongly ADV iajs-3013 10 28 almost almost ADV iajs-3013 10 29 approximately approximately ADV iajs-3013 10 30 nearly nearly ADV iajs-3013 10 31 quasi quasi ADJ iajs-3013 10 32 compactly compactly ADV iajs-3013 10 33 packed pack VERB iajs-3013 10 34 ( ( PUNCT iajs-3013 10 35 submodules submodule NOUN iajs-3013 10 36 ) ) PUNCT iajs-3013 10 37 modules module NOUN iajs-3013 10 38 ] ] PUNCT iajs-3013 10 39 and and CCONJ iajs-3013 10 40 study study VERB iajs-3013 10 41 some some DET iajs-3013 10 42 basic basic ADJ iajs-3013 10 43 properties property NOUN iajs-3013 10 44 of of ADP iajs-3013 10 45 these these DET iajs-3013 10 46 concepts concept NOUN iajs-3013 10 47 . . PUNCT iajs-3013 11 1 this this DET iajs-3013 11 2 paper paper NOUN iajs-3013 11 3 consists consist VERB iajs-3013 11 4 of of ADP iajs-3013 11 5 three three NUM iajs-3013 11 6 sections section NOUN iajs-3013 11 7 . . PUNCT iajs-3013 12 1 section section NOUN iajs-3013 12 2 one one PRON iajs-3013 12 3 covers cover VERB iajs-3013 12 4 some some DET iajs-3013 12 5 basic basic ADJ iajs-3013 12 6 concepts concept NOUN iajs-3013 12 7 , , PUNCT iajs-3013 12 8 recalls recall VERB iajs-3013 12 9 some some DET iajs-3013 12 10 remarks remark NOUN iajs-3013 12 11 and and CCONJ iajs-3013 12 12 propositions proposition NOUN iajs-3013 12 13 needed need VERB iajs-3013 12 14 in in ADP iajs-3013 12 15 the the DET iajs-3013 12 16 sequel sequel NOUN iajs-3013 12 17 . . PUNCT iajs-3013 13 1 section section NOUN iajs-3013 13 2 two two NUM iajs-3013 13 3 introduces introduce NOUN iajs-3013 13 4 and and CCONJ iajs-3013 13 5 studies study NOUN iajs-3013 13 6 the the DET iajs-3013 13 7 concept concept NOUN iajs-3013 13 8 of of ADP iajs-3013 13 9 almost almost ADV iajs-3013 13 10 approximately approximately ADV iajs-3013 13 11 nearly nearly ADV iajs-3013 13 12 quasi quasi ADJ iajs-3013 13 13 compactly compactly ADV iajs-3013 13 14 packed pack VERB iajs-3013 13 15 ( ( PUNCT iajs-3013 13 16 submodules submodule NOUN iajs-3013 13 17 ) ) PUNCT iajs-3013 13 18 modules module NOUN iajs-3013 13 19 and and CCONJ iajs-3013 13 20 gives give VERB iajs-3013 13 21 some some DET iajs-3013 13 22 basic basic ADJ iajs-3013 13 23 properties property NOUN iajs-3013 13 24 . . PUNCT iajs-3013 14 1 section section NOUN iajs-3013 14 2 three three NUM iajs-3013 14 3 , , PUNCT iajs-3013 14 4 devoted devote VERB iajs-3013 14 5 to to ADP iajs-3013 14 6 introducing introduce VERB iajs-3013 14 7 the the DET iajs-3013 14 8 concept concept NOUN iajs-3013 14 9 of of ADP iajs-3013 14 10 strongly strongly ADV iajs-3013 14 11 almost almost ADV iajs-3013 14 12 approximately approximately ADV iajs-3013 14 13 nearly nearly ADV iajs-3013 14 14 quasi quasi ADJ iajs-3013 14 15 compactly compactly ADV iajs-3013 14 16 packed pack VERB iajs-3013 14 17 ( ( PUNCT iajs-3013 14 18 submodules submodule NOUN iajs-3013 14 19 ) ) PUNCT iajs-3013 14 20 modules module NOUN iajs-3013 14 21 . . PUNCT iajs-3013 15 1 also also ADV iajs-3013 15 2 , , PUNCT iajs-3013 15 3 we we PRON iajs-3013 15 4 introduce introduce VERB iajs-3013 15 5 the the DET iajs-3013 15 6 concepts concept NOUN iajs-3013 15 7 of of ADP iajs-3013 15 8 almost almost ADV iajs-3013 15 9 approximately approximately ADV iajs-3013 15 10 nearly nearly ADV iajs-3013 15 11 quasiprime quasiprime ADJ iajs-3013 15 12 radical radical ADJ iajs-3013 15 13 submodules submodule NOUN iajs-3013 15 14 and and CCONJ iajs-3013 15 15 almost almost ADV iajs-3013 15 16 approximately approximately ADV iajs-3013 15 17 nearly nearly ADV iajs-3013 15 18 quasiprime quasiprime ADJ iajs-3013 15 19 radical radical ADJ iajs-3013 15 20 submodules submodule NOUN iajs-3013 15 21 and and CCONJ iajs-3013 15 22 study study VERB iajs-3013 15 23 this this DET iajs-3013 15 24 concept concept NOUN iajs-3013 15 25 in in ADP iajs-3013 15 26 detail detail NOUN iajs-3013 15 27 . . PUNCT iajs-3013 16 1 finally finally ADV iajs-3013 16 2 , , PUNCT iajs-3013 16 3 we we PRON iajs-3013 16 4 doi.org/10.30526/36.1.3013 doi.org/10.30526/36.1.3013 NOUN iajs-3013 16 5 article article NOUN iajs-3013 16 6 history history NOUN iajs-3013 16 7 : : PUNCT iajs-3013 16 8 received receive VERB iajs-3013 16 9 14 14 NUM iajs-3013 16 10 september september PROPN iajs-3013 16 11 2022 2022 NUM iajs-3013 16 12 , , PUNCT iajs-3013 16 13 accepted accept VERB iajs-3013 16 14 6 6 NUM iajs-3013 16 15 november november PROPN iajs-3013 16 16 2022 2022 NUM iajs-3013 16 17 , , PUNCT iajs-3013 16 18 published publish VERB iajs-3013 16 19 in in ADP iajs-3013 16 20 january january PROPN iajs-3013 16 21 2023 2023 NUM iajs-3013 16 22 . . PUNCT iajs-3013 17 1 ibn ibn PROPN iajs-3013 17 2 al al PROPN iajs-3013 17 3 - - PUNCT iajs-3013 17 4 haitham haitham PROPN iajs-3013 17 5 journal journal PROPN iajs-3013 17 6 for for ADP iajs-3013 17 7 pure pure ADJ iajs-3013 17 8 and and CCONJ iajs-3013 17 9 applied applied ADJ iajs-3013 17 10 sciences sciences PROPN iajs-3013 17 11 journal journal PROPN iajs-3013 17 12 homepage homepage NOUN iajs-3013 17 13 : : PUNCT iajs-3013 17 14 jih.uobaghdad.edu.iq jih.uobaghdad.edu.iq PROPN iajs-3013 17 15 ali ali PROPN iajs-3013 17 16 sh sh PROPN iajs-3013 17 17 . . PROPN iajs-3013 17 18 ajeel ajeel PROPN iajs-3013 17 19 department department PROPN iajs-3013 17 20 of of ADP iajs-3013 17 21 mathematics mathematics PROPN iajs-3013 17 22 , , PUNCT iajs-3013 17 23 college college NOUN iajs-3013 17 24 of of ADP iajs-3013 17 25 computer computer NOUN iajs-3013 17 26 sciences sciences PROPN iajs-3013 17 27 and and CCONJ iajs-3013 17 28 mathematics mathematic NOUN iajs-3013 17 29 , , PUNCT iajs-3013 17 30 tikrit tikrit NOUN iajs-3013 17 31 university university PROPN iajs-3013 17 32 , , PUNCT iajs-3013 17 33 iraq iraq PROPN iajs-3013 17 34 . . PUNCT iajs-3013 18 1 ali.shebl@st.tu.edu.iq ali.shebl@st.tu.edu.iq PROPN iajs-3013 18 2 haibat haibat PROPN iajs-3013 18 3 k. k. PROPN iajs-3013 18 4 mohammadali mohammadali PROPN iajs-3013 18 5 department department PROPN iajs-3013 18 6 of of ADP iajs-3013 18 7 mathematics mathematics PROPN iajs-3013 18 8 , , PUNCT iajs-3013 18 9 college college NOUN iajs-3013 18 10 of of ADP iajs-3013 18 11 computer computer NOUN iajs-3013 18 12 sciences sciences PROPN iajs-3013 18 13 and and CCONJ iajs-3013 18 14 mathematics mathematic NOUN iajs-3013 18 15 , , PUNCT iajs-3013 18 16 tikrit tikrit NOUN iajs-3013 18 17 university university PROPN iajs-3013 18 18 , , PUNCT iajs-3013 18 19 iraq iraq PROPN iajs-3013 18 20 . . PUNCT iajs-3013 19 1 h.mohammadali@tu.edu.iq h.mohammadali@tu.edu.iq PROPN iajs-3013 19 2 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3013 19 3 mailto:ali.shebl@st.tu.edu.iq mailto:ali.shebl@st.tu.edu.iq PROPN iajs-3013 19 4 mailto:h.mohammadali@tu.edu.iq mailto:h.mohammadali@tu.edu.iq PROPN iajs-3013 19 5 ihjpas ihjpas PROPN iajs-3013 19 6 . . PUNCT iajs-3013 20 1 36(1)2023 36(1)2023 NUM iajs-3013 20 2 301 301 NUM iajs-3013 20 3 remark remark NOUN iajs-3013 20 4 that that SCONJ iajs-3013 20 5 all all DET iajs-3013 20 6 rings ring NOUN iajs-3013 20 7 in in ADP iajs-3013 20 8 this this DET iajs-3013 20 9 paper paper NOUN iajs-3013 20 10 are be AUX iajs-3013 20 11 commutative commutative ADJ iajs-3013 20 12 with with ADP iajs-3013 20 13 identity identity NOUN iajs-3013 20 14 and and CCONJ iajs-3013 20 15 all all DET iajs-3013 20 16 modules module NOUN iajs-3013 20 17 are be AUX iajs-3013 20 18 unitary unitary ADJ iajs-3013 20 19 left left ADJ iajs-3013 20 20 𝑅-module 𝑅-module NOUN iajs-3013 20 21 . . PUNCT iajs-3013 21 1 2 2 X iajs-3013 21 2 . . X iajs-3013 21 3 preliminaries preliminary NOUN iajs-3013 21 4 this this DET iajs-3013 21 5 section section NOUN iajs-3013 21 6 includes include VERB iajs-3013 21 7 some some DET iajs-3013 21 8 well well ADV iajs-3013 21 9 - - PUNCT iajs-3013 21 10 known know VERB iajs-3013 21 11 definitions definition NOUN iajs-3013 21 12 , , PUNCT iajs-3013 21 13 remarks remark VERB iajs-3013 21 14 , , PUNCT iajs-3013 21 15 and and CCONJ iajs-3013 21 16 propositions proposition NOUN iajs-3013 21 17 needed need VERB iajs-3013 21 18 in in ADP iajs-3013 21 19 our our PRON iajs-3013 21 20 study study NOUN iajs-3013 21 21 of of ADP iajs-3013 21 22 the the DET iajs-3013 21 23 next next ADJ iajs-3013 21 24 sections section NOUN iajs-3013 21 25 . . PUNCT iajs-3013 22 1 remark remark VERB iajs-3013 22 2 2.1 2.1 NUM iajs-3013 23 1 [ [ X iajs-3013 23 2 5 5 NUM iajs-3013 23 3 ] ] PUNCT iajs-3013 23 4 in in ADP iajs-3013 23 5 a a DET iajs-3013 23 6 finitely finitely ADV iajs-3013 23 7 generated generate VERB iajs-3013 23 8 𝑅-module 𝑅-module PROPN iajs-3013 23 9 , , PUNCT iajs-3013 23 10 every every DET iajs-3013 23 11 proper proper ADJ iajs-3013 23 12 submodule submodule NOUN iajs-3013 23 13 contend contend VERB iajs-3013 23 14 in in ADP iajs-3013 23 15 maximal maximal ADJ iajs-3013 23 16 submodule submodule NOUN iajs-3013 23 17 . . PUNCT iajs-3013 24 1 definition definition NOUN iajs-3013 24 2 2.2 2.2 NUM iajs-3013 24 3 [ [ X iajs-3013 24 4 6 6 NUM iajs-3013 24 5 ] ] PUNCT iajs-3013 24 6 an an DET iajs-3013 24 7 𝑅-module 𝑅-module PROPN iajs-3013 24 8 𝑄 𝑄 PROPN iajs-3013 24 9 is be AUX iajs-3013 24 10 multiplication multiplication NOUN iajs-3013 24 11 if if SCONJ iajs-3013 24 12 every every DET iajs-3013 24 13 submodule submodule NOUN iajs-3013 24 14 𝐹 𝐹 PROPN iajs-3013 24 15 of of ADP iajs-3013 24 16 𝑄 𝑄 PROPN iajs-3013 24 17 is be AUX iajs-3013 24 18 of of ADP iajs-3013 24 19 the the DET iajs-3013 24 20 form form NOUN iajs-3013 24 21 𝐹 𝐹 PROPN iajs-3013 24 22 = = SYM iajs-3013 24 23 𝐼𝑄 𝐼𝑄 PROPN iajs-3013 24 24 for for ADP iajs-3013 24 25 some some DET iajs-3013 24 26 ideal ideal ADJ iajs-3013 24 27 𝐼 𝐼 ADP iajs-3013 24 28 of of ADP iajs-3013 24 29 𝑅. 𝑅. NOUN iajs-3013 24 30 proposition proposition NOUN iajs-3013 24 31 2.3 2.3 NUM iajs-3013 24 32 [ [ X iajs-3013 24 33 7 7 NUM iajs-3013 24 34 ] ] PUNCT iajs-3013 24 35 let let VERB iajs-3013 24 36 𝑄 𝑄 PRON iajs-3013 24 37 be be AUX iajs-3013 24 38 a a DET iajs-3013 24 39 non non ADJ iajs-3013 24 40 - - ADJ iajs-3013 24 41 zero zero NUM iajs-3013 24 42 multiplication multiplication NOUN iajs-3013 24 43 𝑅-module 𝑅-module NOUN iajs-3013 24 44 , , PUNCT iajs-3013 24 45 then then ADV iajs-3013 24 46 every every DET iajs-3013 24 47 proper proper ADJ iajs-3013 24 48 submodule submodule NOUN iajs-3013 24 49 of of ADP iajs-3013 24 50 𝑄 𝑄 PRON iajs-3013 24 51 contend contend VERB iajs-3013 24 52 in in ADP iajs-3013 24 53 a a DET iajs-3013 24 54 maximal maximal ADJ iajs-3013 24 55 submodule submodule NOUN iajs-3013 24 56 . . PUNCT iajs-3013 25 1 definition definition NOUN iajs-3013 25 2 2.4 2.4 NUM iajs-3013 25 3 [ [ SYM iajs-3013 25 4 8 8 NUM iajs-3013 25 5 ] ] PUNCT iajs-3013 25 6 a a DET iajs-3013 25 7 submodule submodule NOUN iajs-3013 25 8 𝐹 𝐹 PROPN iajs-3013 25 9 of of ADP iajs-3013 25 10 an an DET iajs-3013 25 11 𝑅-module 𝑅-module PROPN iajs-3013 25 12 𝑄 𝑄 PROPN iajs-3013 25 13 is be AUX iajs-3013 25 14 called call VERB iajs-3013 25 15 small small ADJ iajs-3013 25 16 if if SCONJ iajs-3013 25 17 𝐹 𝐹 PROPN iajs-3013 25 18 + + CCONJ iajs-3013 25 19 𝐾 𝐾 NOUN iajs-3013 25 20 = = SYM iajs-3013 25 21 𝑄 𝑄 PROPN iajs-3013 25 22 implies imply VERB iajs-3013 25 23 that that SCONJ iajs-3013 25 24 𝐾 𝐾 PROPN iajs-3013 25 25 = = PUNCT iajs-3013 25 26 𝑄 𝑄 PROPN iajs-3013 25 27 for for ADP iajs-3013 25 28 any any DET iajs-3013 25 29 proper proper ADJ iajs-3013 25 30 submodule submodule NOUN iajs-3013 25 31 𝐾 𝐾 PROPN iajs-3013 25 32 of of ADP iajs-3013 25 33 𝑄. 𝑄. PROPN iajs-3013 25 34 proposition proposition NOUN iajs-3013 25 35 2.5 2.5 NUM iajs-3013 25 36 [ [ X iajs-3013 25 37 1 1 X iajs-3013 25 38 ] ] PUNCT iajs-3013 25 39 let let VERB iajs-3013 25 40 𝑓 𝑓 X iajs-3013 25 41 : : PUNCT iajs-3013 25 42 𝑄 𝑄 PROPN iajs-3013 25 43 β†’ β†’ SYM iajs-3013 25 44 𝑄′ 𝑄′ ADJ iajs-3013 25 45 be be VERB iajs-3013 25 46 an an DET iajs-3013 25 47 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-3013 25 48 , , PUNCT iajs-3013 25 49 and and CCONJ iajs-3013 25 50 ker ker X iajs-3013 25 51 𝑓 𝑓 PRON iajs-3013 25 52 is be AUX iajs-3013 25 53 small small ADJ iajs-3013 25 54 submodule submodule NOUN iajs-3013 25 55 for for ADP iajs-3013 25 56 𝑄. 𝑄. PROPN iajs-3013 25 57 if if SCONJ iajs-3013 25 58 𝐹 𝐹 PROPN iajs-3013 25 59 is be AUX iajs-3013 25 60 an an DET iajs-3013 25 61 alappnqprime alappnqprime NOUN iajs-3013 25 62 submodule submodule NOUN iajs-3013 25 63 for for ADP iajs-3013 25 64 𝑄′ 𝑄′ PROPN iajs-3013 25 65 then then ADV iajs-3013 25 66 π‘“βˆ’1(𝐹 π‘“βˆ’1(𝐹 X iajs-3013 25 67 ) ) PUNCT iajs-3013 25 68 is be AUX iajs-3013 25 69 alappnq alappnq ADJ iajs-3013 25 70 - - PUNCT iajs-3013 25 71 prime prime NOUN iajs-3013 25 72 submodule submodule NOUN iajs-3013 25 73 for for ADP iajs-3013 25 74 𝑄. 𝑄. NOUN iajs-3013 25 75 proposition proposition NOUN iajs-3013 25 76 2.6 2.6 NUM iajs-3013 25 77 [ [ X iajs-3013 25 78 1 1 NUM iajs-3013 25 79 ] ] PUNCT iajs-3013 25 80 let let VERB iajs-3013 25 81 𝑓 𝑓 X iajs-3013 25 82 : : PUNCT iajs-3013 25 83 𝑄 𝑄 PROPN iajs-3013 25 84 β†’ β†’ SYM iajs-3013 25 85 𝑄′ 𝑄′ ADJ iajs-3013 25 86 be be VERB iajs-3013 25 87 an an DET iajs-3013 25 88 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-3013 25 89 , , PUNCT iajs-3013 25 90 and and CCONJ iajs-3013 25 91 ker ker X iajs-3013 25 92 𝑓 𝑓 PRON iajs-3013 25 93 is be AUX iajs-3013 25 94 small small ADJ iajs-3013 25 95 submodule submodule NOUN iajs-3013 25 96 for for ADP iajs-3013 25 97 𝑄. 𝑄. PROPN iajs-3013 25 98 if if SCONJ iajs-3013 25 99 𝐹 𝐹 PRON iajs-3013 25 100 be be VERB iajs-3013 25 101 an an DET iajs-3013 25 102 alappnqprime alappnqprime NOUN iajs-3013 25 103 submodule submodule NOUN iajs-3013 25 104 for for ADP iajs-3013 25 105 𝑄 𝑄 PRON iajs-3013 25 106 with with ADP iajs-3013 25 107 πΎπ‘’π‘Ÿ πΎπ‘’π‘Ÿ PROPN iajs-3013 26 1 𝑓 𝑓 PRON iajs-3013 26 2 βŠ† βŠ† NUM iajs-3013 26 3 𝐹 𝐹 PROPN iajs-3013 26 4 then then ADV iajs-3013 26 5 𝑓(𝐹 𝑓(𝐹 NOUN iajs-3013 26 6 ) ) PUNCT iajs-3013 26 7 is be AUX iajs-3013 26 8 alappnq alappnq ADJ iajs-3013 26 9 - - PUNCT iajs-3013 26 10 prime prime NOUN iajs-3013 26 11 submodule submodule NOUN iajs-3013 26 12 for for ADP iajs-3013 26 13 𝑄′. 𝑄′. ADJ iajs-3013 26 14 definition definition NOUN iajs-3013 26 15 2.7 2.7 NUM iajs-3013 26 16 [ [ X iajs-3013 26 17 9 9 NUM iajs-3013 26 18 ] ] PUNCT iajs-3013 26 19 a a DET iajs-3013 26 20 subset subset NOUN iajs-3013 26 21 𝑆 𝑆 PROPN iajs-3013 26 22 of of ADP iajs-3013 26 23 a a DET iajs-3013 26 24 ring ring NOUN iajs-3013 26 25 𝑅 𝑅 PROPN iajs-3013 26 26 is be AUX iajs-3013 26 27 called call VERB iajs-3013 26 28 multiplicatively multiplicatively ADV iajs-3013 26 29 closed close VERB iajs-3013 26 30 if if SCONJ iajs-3013 26 31 1 1 NUM iajs-3013 26 32 ∈ ∈ PROPN iajs-3013 26 33 𝑆 𝑆 PROPN iajs-3013 26 34 and and CCONJ iajs-3013 26 35 π‘Žπ‘ π‘Žπ‘ PROPN iajs-3013 26 36 ∈ ∈ PROPN iajs-3013 26 37 𝑆 𝑆 PROPN iajs-3013 26 38 for for ADP iajs-3013 26 39 every every DET iajs-3013 26 40 π‘Ž π‘Ž NOUN iajs-3013 26 41 , , PUNCT iajs-3013 26 42 𝑏 𝑏 PROPN iajs-3013 26 43 ∈ ∈ PROPN iajs-3013 26 44 𝑆. 𝑆. NOUN iajs-3013 26 45 let let VERB iajs-3013 26 46 𝑇 𝑇 PROPN iajs-3013 26 47 be be AUX iajs-3013 26 48 the the DET iajs-3013 26 49 set set NOUN iajs-3013 26 50 of of ADP iajs-3013 26 51 all all DET iajs-3013 26 52 order order NOUN iajs-3013 26 53 pairs pair NOUN iajs-3013 26 54 ( ( PUNCT iajs-3013 26 55 π‘ž π‘ž NOUN iajs-3013 26 56 , , PUNCT iajs-3013 26 57 𝑠 𝑠 PROPN iajs-3013 26 58 ) ) PUNCT iajs-3013 26 59 where where SCONJ iajs-3013 26 60 π‘ž π‘ž PROPN iajs-3013 26 61 ∈ ∈ PROPN iajs-3013 26 62 𝑄 𝑄 PROPN iajs-3013 26 63 and and CCONJ iajs-3013 26 64 𝑠 𝑠 PRON iajs-3013 26 65 ∈ ∈ NOUN iajs-3013 26 66 𝑆. 𝑆. VERB iajs-3013 26 67 the the DET iajs-3013 26 68 relation relation NOUN iajs-3013 26 69 on on ADP iajs-3013 26 70 𝑇 𝑇 PROPN iajs-3013 26 71 is be AUX iajs-3013 26 72 defined define VERB iajs-3013 26 73 by by ADP iajs-3013 26 74 ( ( PUNCT iajs-3013 26 75 π‘ž π‘ž PROPN iajs-3013 26 76 , , PUNCT iajs-3013 26 77 𝑠)~(π‘žβ€² 𝑠)~(π‘žβ€² PROPN iajs-3013 26 78 , , PUNCT iajs-3013 26 79 𝑠′ 𝑠′ NOUN iajs-3013 26 80 ) ) PUNCT iajs-3013 26 81 if if SCONJ iajs-3013 26 82 there there PRON iajs-3013 26 83 exists exist VERB iajs-3013 26 84 𝑑 𝑑 PROPN iajs-3013 26 85 ∈ ∈ PROPN iajs-3013 26 86 𝑆 𝑆 PROPN iajs-3013 26 87 such such ADJ iajs-3013 26 88 that that SCONJ iajs-3013 26 89 𝑑(π‘ π‘žβ€² 𝑑(π‘ π‘žβ€² PUNCT iajs-3013 26 90 βˆ’ βˆ’ PROPN iajs-3013 26 91 π‘ β€²π‘ž π‘ β€²π‘ž NOUN iajs-3013 27 1 ) ) PUNCT iajs-3013 27 2 = = SYM iajs-3013 27 3 0 0 NUM iajs-3013 27 4 is be AUX iajs-3013 27 5 an an DET iajs-3013 27 6 equivalence equivalence NOUN iajs-3013 27 7 relation relation NOUN iajs-3013 27 8 . . PUNCT iajs-3013 28 1 we we PRON iajs-3013 28 2 denote denote VERB iajs-3013 28 3 the the DET iajs-3013 28 4 equivalence equivalence NOUN iajs-3013 28 5 classes class NOUN iajs-3013 28 6 of of ADP iajs-3013 28 7 ( ( PUNCT iajs-3013 28 8 π‘ž π‘ž PROPN iajs-3013 28 9 , , PUNCT iajs-3013 28 10 𝑠 𝑠 PROPN iajs-3013 28 11 ) ) PUNCT iajs-3013 28 12 by by ADP iajs-3013 28 13 π‘ž π‘ž PROPN iajs-3013 28 14 𝑠 𝑠 PROPN iajs-3013 28 15 . . PUNCT iajs-3013 29 1 let let VERB iajs-3013 29 2 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 29 3 denote denote VERB iajs-3013 29 4 the the DET iajs-3013 29 5 set set NOUN iajs-3013 29 6 of of ADP iajs-3013 29 7 all all DET iajs-3013 29 8 equivalence equivalence NOUN iajs-3013 29 9 classes class NOUN iajs-3013 29 10 𝑇 𝑇 PROPN iajs-3013 29 11 with with ADP iajs-3013 29 12 respect respect NOUN iajs-3013 29 13 to to ADP iajs-3013 29 14 this this DET iajs-3013 29 15 relation relation NOUN iajs-3013 29 16 . . PUNCT iajs-3013 30 1 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 30 2 is be AUX iajs-3013 30 3 an an DET iajs-3013 30 4 𝑅-module 𝑅-module PROPN iajs-3013 30 5 . . PUNCT iajs-3013 31 1 definition definition NOUN iajs-3013 31 2 2.8 2.8 NUM iajs-3013 31 3 [ [ SYM iajs-3013 31 4 10 10 NUM iajs-3013 31 5 ] ] PUNCT iajs-3013 31 6 an an DET iajs-3013 31 7 𝑅-module 𝑅-module PROPN iajs-3013 31 8 𝑄 𝑄 PROPN iajs-3013 31 9 is be AUX iajs-3013 31 10 𝑍-regular 𝑍-regular ADJ iajs-3013 31 11 if if SCONJ iajs-3013 31 12 for for ADP iajs-3013 31 13 each each DET iajs-3013 31 14 π‘ž π‘ž PROPN iajs-3013 31 15 ∈ ∈ PROPN iajs-3013 31 16 𝑄 𝑄 PROPN iajs-3013 31 17 there there ADV iajs-3013 31 18 exists exist VERB iajs-3013 31 19 𝑓 𝑓 DET iajs-3013 31 20 ∈ ∈ NOUN iajs-3013 31 21 𝑄′ 𝑄′ ADJ iajs-3013 31 22 = = SYM iajs-3013 31 23 π»π‘œπ‘šπ‘…(𝑄 π»π‘œπ‘šπ‘…(𝑄 PROPN iajs-3013 31 24 , , PUNCT iajs-3013 31 25 𝑅 𝑅 NOUN iajs-3013 31 26 ) ) PUNCT iajs-3013 31 27 such such ADJ iajs-3013 31 28 that that SCONJ iajs-3013 31 29 π‘ž π‘ž X iajs-3013 31 30 = = SYM iajs-3013 31 31 𝑓(π‘ž)π‘ž. 𝑓(π‘ž)π‘ž. PROPN iajs-3013 31 32 proposition proposition NOUN iajs-3013 31 33 2.9 2.9 NUM iajs-3013 32 1 [ [ X iajs-3013 32 2 5 5 NUM iajs-3013 32 3 ] ] PUNCT iajs-3013 32 4 let let VERB iajs-3013 32 5 𝑄 𝑄 PRON iajs-3013 32 6 be be AUX iajs-3013 32 7 an an DET iajs-3013 32 8 𝑅-module 𝑅-module PROPN iajs-3013 32 9 then then ADV iajs-3013 32 10 the the DET iajs-3013 32 11 following following ADJ iajs-3013 32 12 statements statement NOUN iajs-3013 32 13 are be AUX iajs-3013 32 14 equivalent equivalent ADJ iajs-3013 32 15 : : PUNCT iajs-3013 33 1 1 1 X iajs-3013 33 2 . . X iajs-3013 33 3 every every DET iajs-3013 33 4 proper proper ADJ iajs-3013 33 5 submodule submodule NOUN iajs-3013 33 6 of of ADP iajs-3013 33 7 𝑄 𝑄 PROPN iajs-3013 33 8 is be AUX iajs-3013 33 9 a a DET iajs-3013 33 10 semi semi ADJ iajs-3013 33 11 prime prime NOUN iajs-3013 33 12 . . PUNCT iajs-3013 34 1 2 2 X iajs-3013 34 2 . . X iajs-3013 34 3 every every DET iajs-3013 34 4 proper proper ADJ iajs-3013 34 5 submodule submodule NOUN iajs-3013 34 6 of of ADP iajs-3013 34 7 𝑄 𝑄 PROPN iajs-3013 34 8 is be AUX iajs-3013 34 9 the the DET iajs-3013 34 10 intersection intersection NOUN iajs-3013 34 11 of of ADP iajs-3013 34 12 prime prime ADJ iajs-3013 34 13 submodule submodule NOUN iajs-3013 34 14 of of ADP iajs-3013 34 15 𝑄. 𝑄. PROPN iajs-3013 34 16 proposition proposition NOUN iajs-3013 34 17 2.10 2.10 NUM iajs-3013 34 18 [ [ SYM iajs-3013 34 19 5 5 NUM iajs-3013 34 20 ] ] PUNCT iajs-3013 34 21 let let VERB iajs-3013 34 22 𝑄 𝑄 PRON iajs-3013 34 23 be be AUX iajs-3013 34 24 a a DET iajs-3013 34 25 non non ADJ iajs-3013 34 26 - - ADJ iajs-3013 34 27 zero zero ADJ iajs-3013 34 28 𝑍-regular 𝑍-regular PROPN iajs-3013 34 29 𝑅-module 𝑅-module PROPN iajs-3013 34 30 , , PUNCT iajs-3013 34 31 then then ADV iajs-3013 34 32 every every DET iajs-3013 34 33 proper proper ADJ iajs-3013 34 34 submodule submodule NOUN iajs-3013 34 35 of of ADP iajs-3013 34 36 𝑄 𝑄 PROPN iajs-3013 34 37 is be AUX iajs-3013 34 38 a a DET iajs-3013 34 39 semi semi ADJ iajs-3013 34 40 prime prime NOUN iajs-3013 34 41 . . PUNCT iajs-3013 35 1 from from ADP iajs-3013 35 2 propositions proposition NOUN iajs-3013 35 3 2.9 2.9 NUM iajs-3013 35 4 and and CCONJ iajs-3013 35 5 2.10 2.10 NUM iajs-3013 35 6 , , PUNCT iajs-3013 35 7 we we PRON iajs-3013 35 8 get get VERB iajs-3013 35 9 the the DET iajs-3013 35 10 following follow VERB iajs-3013 35 11 corollary corollary NOUN iajs-3013 35 12 . . PUNCT iajs-3013 36 1 ihjpas ihjpas PROPN iajs-3013 36 2 . . PUNCT iajs-3013 37 1 36(1)2023 36(1)2023 NUM iajs-3013 37 2 302 302 NUM iajs-3013 37 3 corollary corollary NOUN iajs-3013 37 4 2.11 2.11 NUM iajs-3013 37 5 let let VERB iajs-3013 37 6 𝑄 𝑄 PRON iajs-3013 37 7 be be AUX iajs-3013 37 8 𝑍-regular 𝑍-regular PROPN iajs-3013 37 9 𝑅-module 𝑅-module PROPN iajs-3013 37 10 , , PUNCT iajs-3013 37 11 then then ADV iajs-3013 37 12 every every DET iajs-3013 37 13 proper proper ADJ iajs-3013 37 14 submodule submodule NOUN iajs-3013 37 15 of of ADP iajs-3013 37 16 𝑄 𝑄 PROPN iajs-3013 37 17 is be AUX iajs-3013 37 18 the the DET iajs-3013 37 19 intersection intersection NOUN iajs-3013 37 20 of of ADP iajs-3013 37 21 a a DET iajs-3013 37 22 prime prime ADJ iajs-3013 37 23 submodule submodule NOUN iajs-3013 37 24 of of ADP iajs-3013 37 25 𝑄. 𝑄. PROPN iajs-3013 37 26 remark remark NOUN iajs-3013 37 27 2.12 2.12 NUM iajs-3013 37 28 [ [ X iajs-3013 37 29 1 1 NUM iajs-3013 37 30 ] ] PUNCT iajs-3013 37 31 every every DET iajs-3013 37 32 prime prime ADJ iajs-3013 37 33 submodule submodule PROPN iajs-3013 37 34 𝐹 𝐹 PROPN iajs-3013 37 35 of of ADP iajs-3013 37 36 an an DET iajs-3013 37 37 𝑅-module 𝑅-module PROPN iajs-3013 37 38 𝑄 𝑄 PROPN iajs-3013 37 39 is be AUX iajs-3013 37 40 an an DET iajs-3013 37 41 alappnq alappnq ADJ iajs-3013 37 42 - - PUNCT iajs-3013 37 43 prime prime NOUN iajs-3013 37 44 submodule submodule NOUN iajs-3013 37 45 of of ADP iajs-3013 37 46 𝑄. 𝑄. PROPN iajs-3013 37 47 definition definition NOUN iajs-3013 37 48 2.13 2.13 NUM iajs-3013 37 49 [ [ X iajs-3013 37 50 11 11 NUM iajs-3013 37 51 ] ] PUNCT iajs-3013 37 52 an an DET iajs-3013 37 53 𝑅-module 𝑅-module PROPN iajs-3013 37 54 𝑄 𝑄 PROPN iajs-3013 37 55 is be AUX iajs-3013 37 56 faithful faithful ADJ iajs-3013 37 57 if if SCONJ iajs-3013 37 58 π‘Žπ‘›π‘›π‘…(𝑄 π‘Žπ‘›π‘›π‘…(𝑄 NOUN iajs-3013 37 59 ) ) PUNCT iajs-3013 37 60 = = SYM iajs-3013 37 61 ( ( PUNCT iajs-3013 37 62 0 0 NUM iajs-3013 37 63 ) ) PUNCT iajs-3013 37 64 . . PUNCT iajs-3013 38 1 proposition proposition NOUN iajs-3013 38 2 2.14 2.14 NUM iajs-3013 39 1 [ [ X iajs-3013 39 2 13 13 NUM iajs-3013 39 3 ] ] X iajs-3013 39 4 a a DET iajs-3013 39 5 proper proper ADJ iajs-3013 39 6 submodule submodule NOUN iajs-3013 39 7 𝐹 𝐹 PROPN iajs-3013 39 8 of of ADP iajs-3013 39 9 faithful faithful ADJ iajs-3013 39 10 multiplication multiplication NOUN iajs-3013 39 11 𝑅-module 𝑅-module PROPN iajs-3013 39 12 𝑄 𝑄 PROPN iajs-3013 39 13 is be AUX iajs-3013 39 14 an an DET iajs-3013 39 15 alappnq alappnq ADJ iajs-3013 39 16 - - PUNCT iajs-3013 39 17 prime prime NOUN iajs-3013 39 18 submodule submodule NOUN iajs-3013 39 19 of of ADP iajs-3013 39 20 𝑄 𝑄 PRON iajs-3013 39 21 if if SCONJ iajs-3013 39 22 and and CCONJ iajs-3013 39 23 only only ADV iajs-3013 39 24 if if SCONJ iajs-3013 39 25 [ [ X iajs-3013 39 26 𝐹:𝑅 𝐹:𝑅 ADP iajs-3013 39 27 𝑄 𝑄 NOUN iajs-3013 39 28 ] ] PUNCT iajs-3013 39 29 is be AUX iajs-3013 39 30 an an DET iajs-3013 39 31 alappnq alappnq ADJ iajs-3013 39 32 - - PUNCT iajs-3013 39 33 prime prime NOUN iajs-3013 39 34 ideal ideal NOUN iajs-3013 39 35 of of ADP iajs-3013 39 36 𝑅. 𝑅. NOUN iajs-3013 39 37 definition definition NOUN iajs-3013 39 38 2.15 2.15 NUM iajs-3013 39 39 [ [ X iajs-3013 39 40 9 9 NUM iajs-3013 39 41 ] ] PUNCT iajs-3013 39 42 β€œ " PUNCT iajs-3013 39 43 an an DET iajs-3013 39 44 𝑅-module 𝑅-module PROPN iajs-3013 39 45 𝑄 𝑄 PROPN iajs-3013 39 46 is be AUX iajs-3013 39 47 called call VERB iajs-3013 39 48 bezout bezout NOUN iajs-3013 39 49 module module NOUN iajs-3013 39 50 if if SCONJ iajs-3013 39 51 every every DET iajs-3013 39 52 finitely finitely ADV iajs-3013 39 53 generated generate VERB iajs-3013 39 54 submodule submodule NOUN iajs-3013 39 55 of of ADP iajs-3013 39 56 𝑄 𝑄 PROPN iajs-3013 39 57 is be AUX iajs-3013 39 58 cyclic cyclic ADJ iajs-3013 39 59 ” " PUNCT iajs-3013 39 60 . . PUNCT iajs-3013 40 1 3 3 X iajs-3013 40 2 . . X iajs-3013 40 3 almost almost ADV iajs-3013 40 4 approximately approximately ADV iajs-3013 40 5 nearly nearly ADV iajs-3013 40 6 quasi quasi ADJ iajs-3013 40 7 compactly compactly ADV iajs-3013 40 8 packed pack VERB iajs-3013 40 9 modules module NOUN iajs-3013 40 10 before before SCONJ iajs-3013 40 11 we we PRON iajs-3013 40 12 introduce introduce VERB iajs-3013 40 13 the the DET iajs-3013 40 14 concept concept NOUN iajs-3013 40 15 of of ADP iajs-3013 40 16 almost almost ADV iajs-3013 40 17 approximately approximately ADV iajs-3013 40 18 nearly nearly ADV iajs-3013 40 19 quasi quasi ADJ iajs-3013 40 20 compactly compactly ADV iajs-3013 40 21 packed pack VERB iajs-3013 40 22 modules module NOUN iajs-3013 40 23 , , PUNCT iajs-3013 40 24 and and CCONJ iajs-3013 40 25 study study VERB iajs-3013 40 26 some some DET iajs-3013 40 27 properties property NOUN iajs-3013 40 28 , , PUNCT iajs-3013 40 29 we we PRON iajs-3013 40 30 need need VERB iajs-3013 40 31 to to PART iajs-3013 40 32 define define VERB iajs-3013 40 33 the the DET iajs-3013 40 34 concept concept NOUN iajs-3013 40 35 of of ADP iajs-3013 40 36 almost almost ADV iajs-3013 40 37 approximately approximately ADV iajs-3013 40 38 nearly nearly ADV iajs-3013 40 39 quasi quasi ADJ iajs-3013 40 40 compactly compactly ADV iajs-3013 40 41 packed pack VERB iajs-3013 40 42 submodules submodule NOUN iajs-3013 40 43 . . PUNCT iajs-3013 41 1 definition definition NOUN iajs-3013 41 2 3.1 3.1 NUM iajs-3013 41 3 a a DET iajs-3013 41 4 proper proper ADJ iajs-3013 41 5 submodule submodule NOUN iajs-3013 41 6 𝐹 𝐹 PROPN iajs-3013 41 7 of of ADP iajs-3013 41 8 an an DET iajs-3013 41 9 𝑅-module 𝑅-module PROPN iajs-3013 41 10 𝑄 𝑄 PROPN iajs-3013 41 11 is be AUX iajs-3013 41 12 called call VERB iajs-3013 41 13 almost almost ADV iajs-3013 41 14 approximately approximately ADV iajs-3013 41 15 nearly nearly ADV iajs-3013 41 16 quasi quasi ADJ iajs-3013 41 17 compactly compactly ADV iajs-3013 41 18 packed pack VERB iajs-3013 41 19 ( ( PUNCT iajs-3013 41 20 simply simply ADV iajs-3013 41 21 alappnq alappnq PROPN iajs-3013 41 22 compactly compactly ADV iajs-3013 41 23 packed pack VERB iajs-3013 41 24 ) ) PUNCT iajs-3013 41 25 if if SCONJ iajs-3013 41 26 for for ADP iajs-3013 41 27 each each DET iajs-3013 41 28 family family NOUN iajs-3013 41 29 { { PUNCT iajs-3013 41 30 𝐹𝛼}π›ΌβˆˆΚŒ 𝐹𝛼}π›ΌβˆˆΚŒ PUNCT iajs-3013 41 31 of of ADP iajs-3013 41 32 alappnq alappnq ADJ iajs-3013 41 33 - - PUNCT iajs-3013 41 34 prime prime NOUN iajs-3013 41 35 submodules submodule NOUN iajs-3013 41 36 of of ADP iajs-3013 41 37 𝑄 𝑄 PRON iajs-3013 41 38 with with ADP iajs-3013 41 39 𝐹 𝐹 PROPN iajs-3013 41 40 βŠ† βŠ† NUM iajs-3013 41 41 ⋃ ⋃ ADP iajs-3013 41 42 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 41 43 there there PRON iajs-3013 41 44 exists exist VERB iajs-3013 41 45 𝛼1 𝛼1 NOUN iajs-3013 41 46 , , PUNCT iajs-3013 41 47 𝛼2 𝛼2 VERB iajs-3013 41 48 , , PUNCT iajs-3013 41 49 … … PUNCT iajs-3013 41 50 , , PUNCT iajs-3013 41 51 𝛼𝑛 𝛼𝑛 PROPN iajs-3013 41 52 ∈ ∈ PROPN iajs-3013 41 53 ʌ ʌ NOUN iajs-3013 41 54 such such ADJ iajs-3013 41 55 that that SCONJ iajs-3013 41 56 𝐹 𝐹 PROPN iajs-3013 41 57 βŠ† βŠ† NUM iajs-3013 41 58 ⋃ ⋃ ADP iajs-3013 41 59 𝐹𝛼𝑖 𝐹𝛼𝑖 PROPN iajs-3013 41 60 . . PUNCT iajs-3013 42 1 𝑛 𝑛 PRON iajs-3013 42 2 𝑖=1 𝑖=1 PROPN iajs-3013 42 3 definition definition NOUN iajs-3013 42 4 3.2 3.2 NUM iajs-3013 42 5 an an DET iajs-3013 42 6 𝑅-module 𝑅-module PROPN iajs-3013 42 7 𝑄 𝑄 PROPN iajs-3013 42 8 is be AUX iajs-3013 42 9 called call VERB iajs-3013 42 10 alappnq alappnq NOUN iajs-3013 42 11 compactly compactly ADV iajs-3013 42 12 packed pack VERB iajs-3013 42 13 if if SCONJ iajs-3013 42 14 every every DET iajs-3013 42 15 proper proper ADJ iajs-3013 42 16 submodule submodule NOUN iajs-3013 42 17 of of ADP iajs-3013 42 18 𝑄 𝑄 PROPN iajs-3013 42 19 is be AUX iajs-3013 42 20 alappnq alappnq NOUN iajs-3013 42 21 compactly compactly ADV iajs-3013 42 22 packed pack VERB iajs-3013 42 23 . . PUNCT iajs-3013 43 1 remarks remark NOUN iajs-3013 43 2 and and CCONJ iajs-3013 43 3 examples example NOUN iajs-3013 43 4 3.3 3.3 NUM iajs-3013 43 5 1 1 NUM iajs-3013 43 6 . . PUNCT iajs-3013 44 1 𝑍6 𝑍6 PROPN iajs-3013 44 2 as as SCONJ iajs-3013 44 3 𝑍-module 𝑍-module ADJ iajs-3013 44 4 alappnq alappnq NOUN iajs-3013 44 5 compactly compactly ADV iajs-3013 44 6 packed pack VERB iajs-3013 44 7 𝑍-module 𝑍-module PROPN iajs-3013 44 8 . . PROPN iajs-3013 44 9 2 2 NUM iajs-3013 44 10 . . X iajs-3013 45 1 every every DET iajs-3013 45 2 module module NOUN iajs-3013 45 3 contains contain VERB iajs-3013 45 4 a a DET iajs-3013 45 5 finite finite ADJ iajs-3013 45 6 number number NOUN iajs-3013 45 7 of of ADP iajs-3013 45 8 alappnq alappnq ADJ iajs-3013 45 9 - - PUNCT iajs-3013 45 10 prime prime NOUN iajs-3013 45 11 submodules submodule NOUN iajs-3013 45 12 is be AUX iajs-3013 45 13 alappnq alappnq NOUN iajs-3013 45 14 compactly compactly ADV iajs-3013 45 15 packed pack VERB iajs-3013 45 16 . . PUNCT iajs-3013 46 1 3 3 X iajs-3013 46 2 . . X iajs-3013 46 3 every every DET iajs-3013 46 4 proper proper ADJ iajs-3013 46 5 finite finite ADJ iajs-3013 46 6 submodule submodule NOUN iajs-3013 46 7 of of ADP iajs-3013 46 8 an an DET iajs-3013 46 9 𝑅-module 𝑅-module PROPN iajs-3013 46 10 𝑄 𝑄 PROPN iajs-3013 46 11 is be AUX iajs-3013 46 12 an an DET iajs-3013 46 13 alappnq alappnq NOUN iajs-3013 46 14 compactly compactly ADV iajs-3013 46 15 packed pack VERB iajs-3013 46 16 . . PUNCT iajs-3013 47 1 proposition proposition NOUN iajs-3013 47 2 3.4 3.4 NUM iajs-3013 47 3 let let VERB iajs-3013 47 4 𝑄 𝑄 PRON iajs-3013 47 5 be be AUX iajs-3013 47 6 alappnq alappnq NOUN iajs-3013 47 7 compactly compactly ADV iajs-3013 47 8 packed pack VERB iajs-3013 47 9 𝑅-module 𝑅-module PROPN iajs-3013 47 10 with with ADP iajs-3013 47 11 𝐽(𝑄 𝐽(𝑄 NOUN iajs-3013 47 12 ) ) PUNCT iajs-3013 47 13 β‰  β‰  PROPN iajs-3013 47 14 𝑄 𝑄 PROPN iajs-3013 47 15 , , PUNCT iajs-3013 47 16 then then ADV iajs-3013 47 17 𝑄 𝑄 PRON iajs-3013 47 18 satisfies satisfy VERB iajs-3013 47 19 the the DET iajs-3013 47 20 ascending ascend VERB iajs-3013 47 21 chain chain NOUN iajs-3013 47 22 condition condition NOUN iajs-3013 47 23 for for ADP iajs-3013 47 24 alappnq alappnq ADJ iajs-3013 47 25 - - PUNCT iajs-3013 47 26 prime prime NOUN iajs-3013 47 27 submodules submodule NOUN iajs-3013 47 28 . . PUNCT iajs-3013 48 1 proof proof NOUN iajs-3013 48 2 let let VERB iajs-3013 48 3 𝐿1 𝐿1 VERB iajs-3013 48 4 βŠ† βŠ† NUM iajs-3013 48 5 𝐿2 𝐿2 PROPN iajs-3013 48 6 βŠ† βŠ† NUM iajs-3013 48 7 𝐿3 𝐿3 NOUN iajs-3013 48 8 βŠ† βŠ† NUM iajs-3013 48 9 β‹― β‹― NOUN iajs-3013 48 10 be be AUX iajs-3013 48 11 ascending ascend VERB iajs-3013 48 12 chain chain NOUN iajs-3013 48 13 of of ADP iajs-3013 48 14 alappnq alappnq ADJ iajs-3013 48 15 - - PUNCT iajs-3013 48 16 prime prime NOUN iajs-3013 48 17 submodules submodule NOUN iajs-3013 48 18 of of ADP iajs-3013 48 19 𝑄. 𝑄. NOUN iajs-3013 48 20 let let VERB iajs-3013 48 21 𝐿 𝐿 PROPN iajs-3013 48 22 = = PROPN iajs-3013 48 23 ⋃ ⋃ PROPN iajs-3013 48 24 𝐿𝑖𝑖 𝐿𝑖𝑖 PROPN iajs-3013 48 25 , , PUNCT iajs-3013 48 26 we we PRON iajs-3013 48 27 claim claim VERB iajs-3013 48 28 that that SCONJ iajs-3013 48 29 𝐿 𝐿 PROPN iajs-3013 48 30 β‰  β‰  PROPN iajs-3013 48 31 𝑄. 𝑄. NOUN iajs-3013 48 32 in in ADP iajs-3013 48 33 fact fact NOUN iajs-3013 48 34 if if SCONJ iajs-3013 48 35 𝐿 𝐿 PROPN iajs-3013 48 36 = = SYM iajs-3013 48 37 𝑄 𝑄 PROPN iajs-3013 48 38 and and CCONJ iajs-3013 48 39 𝐻 𝐻 PROPN iajs-3013 48 40 is be AUX iajs-3013 48 41 a a DET iajs-3013 48 42 maximal maximal ADJ iajs-3013 48 43 submodule submodule NOUN iajs-3013 48 44 of of ADP iajs-3013 48 45 𝑄 𝑄 PROPN iajs-3013 48 46 , , PUNCT iajs-3013 48 47 then then ADV iajs-3013 48 48 𝐻 𝐻 PROPN iajs-3013 48 49 ⊊ ⊊ VERB iajs-3013 48 50 ⋃ ⋃ PROPN iajs-3013 48 51 𝐿𝑖𝑖 𝐿𝑖𝑖 PROPN iajs-3013 48 52 , , PUNCT iajs-3013 48 53 but but CCONJ iajs-3013 48 54 𝑄 𝑄 PRON iajs-3013 48 55 is be AUX iajs-3013 48 56 alappnq alappnq ADJ iajs-3013 48 57 compactly compactly ADV iajs-3013 48 58 packed pack VERB iajs-3013 48 59 module module NOUN iajs-3013 48 60 , , PUNCT iajs-3013 48 61 then then ADV iajs-3013 48 62 there there PRON iajs-3013 48 63 exists exist VERB iajs-3013 48 64 𝛼1 𝛼1 NOUN iajs-3013 48 65 , , PUNCT iajs-3013 48 66 𝛼2 𝛼2 VERB iajs-3013 48 67 , , PUNCT iajs-3013 48 68 … … PUNCT iajs-3013 48 69 , , PUNCT iajs-3013 48 70 𝛼𝑛 𝛼𝑛 SCONJ iajs-3013 48 71 such such ADJ iajs-3013 48 72 that that SCONJ iajs-3013 48 73 𝐻 𝐻 PROPN iajs-3013 48 74 βŠ† βŠ† NUM iajs-3013 48 75 ⋃ ⋃ ADJ iajs-3013 48 76 𝐿𝛼𝑖 𝐿𝛼𝑖 PROPN iajs-3013 48 77 𝑛 𝑛 PRON iajs-3013 48 78 𝑖=1 𝑖=1 PROPN iajs-3013 48 79 and and CCONJ iajs-3013 48 80 since since SCONJ iajs-3013 48 81 𝐿1 𝐿1 PROPN iajs-3013 48 82 βŠ† βŠ† NUM iajs-3013 48 83 𝐿2 𝐿2 PROPN iajs-3013 48 84 βŠ† βŠ† NUM iajs-3013 48 85 𝐿3 𝐿3 NOUN iajs-3013 48 86 βŠ† βŠ† NUM iajs-3013 48 87 β‹― β‹― PROPN iajs-3013 48 88 is be AUX iajs-3013 48 89 ascending ascend VERB iajs-3013 48 90 chain chain NOUN iajs-3013 48 91 then then ADV iajs-3013 48 92 there there PRON iajs-3013 48 93 exists exist VERB iajs-3013 48 94 π‘ž π‘ž X iajs-3013 48 95 ∈ ∈ PROPN iajs-3013 48 96 { { PUNCT iajs-3013 48 97 1 1 NUM iajs-3013 48 98 , , PUNCT iajs-3013 48 99 2 2 NUM iajs-3013 48 100 , , PUNCT iajs-3013 48 101 … … PUNCT iajs-3013 48 102 , , PUNCT iajs-3013 48 103 𝑛 𝑛 X iajs-3013 48 104 } } PUNCT iajs-3013 48 105 such such ADJ iajs-3013 48 106 that that SCONJ iajs-3013 48 107 ⋃ ⋃ PROPN iajs-3013 48 108 𝐿𝛼𝑖 𝐿𝛼𝑖 PROPN iajs-3013 48 109 𝑛 𝑛 PRON iajs-3013 48 110 𝑖=1 𝑖=1 PUNCT iajs-3013 48 111 = = SYM iajs-3013 48 112 πΏπ›Όπ‘ž πΏπ›Όπ‘ž PROPN iajs-3013 48 113 then then ADV iajs-3013 48 114 𝐻 𝐻 PROPN iajs-3013 48 115 βŠ† βŠ† PROPN iajs-3013 48 116 πΏπ›Όπ‘ž πΏπ›Όπ‘ž PROPN iajs-3013 48 117 , , PUNCT iajs-3013 48 118 and and CCONJ iajs-3013 48 119 since since SCONJ iajs-3013 48 120 𝐻 𝐻 PROPN iajs-3013 48 121 is be AUX iajs-3013 48 122 maximal maximal ADJ iajs-3013 48 123 submodule submodule NOUN iajs-3013 48 124 then then ADV iajs-3013 48 125 𝐻 𝐻 PROPN iajs-3013 48 126 = = PROPN iajs-3013 48 127 πΏπ›Όπ‘ž πΏπ›Όπ‘ž PROPN iajs-3013 48 128 and and CCONJ iajs-3013 48 129 consequently consequently ADV iajs-3013 48 130 𝑄 𝑄 PROPN iajs-3013 48 131 = = PUNCT iajs-3013 48 132 ⋃ ⋃ NOUN iajs-3013 48 133 𝐿𝑖𝑖 𝐿𝑖𝑖 PROPN iajs-3013 48 134 = = SYM iajs-3013 48 135 πΏπ›Όπ‘ž πΏπ›Όπ‘ž PROPN iajs-3013 48 136 which which PRON iajs-3013 48 137 is be AUX iajs-3013 48 138 a a DET iajs-3013 48 139 contradiction contradiction NOUN iajs-3013 48 140 . . PUNCT iajs-3013 49 1 so so ADV iajs-3013 49 2 𝐿 𝐿 PROPN iajs-3013 49 3 is be AUX iajs-3013 49 4 a a DET iajs-3013 49 5 proper proper ADJ iajs-3013 49 6 submodule submodule NOUN iajs-3013 49 7 of of ADP iajs-3013 49 8 𝑄 𝑄 PROPN iajs-3013 49 9 , , PUNCT iajs-3013 49 10 thus thus ADV iajs-3013 49 11 there there PRON iajs-3013 49 12 exists exist VERB iajs-3013 49 13 𝛼1 𝛼1 NOUN iajs-3013 49 14 , , PUNCT iajs-3013 49 15 𝛼2 𝛼2 VERB iajs-3013 49 16 , , PUNCT iajs-3013 49 17 … … PUNCT iajs-3013 49 18 , , PUNCT iajs-3013 49 19 𝛼𝑛 𝛼𝑛 ADP iajs-3013 49 20 such such ADJ iajs-3013 49 21 that that SCONJ iajs-3013 49 22 𝐿 𝐿 PROPN iajs-3013 49 23 βŠ† βŠ† NUM iajs-3013 49 24 ⋃ ⋃ PROPN iajs-3013 49 25 𝐿𝛼𝑖 𝐿𝛼𝑖 PROPN iajs-3013 49 26 𝑛 𝑛 PRON iajs-3013 49 27 𝑖=1 𝑖=1 PUNCT iajs-3013 49 28 , , PUNCT iajs-3013 49 29 and and CCONJ iajs-3013 49 30 since since SCONJ iajs-3013 49 31 𝐿1 𝐿1 PROPN iajs-3013 49 32 βŠ† βŠ† NUM iajs-3013 49 33 𝐿2 𝐿2 PROPN iajs-3013 49 34 βŠ† βŠ† NUM iajs-3013 49 35 𝐿3 𝐿3 NOUN iajs-3013 49 36 βŠ† βŠ† NUM iajs-3013 49 37 β‹― β‹― NOUN iajs-3013 49 38 is be AUX iajs-3013 49 39 an an DET iajs-3013 49 40 ascending ascend VERB iajs-3013 49 41 chain chain NOUN iajs-3013 49 42 then then ADV iajs-3013 49 43 there there PRON iajs-3013 49 44 exists exist VERB iajs-3013 49 45 π‘ž π‘ž X iajs-3013 49 46 ∈ ∈ PROPN iajs-3013 49 47 { { PUNCT iajs-3013 49 48 1 1 NUM iajs-3013 49 49 , , PUNCT iajs-3013 49 50 2 2 NUM iajs-3013 49 51 , , PUNCT iajs-3013 49 52 … … PUNCT iajs-3013 49 53 , , PUNCT iajs-3013 49 54 𝑛 𝑛 X iajs-3013 49 55 } } PUNCT iajs-3013 49 56 such such ADJ iajs-3013 49 57 that that SCONJ iajs-3013 49 58 ⋃ ⋃ PROPN iajs-3013 49 59 𝐿𝛼𝑖 𝐿𝛼𝑖 PROPN iajs-3013 49 60 𝑛 𝑛 PRON iajs-3013 49 61 𝑖=1 𝑖=1 PUNCT iajs-3013 49 62 = = SYM iajs-3013 49 63 πΏπ›Όπ‘ž πΏπ›Όπ‘ž PROPN iajs-3013 49 64 that that PRON iajs-3013 49 65 is be AUX iajs-3013 49 66 ⋃ ⋃ PROPN iajs-3013 49 67 𝐿𝑖𝑖 𝐿𝑖𝑖 PROPN iajs-3013 49 68 βŠ† βŠ† NUM iajs-3013 49 69 πΏπ›Όπ‘ž πΏπ›Όπ‘ž PROPN iajs-3013 49 70 , , PUNCT iajs-3013 49 71 so𝐿1 so𝐿1 PROPN iajs-3013 49 72 βŠ† βŠ† NUM iajs-3013 49 73 𝐿2 𝐿2 PROPN iajs-3013 49 74 βŠ† βŠ† NUM iajs-3013 49 75 𝐿3 𝐿3 NOUN iajs-3013 49 76 βŠ† βŠ† NUM iajs-3013 49 77 β‹― β‹― ADP iajs-3013 49 78 βŠ† βŠ† NUM iajs-3013 49 79 πΏπ›Όπ‘ž πΏπ›Όπ‘ž PROPN iajs-3013 49 80 . . PUNCT iajs-3013 50 1 therefore therefore ADV iajs-3013 50 2 𝑄 𝑄 PRON iajs-3013 50 3 satisfies satisfy VERB iajs-3013 50 4 the the DET iajs-3013 50 5 ascending ascend VERB iajs-3013 50 6 chain chain NOUN iajs-3013 50 7 condition condition NOUN iajs-3013 50 8 on on ADP iajs-3013 50 9 alappnq alappnq ADJ iajs-3013 50 10 - - PUNCT iajs-3013 50 11 prime prime NOUN iajs-3013 50 12 submodules submodule NOUN iajs-3013 50 13 . . PUNCT iajs-3013 51 1 ihjpas ihjpas PROPN iajs-3013 51 2 . . PUNCT iajs-3013 52 1 36(1)2023 36(1)2023 NUM iajs-3013 52 2 303 303 NUM iajs-3013 52 3 since since SCONJ iajs-3013 52 4 every every DET iajs-3013 52 5 proper proper ADJ iajs-3013 52 6 submodule submodule NOUN iajs-3013 52 7 of of ADP iajs-3013 52 8 finitely finitely ADV iajs-3013 52 9 generated generate VERB iajs-3013 52 10 module module NOUN iajs-3013 52 11 contained contain VERB iajs-3013 52 12 in in ADP iajs-3013 52 13 maximal maximal ADJ iajs-3013 52 14 submodule submodule NOUN iajs-3013 52 15 , , PUNCT iajs-3013 52 16 so so SCONJ iajs-3013 52 17 from from ADP iajs-3013 52 18 the the DET iajs-3013 52 19 previous previous ADJ iajs-3013 52 20 proposition proposition NOUN iajs-3013 52 21 , , PUNCT iajs-3013 52 22 we we PRON iajs-3013 52 23 have have VERB iajs-3013 52 24 the the DET iajs-3013 52 25 following follow VERB iajs-3013 52 26 corollary corollary NOUN iajs-3013 52 27 . . PUNCT iajs-3013 53 1 corollary corollary ADJ iajs-3013 53 2 3.5 3.5 NUM iajs-3013 53 3 if if SCONJ iajs-3013 53 4 𝑄 𝑄 PRON iajs-3013 53 5 is be AUX iajs-3013 53 6 an an DET iajs-3013 53 7 alappnq alappnq NOUN iajs-3013 53 8 compactly compactly ADV iajs-3013 53 9 packed pack VERB iajs-3013 53 10 finitely finitely ADV iajs-3013 53 11 generated generate VERB iajs-3013 53 12 module module NOUN iajs-3013 53 13 , , PUNCT iajs-3013 53 14 then then ADV iajs-3013 53 15 𝑄 𝑄 PRON iajs-3013 53 16 satisfies satisfy VERB iajs-3013 53 17 the the DET iajs-3013 53 18 ascending ascend VERB iajs-3013 53 19 chain chain NOUN iajs-3013 53 20 condition condition NOUN iajs-3013 53 21 for for ADP iajs-3013 53 22 alappnq alappnq ADJ iajs-3013 53 23 - - PUNCT iajs-3013 53 24 prime prime NOUN iajs-3013 53 25 submodules submodule NOUN iajs-3013 53 26 . . PUNCT iajs-3013 54 1 also also ADV iajs-3013 54 2 since since SCONJ iajs-3013 54 3 every every DET iajs-3013 54 4 proper proper ADJ iajs-3013 54 5 submodule submodule NOUN iajs-3013 54 6 of of ADP iajs-3013 54 7 multiplication multiplication NOUN iajs-3013 54 8 module module NOUN iajs-3013 54 9 contained contain VERB iajs-3013 54 10 in in ADP iajs-3013 54 11 maximal maximal ADJ iajs-3013 54 12 submodule submodule NOUN iajs-3013 54 13 , , PUNCT iajs-3013 54 14 so so SCONJ iajs-3013 54 15 from from ADP iajs-3013 54 16 the the DET iajs-3013 54 17 previous previous ADJ iajs-3013 54 18 proposition proposition NOUN iajs-3013 54 19 , , PUNCT iajs-3013 54 20 we we PRON iajs-3013 54 21 have have VERB iajs-3013 54 22 the the DET iajs-3013 54 23 following follow VERB iajs-3013 54 24 corollary corollary NOUN iajs-3013 54 25 . . PUNCT iajs-3013 55 1 corollary corollary ADJ iajs-3013 55 2 3.6 3.6 NUM iajs-3013 55 3 if if SCONJ iajs-3013 55 4 𝑄 𝑄 PRON iajs-3013 55 5 is be AUX iajs-3013 55 6 alappnq alappnq ADJ iajs-3013 55 7 compactly compactly ADV iajs-3013 55 8 packed pack VERB iajs-3013 55 9 multiplication multiplication NOUN iajs-3013 55 10 module module NOUN iajs-3013 55 11 , , PUNCT iajs-3013 55 12 then then ADV iajs-3013 55 13 𝑄 𝑄 PRON iajs-3013 55 14 satisfies satisfy VERB iajs-3013 55 15 the the DET iajs-3013 55 16 ascending ascend VERB iajs-3013 55 17 chain chain NOUN iajs-3013 55 18 condition condition NOUN iajs-3013 55 19 for for ADP iajs-3013 55 20 alappnq alappnq ADJ iajs-3013 55 21 - - PUNCT iajs-3013 55 22 prime prime NOUN iajs-3013 55 23 submodules submodule NOUN iajs-3013 55 24 . . PUNCT iajs-3013 56 1 proposition proposition NOUN iajs-3013 56 2 3.7 3.7 NUM iajs-3013 56 3 let let VERB iajs-3013 56 4 𝑓 𝑓 PRON iajs-3013 56 5 : : PUNCT iajs-3013 56 6 𝑄 𝑄 PROPN iajs-3013 56 7 β†’ β†’ SYM iajs-3013 56 8 𝑄′ 𝑄′ ADJ iajs-3013 56 9 be be VERB iajs-3013 56 10 an an DET iajs-3013 56 11 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-3013 56 12 , , PUNCT iajs-3013 56 13 and and CCONJ iajs-3013 56 14 ker ker X iajs-3013 56 15 𝑓 𝑓 PRON iajs-3013 56 16 is be AUX iajs-3013 56 17 a a DET iajs-3013 56 18 small small ADJ iajs-3013 56 19 submodule submodule NOUN iajs-3013 56 20 of of ADP iajs-3013 56 21 𝑄 𝑄 PROPN iajs-3013 56 22 , , PUNCT iajs-3013 56 23 such such ADJ iajs-3013 56 24 that that DET iajs-3013 56 25 ker ker NOUN iajs-3013 57 1 𝑓 𝑓 PRON iajs-3013 57 2 βŠ† βŠ† NUM iajs-3013 57 3 𝑃 𝑃 NOUN iajs-3013 57 4 for for ADP iajs-3013 57 5 each each DET iajs-3013 57 6 alappnq alappnq ADJ iajs-3013 57 7 - - PUNCT iajs-3013 57 8 prime prime NOUN iajs-3013 57 9 submodule submodule NOUN iajs-3013 57 10 𝑃 𝑃 PROPN iajs-3013 57 11 of of ADP iajs-3013 57 12 𝑄. 𝑄. NOUN iajs-3013 57 13 then then ADV iajs-3013 57 14 𝑄 𝑄 PROPN iajs-3013 57 15 is be AUX iajs-3013 57 16 an an DET iajs-3013 57 17 alappnq alappnq NOUN iajs-3013 57 18 compactly compactly ADV iajs-3013 57 19 packed pack VERB iajs-3013 57 20 if if SCONJ iajs-3013 57 21 and and CCONJ iajs-3013 57 22 only only ADV iajs-3013 57 23 if if SCONJ iajs-3013 57 24 𝑄′ 𝑄′ ADJ iajs-3013 57 25 is be AUX iajs-3013 57 26 an an DET iajs-3013 57 27 alappnq alappnq NOUN iajs-3013 57 28 compactly compactly ADV iajs-3013 57 29 packed pack VERB iajs-3013 57 30 . . PUNCT iajs-3013 58 1 proof proof NOUN iajs-3013 58 2 ( ( PUNCT iajs-3013 58 3 ⟾ ⟾ ADJ iajs-3013 58 4 ) ) PUNCT iajs-3013 58 5 suppose suppose VERB iajs-3013 58 6 that that SCONJ iajs-3013 58 7 𝑄 𝑄 PROPN iajs-3013 58 8 is be AUX iajs-3013 58 9 an an DET iajs-3013 58 10 alappnq alappnq NOUN iajs-3013 58 11 compactly compactly ADV iajs-3013 58 12 packed pack VERB iajs-3013 58 13 𝑅-module 𝑅-module PROPN iajs-3013 58 14 , , PUNCT iajs-3013 58 15 and and CCONJ iajs-3013 58 16 𝐹′ 𝐹′ NUM iajs-3013 58 17 βŠ† βŠ† NUM iajs-3013 58 18 ⋃ ⋃ NOUN iajs-3013 58 19 π‘ƒβ€²π›Όπ›ΌβˆˆΚŒ π‘ƒβ€²π›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 58 20 , , PUNCT iajs-3013 58 21 where where SCONJ iajs-3013 58 22 𝐹′ 𝐹′ PROPN iajs-3013 58 23 is be AUX iajs-3013 58 24 a a DET iajs-3013 58 25 proper proper ADJ iajs-3013 58 26 submodule submodule NOUN iajs-3013 58 27 of of ADP iajs-3013 58 28 𝑄′ 𝑄′ PROPN iajs-3013 58 29 and and CCONJ iajs-3013 58 30 𝑃′ 𝑃′ NOUN iajs-3013 58 31 is be AUX iajs-3013 58 32 an an DET iajs-3013 58 33 alappnq alappnq ADJ iajs-3013 58 34 - - PUNCT iajs-3013 58 35 prime prime NOUN iajs-3013 58 36 submodule submodule NOUN iajs-3013 58 37 of of ADP iajs-3013 58 38 𝑄′ 𝑄′ PROPN iajs-3013 58 39 for for ADP iajs-3013 58 40 all all DET iajs-3013 58 41 𝛼 𝛼 PROPN iajs-3013 58 42 ∈ ∈ PROPN iajs-3013 58 43 ʌ ʌ X iajs-3013 58 44 . . PUNCT iajs-3013 59 1 then then ADV iajs-3013 59 2 , , PUNCT iajs-3013 59 3 π‘“βˆ’1(𝐹′ π‘“βˆ’1(𝐹′ NOUN iajs-3013 59 4 ) ) PUNCT iajs-3013 59 5 βŠ† βŠ† NUM iajs-3013 59 6 π‘“βˆ’1(⋃ π‘“βˆ’1(⋃ NOUN iajs-3013 59 7 π‘ƒβ€²π›Όπ›ΌβˆˆΚŒ π‘ƒβ€²π›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 59 8 ) ) PUNCT iajs-3013 59 9 and and CCONJ iajs-3013 59 10 hence hence ADV iajs-3013 59 11 π‘“βˆ’1(𝐹′ π‘“βˆ’1(𝐹′ NUM iajs-3013 59 12 ) ) PUNCT iajs-3013 59 13 βŠ† βŠ† NUM iajs-3013 59 14 ⋃ ⋃ NOUN iajs-3013 59 15 π‘“βˆ’1(𝑃′𝛼)π›ΌβˆˆΚŒ π‘“βˆ’1(𝑃′𝛼)π›ΌβˆˆΚŒ NOUN iajs-3013 59 16 . . PUNCT iajs-3013 60 1 but but CCONJ iajs-3013 60 2 by by ADP iajs-3013 60 3 proposition proposition NOUN iajs-3013 60 4 2.5 2.5 NUM iajs-3013 60 5 we we PRON iajs-3013 60 6 have have VERB iajs-3013 60 7 π‘“βˆ’1(𝑃′𝛼 π‘“βˆ’1(𝑃′𝛼 NOUN iajs-3013 60 8 ) ) PUNCT iajs-3013 60 9 is be AUX iajs-3013 60 10 an an DET iajs-3013 60 11 alappnq alappnq ADJ iajs-3013 60 12 - - PUNCT iajs-3013 60 13 prime prime NOUN iajs-3013 60 14 submodule submodule NOUN iajs-3013 60 15 of of ADP iajs-3013 60 16 𝑄 𝑄 PROPN iajs-3013 60 17 for for ADP iajs-3013 60 18 all all DET iajs-3013 60 19 𝛼 𝛼 PRON iajs-3013 60 20 ∈ ∈ PROPN iajs-3013 60 21 ʌ ʌ X iajs-3013 60 22 . . PUNCT iajs-3013 61 1 since since SCONJ iajs-3013 61 2 𝑄 𝑄 PRON iajs-3013 61 3 is be AUX iajs-3013 61 4 an an DET iajs-3013 61 5 alappnq alappnq NOUN iajs-3013 61 6 compactly compactly ADV iajs-3013 61 7 packed pack VERB iajs-3013 61 8 then then ADV iajs-3013 61 9 there there PRON iajs-3013 61 10 exists exist VERB iajs-3013 61 11 𝛼1 𝛼1 NOUN iajs-3013 61 12 , , PUNCT iajs-3013 61 13 𝛼2 𝛼2 VERB iajs-3013 61 14 , , PUNCT iajs-3013 61 15 … … PUNCT iajs-3013 61 16 , , PUNCT iajs-3013 61 17 𝛼𝑛 𝛼𝑛 PROPN iajs-3013 61 18 ∈ ∈ PROPN iajs-3013 61 19 ʌ ʌ NOUN iajs-3013 61 20 such such ADJ iajs-3013 61 21 that that PRON iajs-3013 61 22 π‘“βˆ’1(𝐹′ π‘“βˆ’1(𝐹′ NOUN iajs-3013 61 23 ) ) PUNCT iajs-3013 61 24 βŠ† βŠ† NUM iajs-3013 61 25 ⋃ ⋃ NOUN iajs-3013 61 26 π‘“βˆ’1(𝑃′𝛼𝑖 π‘“βˆ’1(𝑃′𝛼𝑖 NOUN iajs-3013 61 27 ) ) PUNCT iajs-3013 61 28 𝑛 𝑛 PROPN iajs-3013 61 29 𝑖=1 𝑖=1 PROPN iajs-3013 61 30 implies imply VERB iajs-3013 61 31 that that SCONJ iajs-3013 61 32 π‘“βˆ’1(𝐹′ π‘“βˆ’1(𝐹′ NOUN iajs-3013 61 33 ) ) PUNCT iajs-3013 61 34 βŠ† βŠ† NUM iajs-3013 61 35 π‘“βˆ’1(⋃ π‘“βˆ’1(⋃ NOUN iajs-3013 61 36 𝑃′𝛼𝑖 𝑃′𝛼𝑖 PROPN iajs-3013 61 37 𝑛 𝑛 PRON iajs-3013 61 38 𝑖=1 𝑖=1 PROPN iajs-3013 61 39 ) ) PUNCT iajs-3013 61 40 . . PUNCT iajs-3013 62 1 but but CCONJ iajs-3013 62 2 𝑓 𝑓 PRON iajs-3013 62 3 is be AUX iajs-3013 62 4 an an DET iajs-3013 62 5 epimorphism epimorphism NOUN iajs-3013 62 6 then then ADV iajs-3013 62 7 𝐹′ 𝐹′ NUM iajs-3013 62 8 βŠ† βŠ† NUM iajs-3013 62 9 ⋃ ⋃ ADP iajs-3013 62 10 𝑃′𝛼𝑖 𝑃′𝛼𝑖 PROPN iajs-3013 62 11 𝑛 𝑛 PRON iajs-3013 62 12 𝑖=1 𝑖=1 PROPN iajs-3013 62 13 . . PUNCT iajs-3013 63 1 thus thus ADV iajs-3013 63 2 𝑄′ 𝑄′ PROPN iajs-3013 63 3 is be AUX iajs-3013 63 4 an an DET iajs-3013 63 5 alappnq alappnq NOUN iajs-3013 63 6 compactly compactly ADV iajs-3013 63 7 packed pack VERB iajs-3013 63 8 . . PUNCT iajs-3013 64 1 ( ( PUNCT iajs-3013 64 2 ⟽ ⟽ X iajs-3013 64 3 ) ) PUNCT iajs-3013 64 4 suppose suppose VERB iajs-3013 64 5 that that SCONJ iajs-3013 64 6 𝑄′ 𝑄′ PROPN iajs-3013 64 7 is be AUX iajs-3013 64 8 an an DET iajs-3013 64 9 alappnq alappnq NOUN iajs-3013 64 10 compactly compactly ADV iajs-3013 64 11 packed pack VERB iajs-3013 64 12 𝑅-module 𝑅-module PROPN iajs-3013 64 13 and and CCONJ iajs-3013 64 14 ker ker VERB iajs-3013 64 15 𝑓 𝑓 PRON iajs-3013 64 16 βŠ† βŠ† NUM iajs-3013 64 17 𝑃 𝑃 NOUN iajs-3013 64 18 for for ADP iajs-3013 64 19 each each DET iajs-3013 64 20 alappnq alappnq ADJ iajs-3013 64 21 - - PUNCT iajs-3013 64 22 prime prime NOUN iajs-3013 64 23 submodule submodule NOUN iajs-3013 64 24 𝑃 𝑃 PROPN iajs-3013 64 25 of of ADP iajs-3013 64 26 𝑄. 𝑄. NOUN iajs-3013 64 27 let let VERB iajs-3013 64 28 𝐹 𝐹 PRON iajs-3013 64 29 be be AUX iajs-3013 64 30 aproper aproper NOUN iajs-3013 64 31 submodule submodule NOUN iajs-3013 64 32 of of ADP iajs-3013 64 33 𝑄 𝑄 PRON iajs-3013 64 34 such such ADJ iajs-3013 64 35 that that SCONJ iajs-3013 64 36 𝐹 𝐹 PROPN iajs-3013 64 37 βŠ† βŠ† NUM iajs-3013 64 38 ⋃ ⋃ ADP iajs-3013 64 39 π‘ƒπ›Όπ›ΌβˆˆΚŒ π‘ƒπ›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 64 40 , , PUNCT iajs-3013 64 41 where where SCONJ iajs-3013 64 42 𝑃𝛼 𝑃𝛼 PROPN iajs-3013 64 43 is be AUX iajs-3013 64 44 an an DET iajs-3013 64 45 alappnq alappnq ADJ iajs-3013 64 46 - - PUNCT iajs-3013 64 47 prime prime NOUN iajs-3013 64 48 submodule submodule NOUN iajs-3013 64 49 of of ADP iajs-3013 64 50 𝑄 𝑄 PROPN iajs-3013 64 51 for for ADP iajs-3013 64 52 all all DET iajs-3013 64 53 𝛼 𝛼 PRON iajs-3013 64 54 ∈ ∈ PROPN iajs-3013 64 55 ʌ ʌ X iajs-3013 64 56 . . PUNCT iajs-3013 65 1 then then ADV iajs-3013 65 2 𝑓(𝐹 𝑓(𝐹 NOUN iajs-3013 65 3 ) ) PUNCT iajs-3013 66 1 βŠ† βŠ† NUM iajs-3013 66 2 𝑓(⋃ 𝑓(⋃ PROPN iajs-3013 66 3 𝑃𝛼)π›ΌβˆˆΚŒ 𝑃𝛼)π›ΌβˆˆΚŒ NOUN iajs-3013 66 4 implies imply VERB iajs-3013 66 5 that that SCONJ iajs-3013 66 6 𝑓(𝐹 𝑓(𝐹 NOUN iajs-3013 66 7 ) ) PUNCT iajs-3013 66 8 βŠ† βŠ† NUM iajs-3013 66 9 ⋃ ⋃ NOUN iajs-3013 66 10 𝑓(𝑃𝛼)π›ΌβˆˆΚŒ 𝑓(𝑃𝛼)π›ΌβˆˆΚŒ NOUN iajs-3013 66 11 . . PUNCT iajs-3013 67 1 but but CCONJ iajs-3013 67 2 ker ker VERB iajs-3013 67 3 𝑓 𝑓 PRON iajs-3013 67 4 βŠ† βŠ† NUM iajs-3013 67 5 𝑃𝛼 𝑃𝛼 PROPN iajs-3013 67 6 for for ADP iajs-3013 67 7 each each DET iajs-3013 67 8 𝛼. 𝛼. NOUN iajs-3013 67 9 then then ADV iajs-3013 67 10 by by ADP iajs-3013 67 11 proposition proposition NOUN iajs-3013 67 12 2.6 2.6 NUM iajs-3013 67 13 𝑓(𝑃𝛼 𝑓(𝑃𝛼 NOUN iajs-3013 67 14 ) ) PUNCT iajs-3013 67 15 is be AUX iajs-3013 67 16 an an DET iajs-3013 67 17 alappnqprime alappnqprime NOUN iajs-3013 67 18 submodule submodule NOUN iajs-3013 67 19 of of ADP iajs-3013 67 20 𝑄′ 𝑄′ PROPN iajs-3013 67 21 for for ADP iajs-3013 67 22 all all DET iajs-3013 67 23 𝛼 𝛼 PROPN iajs-3013 67 24 ∈ ∈ PROPN iajs-3013 67 25 ʌ ʌ X iajs-3013 67 26 . . PUNCT iajs-3013 68 1 since since SCONJ iajs-3013 68 2 𝑄′ 𝑄′ PROPN iajs-3013 68 3 is be AUX iajs-3013 68 4 an an DET iajs-3013 68 5 alappnq alappnq NOUN iajs-3013 68 6 compactly compactly ADV iajs-3013 68 7 packed pack VERB iajs-3013 68 8 𝑅-module 𝑅-module PROPN iajs-3013 68 9 , , PUNCT iajs-3013 68 10 then then ADV iajs-3013 68 11 there there PRON iajs-3013 68 12 exists exist VERB iajs-3013 68 13 𝛼1 𝛼1 NOUN iajs-3013 68 14 , , PUNCT iajs-3013 68 15 𝛼2 𝛼2 VERB iajs-3013 68 16 , , PUNCT iajs-3013 68 17 … … PUNCT iajs-3013 68 18 , , PUNCT iajs-3013 68 19 𝛼𝑛 𝛼𝑛 PROPN iajs-3013 68 20 ∈ ∈ PROPN iajs-3013 68 21 ʌ ʌ NOUN iajs-3013 68 22 such such ADJ iajs-3013 68 23 that that DET iajs-3013 68 24 𝑓(𝐹 𝑓(𝐹 NOUN iajs-3013 68 25 ) ) PUNCT iajs-3013 69 1 βŠ† βŠ† NUM iajs-3013 69 2 ⋃ ⋃ PROPN iajs-3013 69 3 𝑓(𝑃𝛼𝑖 𝑓(𝑃𝛼𝑖 ADJ iajs-3013 69 4 ) ) PUNCT iajs-3013 69 5 𝑛 𝑛 PRON iajs-3013 69 6 𝑖=1 𝑖=1 PROPN iajs-3013 69 7 . . PUNCT iajs-3013 70 1 now now ADV iajs-3013 70 2 , , PUNCT iajs-3013 70 3 let let VERB iajs-3013 70 4 π‘₯ π‘₯ PRON iajs-3013 70 5 ∈ ∈ VERB iajs-3013 70 6 𝐹 𝐹 PROPN iajs-3013 70 7 then then ADV iajs-3013 70 8 𝑓(π‘₯ 𝑓(π‘₯ VERB iajs-3013 70 9 ) ) PUNCT iajs-3013 70 10 ∈ ∈ PROPN iajs-3013 70 11 𝑓(𝐹 𝑓(𝐹 NOUN iajs-3013 70 12 ) ) PUNCT iajs-3013 70 13 βŠ† βŠ† NUM iajs-3013 70 14 ⋃ ⋃ PROPN iajs-3013 70 15 𝑓(𝑃𝛼𝑖 𝑓(𝑃𝛼𝑖 ADJ iajs-3013 70 16 ) ) PUNCT iajs-3013 70 17 𝑛 𝑛 PRON iajs-3013 70 18 𝑖=1 𝑖=1 PROPN iajs-3013 70 19 , , PUNCT iajs-3013 70 20 then then ADV iajs-3013 70 21 there there PRON iajs-3013 70 22 exists exist VERB iajs-3013 70 23 𝑗 𝑗 PRON iajs-3013 70 24 ∈ ∈ PROPN iajs-3013 70 25 { { PUNCT iajs-3013 70 26 1 1 NUM iajs-3013 70 27 , , PUNCT iajs-3013 70 28 2 2 NUM iajs-3013 70 29 , , PUNCT iajs-3013 70 30 … … PUNCT iajs-3013 70 31 , , PUNCT iajs-3013 70 32 𝑛 𝑛 X iajs-3013 70 33 } } PUNCT iajs-3013 70 34 such such ADJ iajs-3013 70 35 that that SCONJ iajs-3013 70 36 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3013 70 37 ) ) PUNCT iajs-3013 70 38 ∈ ∈ PROPN iajs-3013 70 39 𝑓(𝑃𝛼𝑗 𝑓(𝑃𝛼𝑗 NOUN iajs-3013 70 40 ) ) PUNCT iajs-3013 70 41 , , PUNCT iajs-3013 70 42 implies imply VERB iajs-3013 70 43 that that SCONJ iajs-3013 70 44 there there PRON iajs-3013 70 45 exists exist VERB iajs-3013 70 46 𝑏 𝑏 PRON iajs-3013 70 47 ∈ ∈ PROPN iajs-3013 70 48 𝑃𝛼𝑗 𝑃𝛼𝑗 PROPN iajs-3013 70 49 such such ADJ iajs-3013 70 50 that that SCONJ iajs-3013 70 51 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3013 70 52 ) ) PUNCT iajs-3013 70 53 = = SYM iajs-3013 70 54 𝑓(𝑏 𝑓(𝑏 NOUN iajs-3013 70 55 ) ) PUNCT iajs-3013 70 56 , , PUNCT iajs-3013 70 57 then then ADV iajs-3013 70 58 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3013 70 59 ) ) PUNCT iajs-3013 70 60 βˆ’ βˆ’ PROPN iajs-3013 70 61 𝑓(𝑏 𝑓(𝑏 NOUN iajs-3013 70 62 ) ) PUNCT iajs-3013 71 1 = = SYM iajs-3013 71 2 0 0 NUM iajs-3013 71 3 , , PUNCT iajs-3013 71 4 and and CCONJ iajs-3013 71 5 𝑓(π‘₯ 𝑓(π‘₯ VERB iajs-3013 71 6 βˆ’ βˆ’ PROPN iajs-3013 71 7 𝑏 𝑏 NOUN iajs-3013 71 8 ) ) PUNCT iajs-3013 72 1 = = SYM iajs-3013 72 2 0 0 NUM iajs-3013 73 1 so so ADV iajs-3013 73 2 π‘₯ π‘₯ ADP iajs-3013 73 3 βˆ’ βˆ’ PROPN iajs-3013 74 1 𝑏 𝑏 PROPN iajs-3013 74 2 ∈ ∈ PROPN iajs-3013 74 3 ker ker NOUN iajs-3013 74 4 𝑓 𝑓 PRON iajs-3013 74 5 βŠ† βŠ† NUM iajs-3013 74 6 𝑃𝛼𝑗. 𝑃𝛼𝑗. PROPN iajs-3013 74 7 that that PRON iajs-3013 74 8 is be AUX iajs-3013 74 9 π‘₯ π‘₯ DET iajs-3013 74 10 ∈ ∈ PROPN iajs-3013 74 11 𝑃𝛼𝑗. 𝑃𝛼𝑗. PROPN iajs-3013 74 12 hence hence ADV iajs-3013 74 13 , , PUNCT iajs-3013 74 14 𝐹 𝐹 PROPN iajs-3013 74 15 βŠ† βŠ† NUM iajs-3013 74 16 ⋃ ⋃ ADP iajs-3013 74 17 𝑃𝛼𝑖 𝑃𝛼𝑖 PROPN iajs-3013 74 18 𝑛 𝑛 PRON iajs-3013 74 19 𝑖=1 𝑖=1 PUNCT iajs-3013 74 20 , , PUNCT iajs-3013 74 21 that that PRON iajs-3013 74 22 is is ADV iajs-3013 74 23 𝐹 𝐹 PROPN iajs-3013 74 24 is be AUX iajs-3013 74 25 an an DET iajs-3013 74 26 alappnq alappnq NOUN iajs-3013 74 27 compactly compactly ADV iajs-3013 74 28 packed pack VERB iajs-3013 74 29 submodule submodule NOUN iajs-3013 74 30 . . PUNCT iajs-3013 75 1 therefore therefore ADV iajs-3013 75 2 , , PUNCT iajs-3013 75 3 𝑄 𝑄 PROPN iajs-3013 75 4 is be AUX iajs-3013 75 5 an an DET iajs-3013 75 6 alappnq alappnq NOUN iajs-3013 75 7 compactly compactly ADV iajs-3013 75 8 packed pack VERB iajs-3013 75 9 𝑅-module 𝑅-module PROPN iajs-3013 75 10 . . PUNCT iajs-3013 76 1 the the DET iajs-3013 76 2 following follow VERB iajs-3013 76 3 proposition proposition NOUN iajs-3013 76 4 gives give VERB iajs-3013 76 5 a a DET iajs-3013 76 6 relation relation NOUN iajs-3013 76 7 between between ADP iajs-3013 76 8 an an DET iajs-3013 76 9 alappnq alappnq NOUN iajs-3013 76 10 compactly compactly ADV iajs-3013 76 11 packed pack VERB iajs-3013 76 12 module module NOUN iajs-3013 76 13 𝑄 𝑄 NOUN iajs-3013 76 14 and and CCONJ iajs-3013 76 15 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 76 16 proposition proposition NOUN iajs-3013 76 17 3.9 3.9 NUM iajs-3013 76 18 let let VERB iajs-3013 76 19 𝑄 𝑄 PRON iajs-3013 76 20 be be AUX iajs-3013 76 21 an an DET iajs-3013 76 22 𝑅-module 𝑅-module NOUN iajs-3013 76 23 , , PUNCT iajs-3013 76 24 and and CCONJ iajs-3013 76 25 𝑆 𝑆 PROPN iajs-3013 76 26 a a DET iajs-3013 76 27 multiplicatively multiplicatively ADV iajs-3013 76 28 closed close VERB iajs-3013 76 29 set set VERB iajs-3013 76 30 in in ADP iajs-3013 76 31 𝑅. 𝑅. NOUN iajs-3013 76 32 if if SCONJ iajs-3013 76 33 𝑄 𝑄 PRON iajs-3013 76 34 is be AUX iajs-3013 76 35 an an DET iajs-3013 76 36 alappnq alappnq NOUN iajs-3013 76 37 compactly compactly ADV iajs-3013 76 38 packed pack VERB iajs-3013 76 39 module module NOUN iajs-3013 76 40 , , PUNCT iajs-3013 76 41 then then ADV iajs-3013 76 42 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 76 43 is be AUX iajs-3013 76 44 an an DET iajs-3013 76 45 alappnq alappnq NOUN iajs-3013 76 46 compactly compactly ADV iajs-3013 76 47 packed pack VERB iajs-3013 76 48 module module NOUN iajs-3013 76 49 . . PUNCT iajs-3013 77 1 proof proof NOUN iajs-3013 77 2 let let VERB iajs-3013 77 3 𝐹 𝐹 PRON iajs-3013 77 4 be be AUX iajs-3013 77 5 a a DET iajs-3013 77 6 proper proper ADJ iajs-3013 77 7 submodule submodule NOUN iajs-3013 77 8 of of ADP iajs-3013 77 9 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 77 10 , , PUNCT iajs-3013 77 11 and and CCONJ iajs-3013 77 12 𝐹 𝐹 PROPN iajs-3013 77 13 βŠ† βŠ† NUM iajs-3013 77 14 ⋃ ⋃ ADP iajs-3013 77 15 π‘ƒπ›Όπ›ΌβˆˆΚŒ π‘ƒπ›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 77 16 , , PUNCT iajs-3013 77 17 where where SCONJ iajs-3013 77 18 𝑃𝛼 𝑃𝛼 PROPN iajs-3013 77 19 is be AUX iajs-3013 77 20 an an DET iajs-3013 77 21 alappnq alappnq ADJ iajs-3013 77 22 - - PUNCT iajs-3013 77 23 prime prime NOUN iajs-3013 77 24 submodule submodule NOUN iajs-3013 77 25 of of ADP iajs-3013 77 26 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 77 27 for for ADP iajs-3013 77 28 all all DET iajs-3013 77 29 𝛼 𝛼 PROPN iajs-3013 77 30 ∈ ∈ PROPN iajs-3013 77 31 ʌ ʌ X iajs-3013 77 32 . . PUNCT iajs-3013 77 33 define define VERB iajs-3013 77 34 𝑓 𝑓 DET iajs-3013 77 35 : : PUNCT iajs-3013 77 36 𝑄 𝑄 PROPN iajs-3013 77 37 β†’ β†’ SYM iajs-3013 77 38 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 77 39 by by ADP iajs-3013 77 40 𝑓(π‘ž 𝑓(π‘ž NOUN iajs-3013 77 41 ) ) PUNCT iajs-3013 77 42 = = PUNCT iajs-3013 77 43 π‘ž π‘ž X iajs-3013 77 44 1 1 NUM iajs-3013 77 45 for for ADP iajs-3013 77 46 every every DET iajs-3013 77 47 π‘ž π‘ž PROPN iajs-3013 77 48 ∈ ∈ PROPN iajs-3013 77 49 𝑄. 𝑄. PROPN iajs-3013 77 50 thus thus ADV iajs-3013 77 51 𝑓 𝑓 PRON iajs-3013 77 52 is be AUX iajs-3013 77 53 an an DET iajs-3013 77 54 epimorphism epimorphism NOUN iajs-3013 77 55 . . PUNCT iajs-3013 78 1 therefore therefore ADV iajs-3013 78 2 π‘“βˆ’1(𝐹 π‘“βˆ’1(𝐹 NUM iajs-3013 78 3 ) ) PUNCT iajs-3013 78 4 βŠ† βŠ† NUM iajs-3013 78 5 π‘“βˆ’1(⋃ π‘“βˆ’1(⋃ NOUN iajs-3013 78 6 π‘ƒπ›Όπ›ΌβˆˆΚŒ π‘ƒπ›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 78 7 ) ) PUNCT iajs-3013 78 8 , , PUNCT iajs-3013 78 9 implies imply VERB iajs-3013 78 10 that that SCONJ iajs-3013 78 11 π‘“βˆ’1(𝐹 π‘“βˆ’1(𝐹 NOUN iajs-3013 78 12 ) ) PUNCT iajs-3013 78 13 βŠ† βŠ† NUM iajs-3013 78 14 ⋃ ⋃ NOUN iajs-3013 78 15 π‘“βˆ’1(𝑃𝛼)π›ΌβˆˆΚŒ π‘“βˆ’1(𝑃𝛼)π›ΌβˆˆΚŒ NOUN iajs-3013 78 16 . . PUNCT iajs-3013 79 1 since since SCONJ iajs-3013 79 2 𝑃𝛼 𝑃𝛼 PROPN iajs-3013 79 3 is be AUX iajs-3013 79 4 an an DET iajs-3013 79 5 alappnq alappnq ADJ iajs-3013 79 6 - - PUNCT iajs-3013 79 7 prime prime NOUN iajs-3013 79 8 submodule submodule NOUN iajs-3013 79 9 of of ADP iajs-3013 79 10 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 79 11 for for ADP iajs-3013 79 12 all all DET iajs-3013 79 13 𝛼 𝛼 PRON iajs-3013 79 14 ∈ ∈ PROPN iajs-3013 79 15 ʌ ʌ NOUN iajs-3013 79 16 and and CCONJ iajs-3013 79 17 𝑓 𝑓 PRON iajs-3013 79 18 is be AUX iajs-3013 79 19 an an DET iajs-3013 79 20 epimorphism epimorphism NOUN iajs-3013 79 21 , , PUNCT iajs-3013 79 22 then then ADV iajs-3013 79 23 by by ADP iajs-3013 79 24 proposition proposition NOUN iajs-3013 79 25 2.5 2.5 NUM iajs-3013 79 26 we we PRON iajs-3013 79 27 have have VERB iajs-3013 79 28 π‘“βˆ’1(𝑃𝛼 π‘“βˆ’1(𝑃𝛼 ADV iajs-3013 79 29 ) ) PUNCT iajs-3013 79 30 is be AUX iajs-3013 79 31 an an DET iajs-3013 79 32 alappnq alappnq ADJ iajs-3013 79 33 - - PUNCT iajs-3013 79 34 prime prime NOUN iajs-3013 79 35 submodule submodule NOUN iajs-3013 79 36 of of ADP iajs-3013 79 37 𝑄 𝑄 PROPN iajs-3013 79 38 for for ADP iajs-3013 79 39 all all DET iajs-3013 79 40 𝛼 𝛼 PRON iajs-3013 79 41 ∈ ∈ PROPN iajs-3013 79 42 ʌ ʌ X iajs-3013 79 43 . . PUNCT iajs-3013 80 1 but but CCONJ iajs-3013 80 2 𝑄 𝑄 PROPN iajs-3013 80 3 is be AUX iajs-3013 80 4 an an DET iajs-3013 80 5 alappnq alappnq NOUN iajs-3013 80 6 compactly compactly ADV iajs-3013 80 7 packed pack VERB iajs-3013 80 8 then then ADV iajs-3013 80 9 there there PRON iajs-3013 80 10 exists exist VERB iajs-3013 80 11 𝛼1 𝛼1 NOUN iajs-3013 80 12 , , PUNCT iajs-3013 80 13 𝛼2 𝛼2 VERB iajs-3013 80 14 , , PUNCT iajs-3013 80 15 … … PUNCT iajs-3013 80 16 , , PUNCT iajs-3013 80 17 𝛼𝑛 𝛼𝑛 PROPN iajs-3013 80 18 ∈ ∈ PROPN iajs-3013 80 19 ʌ ʌ NOUN iajs-3013 80 20 such such ADJ iajs-3013 80 21 that that SCONJ iajs-3013 80 22 π‘“βˆ’1(𝐹 π‘“βˆ’1(𝐹 NOUN iajs-3013 80 23 ) ) PUNCT iajs-3013 80 24 βŠ† βŠ† NUM iajs-3013 80 25 ⋃ ⋃ NOUN iajs-3013 80 26 π‘“βˆ’1(𝑃𝛼𝑖 π‘“βˆ’1(𝑃𝛼𝑖 NOUN iajs-3013 80 27 ) ) PUNCT iajs-3013 80 28 𝑛 𝑛 PRON iajs-3013 81 1 𝑖=1 𝑖=1 PROPN iajs-3013 81 2 . . PUNCT iajs-3013 82 1 hence hence ADV iajs-3013 82 2 , , PUNCT iajs-3013 82 3 ihjpas ihjpas PROPN iajs-3013 82 4 . . PUNCT iajs-3013 83 1 36(1)2023 36(1)2023 NUM iajs-3013 83 2 304 304 NUM iajs-3013 83 3 ( ( PUNCT iajs-3013 83 4 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 NOUN iajs-3013 83 5 βŠ† βŠ† NUM iajs-3013 83 6 ( ( PUNCT iajs-3013 83 7 ⋃ ⋃ NOUN iajs-3013 83 8 π‘“βˆ’1(𝑃𝛼𝑖 π‘“βˆ’1(𝑃𝛼𝑖 NOUN iajs-3013 83 9 ) ) PUNCT iajs-3013 83 10 𝑛 𝑛 PRON iajs-3013 83 11 𝑖=1 𝑖=1 PROPN iajs-3013 83 12 ) ) PUNCT iajs-3013 83 13 𝑆 𝑆 PROPN iajs-3013 83 14 = = SYM iajs-3013 83 15 ⋃ ⋃ PROPN iajs-3013 83 16 ( ( PUNCT iajs-3013 83 17 π‘“βˆ’1(𝑃𝛼𝑖))𝑆 π‘“βˆ’1(𝑃𝛼𝑖))𝑆 NOUN iajs-3013 83 18 𝑛 𝑛 PRON iajs-3013 83 19 𝑖=1 𝑖=1 PROPN iajs-3013 83 20 . . PUNCT iajs-3013 84 1 to to PART iajs-3013 84 2 prove prove VERB iajs-3013 84 3 the the DET iajs-3013 84 4 last last ADJ iajs-3013 84 5 equality equality NOUN iajs-3013 84 6 , , PUNCT iajs-3013 84 7 let let VERB iajs-3013 84 8 π‘ž π‘ž PRON iajs-3013 84 9 𝑠 𝑠 PROPN iajs-3013 84 10 ∈ ∈ PROPN iajs-3013 84 11 ( ( PUNCT iajs-3013 84 12 ⋃ ⋃ NOUN iajs-3013 84 13 π‘“βˆ’1(𝑃𝛼𝑖 π‘“βˆ’1(𝑃𝛼𝑖 NOUN iajs-3013 84 14 ) ) PUNCT iajs-3013 84 15 𝑛 𝑛 PRON iajs-3013 84 16 𝑖=1 𝑖=1 PROPN iajs-3013 84 17 ) ) PUNCT iajs-3013 84 18 𝑆 𝑆 PROPN iajs-3013 84 19 , , PUNCT iajs-3013 84 20 where where SCONJ iajs-3013 84 21 𝑠 𝑠 PROPN iajs-3013 84 22 ∈ ∈ PROPN iajs-3013 84 23 𝑆 𝑆 PROPN iajs-3013 84 24 , , PUNCT iajs-3013 84 25 π‘ž π‘ž X iajs-3013 84 26 ∈ ∈ PROPN iajs-3013 84 27 ⋃ ⋃ NOUN iajs-3013 84 28 π‘“βˆ’1(𝑃𝛼𝑖 π‘“βˆ’1(𝑃𝛼𝑖 NOUN iajs-3013 84 29 ) ) PUNCT iajs-3013 84 30 𝑛 𝑛 PRON iajs-3013 84 31 𝑖=1 𝑖=1 PUNCT iajs-3013 85 1 so so ADV iajs-3013 85 2 there there PRON iajs-3013 85 3 exists exist VERB iajs-3013 85 4 𝑗 𝑗 PRON iajs-3013 85 5 ∈ ∈ PROPN iajs-3013 85 6 { { PUNCT iajs-3013 85 7 1 1 NUM iajs-3013 85 8 , , PUNCT iajs-3013 85 9 2 2 NUM iajs-3013 85 10 , , PUNCT iajs-3013 85 11 … … PUNCT iajs-3013 85 12 , , PUNCT iajs-3013 85 13 𝑛 𝑛 X iajs-3013 85 14 } } PUNCT iajs-3013 85 15 such such ADJ iajs-3013 85 16 that that SCONJ iajs-3013 85 17 π‘ž π‘ž PROPN iajs-3013 85 18 ∈ ∈ PROPN iajs-3013 85 19 π‘“βˆ’1 π‘“βˆ’1 PROPN iajs-3013 85 20 ( ( PUNCT iajs-3013 85 21 𝑃𝛼𝑗 𝑃𝛼𝑗 PROPN iajs-3013 85 22 ) ) PUNCT iajs-3013 85 23 , , PUNCT iajs-3013 85 24 thus thus ADV iajs-3013 85 25 π‘ž π‘ž X iajs-3013 85 26 𝑠 𝑠 X iajs-3013 85 27 ∈ ∈ PROPN iajs-3013 85 28 ( ( PUNCT iajs-3013 85 29 π‘“βˆ’1(𝑃𝛼𝑗))𝑆 π‘“βˆ’1(𝑃𝛼𝑗))𝑆 NOUN iajs-3013 85 30 , , PUNCT iajs-3013 85 31 hence hence ADV iajs-3013 85 32 π‘ž π‘ž X iajs-3013 85 33 𝑠 𝑠 X iajs-3013 85 34 ∈ ∈ PROPN iajs-3013 85 35 ⋃ ⋃ PROPN iajs-3013 85 36 ( ( PUNCT iajs-3013 85 37 π‘“βˆ’1(𝑃𝛼𝑖))𝑆 π‘“βˆ’1(𝑃𝛼𝑖))𝑆 NOUN iajs-3013 85 38 𝑛 𝑛 PRON iajs-3013 85 39 𝑖=1 𝑖=1 PROPN iajs-3013 85 40 . . PUNCT iajs-3013 86 1 it it PRON iajs-3013 86 2 follows follow VERB iajs-3013 86 3 ( ( PUNCT iajs-3013 86 4 ⋃ ⋃ NOUN iajs-3013 86 5 π‘“βˆ’1(𝑃𝛼𝑖 π‘“βˆ’1(𝑃𝛼𝑖 NOUN iajs-3013 86 6 ) ) PUNCT iajs-3013 86 7 𝑛 𝑛 PRON iajs-3013 86 8 𝑖=1 𝑖=1 PROPN iajs-3013 86 9 ) ) PUNCT iajs-3013 86 10 𝑆 𝑆 PROPN iajs-3013 86 11 βŠ† βŠ† NUM iajs-3013 86 12 ⋃ ⋃ PROPN iajs-3013 86 13 ( ( PUNCT iajs-3013 86 14 π‘“βˆ’1(𝑃𝛼𝑖))𝑆 π‘“βˆ’1(𝑃𝛼𝑖))𝑆 NOUN iajs-3013 86 15 𝑛 𝑛 PRON iajs-3013 86 16 𝑖=1 𝑖=1 PROPN iajs-3013 86 17 . . PUNCT iajs-3013 87 1 now now ADV iajs-3013 87 2 , , PUNCT iajs-3013 87 3 let let VERB iajs-3013 87 4 π‘ž π‘ž PRON iajs-3013 87 5 𝑠 𝑠 VERB iajs-3013 87 6 ∈ ∈ PROPN iajs-3013 87 7 ⋃ ⋃ PROPN iajs-3013 87 8 ( ( PUNCT iajs-3013 87 9 π‘“βˆ’1(𝑃𝛼𝑖))𝑆 π‘“βˆ’1(𝑃𝛼𝑖))𝑆 NOUN iajs-3013 87 10 𝑛 𝑛 PRON iajs-3013 87 11 𝑖=1 𝑖=1 PUNCT iajs-3013 87 12 , , PUNCT iajs-3013 87 13 so so ADV iajs-3013 87 14 π‘ž π‘ž X iajs-3013 87 15 𝑠 𝑠 PROPN iajs-3013 87 16 ∈ ∈ PROPN iajs-3013 87 17 ( ( PUNCT iajs-3013 87 18 π‘“βˆ’1(𝑃𝛼𝑗))𝑆 π‘“βˆ’1(𝑃𝛼𝑗))𝑆 NOUN iajs-3013 87 19 for for ADP iajs-3013 87 20 some some DET iajs-3013 87 21 𝑗 𝑗 PRON iajs-3013 87 22 ∈ ∈ NOUN iajs-3013 87 23 { { PUNCT iajs-3013 87 24 1 1 NUM iajs-3013 87 25 , , PUNCT iajs-3013 87 26 2 2 NUM iajs-3013 87 27 , , PUNCT iajs-3013 87 28 … … PUNCT iajs-3013 87 29 , , PUNCT iajs-3013 87 30 𝑛 𝑛 PROPN iajs-3013 87 31 } } PUNCT iajs-3013 87 32 , , PUNCT iajs-3013 87 33 where where SCONJ iajs-3013 87 34 𝑠 𝑠 PROPN iajs-3013 87 35 ∈ ∈ PROPN iajs-3013 87 36 𝑆 𝑆 PROPN iajs-3013 87 37 , , PUNCT iajs-3013 87 38 π‘ž π‘ž PROPN iajs-3013 87 39 ∈ ∈ PROPN iajs-3013 87 40 π‘“βˆ’1 π‘“βˆ’1 PROPN iajs-3013 87 41 ( ( PUNCT iajs-3013 87 42 𝑃𝛼𝑗 𝑃𝛼𝑗 PROPN iajs-3013 87 43 ) ) PUNCT iajs-3013 87 44 . . PUNCT iajs-3013 88 1 hence hence ADV iajs-3013 88 2 , , PUNCT iajs-3013 88 3 π‘ž π‘ž PROPN iajs-3013 88 4 ∈ ∈ PROPN iajs-3013 88 5 ⋃ ⋃ NOUN iajs-3013 88 6 π‘“βˆ’1(𝑃𝛼𝑖 π‘“βˆ’1(𝑃𝛼𝑖 NOUN iajs-3013 88 7 ) ) PUNCT iajs-3013 88 8 𝑛 𝑛 PRON iajs-3013 88 9 𝑖=1 𝑖=1 PROPN iajs-3013 88 10 , , PUNCT iajs-3013 88 11 thus thus ADV iajs-3013 88 12 π‘ž π‘ž X iajs-3013 88 13 𝑠 𝑠 PROPN iajs-3013 88 14 ∈ ∈ PROPN iajs-3013 88 15 ( ( PUNCT iajs-3013 88 16 ⋃ ⋃ NOUN iajs-3013 88 17 π‘“βˆ’1(𝑃𝛼𝑖 π‘“βˆ’1(𝑃𝛼𝑖 NOUN iajs-3013 88 18 ) ) PUNCT iajs-3013 88 19 𝑛 𝑛 PRON iajs-3013 88 20 𝑖=1 𝑖=1 PROPN iajs-3013 88 21 ) ) PUNCT iajs-3013 88 22 𝑆 𝑆 PROPN iajs-3013 88 23 . . PUNCT iajs-3013 89 1 therefore therefore ADV iajs-3013 89 2 , , PUNCT iajs-3013 89 3 ⋃ ⋃ PROPN iajs-3013 89 4 ( ( PUNCT iajs-3013 89 5 π‘“βˆ’1(𝑃𝛼𝑖))𝑆 π‘“βˆ’1(𝑃𝛼𝑖))𝑆 NOUN iajs-3013 89 6 𝑛 𝑛 PRON iajs-3013 89 7 𝑖=1 𝑖=1 PROPN iajs-3013 89 8 βŠ† βŠ† NUM iajs-3013 89 9 ( ( PUNCT iajs-3013 89 10 ⋃ ⋃ NOUN iajs-3013 89 11 π‘“βˆ’1(𝑃𝛼𝑖 π‘“βˆ’1(𝑃𝛼𝑖 NOUN iajs-3013 89 12 ) ) PUNCT iajs-3013 89 13 𝑛 𝑛 PRON iajs-3013 89 14 𝑖=1 𝑖=1 PROPN iajs-3013 89 15 ) ) PUNCT iajs-3013 89 16 𝑆 𝑆 PROPN iajs-3013 89 17 . . PUNCT iajs-3013 90 1 now now ADV iajs-3013 90 2 we we PRON iajs-3013 90 3 prove prove VERB iajs-3013 90 4 that that SCONJ iajs-3013 90 5 ( ( PUNCT iajs-3013 90 6 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 PROPN iajs-3013 90 7 = = SYM iajs-3013 90 8 𝐹 𝐹 PROPN iajs-3013 90 9 for for ADP iajs-3013 90 10 any any DET iajs-3013 90 11 submodule submodule NOUN iajs-3013 90 12 𝐹 𝐹 PROPN iajs-3013 90 13 of of ADP iajs-3013 90 14 𝑄𝑆. 𝑄𝑆. NOUN iajs-3013 90 15 let let VERB iajs-3013 90 16 π‘₯ π‘₯ PRON iajs-3013 90 17 𝑠 𝑠 X iajs-3013 90 18 ∈ ∈ PROPN iajs-3013 90 19 ( ( PUNCT iajs-3013 90 20 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 NOUN iajs-3013 90 21 , , PUNCT iajs-3013 90 22 where where SCONJ iajs-3013 90 23 π‘₯ π‘₯ DET iajs-3013 90 24 ∈ ∈ PROPN iajs-3013 90 25 π‘“βˆ’1(𝐹 π‘“βˆ’1(𝐹 NUM iajs-3013 90 26 ) ) PUNCT iajs-3013 90 27 and and CCONJ iajs-3013 90 28 𝑠 𝑠 PROPN iajs-3013 90 29 ∈ ∈ PROPN iajs-3013 90 30 𝑆. 𝑆. PROPN iajs-3013 90 31 then then ADV iajs-3013 90 32 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3013 90 33 ) ) PUNCT iajs-3013 90 34 ∈ ∈ PROPN iajs-3013 90 35 𝐹 𝐹 PROPN iajs-3013 90 36 , , PUNCT iajs-3013 90 37 therefore therefore ADV iajs-3013 90 38 π‘₯ π‘₯ PROPN iajs-3013 90 39 1 1 NUM iajs-3013 90 40 ∈ ∈ PROPN iajs-3013 90 41 𝐹 𝐹 PROPN iajs-3013 90 42 , , PUNCT iajs-3013 90 43 hence hence ADV iajs-3013 90 44 π‘₯ π‘₯ X iajs-3013 90 45 𝑠 𝑠 NOUN iajs-3013 90 46 = = SYM iajs-3013 91 1 1 1 NUM iajs-3013 91 2 𝑠 𝑠 PART iajs-3013 91 3 π‘₯ π‘₯ PROPN iajs-3013 91 4 1 1 NUM iajs-3013 91 5 ∈ ∈ PROPN iajs-3013 91 6 𝐹. 𝐹. NOUN iajs-3013 91 7 thus thus ADV iajs-3013 91 8 ( ( PUNCT iajs-3013 91 9 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 NOUN iajs-3013 91 10 βŠ† βŠ† NUM iajs-3013 91 11 𝐹. 𝐹. NOUN iajs-3013 91 12 now now ADV iajs-3013 91 13 , , PUNCT iajs-3013 91 14 let let VERB iajs-3013 91 15 π‘₯ π‘₯ PRON iajs-3013 91 16 𝑠 𝑠 PRON iajs-3013 91 17 ∈ ∈ PROPN iajs-3013 91 18 𝐹 𝐹 PROPN iajs-3013 91 19 , , PUNCT iajs-3013 91 20 then then ADV iajs-3013 91 21 , , PUNCT iajs-3013 91 22 1 1 NUM iajs-3013 91 23 𝑠 𝑠 PART iajs-3013 91 24 π‘₯ π‘₯ NOUN iajs-3013 91 25 1 1 NUM iajs-3013 91 26 ∈ ∈ NOUN iajs-3013 91 27 𝐹 𝐹 PROPN iajs-3013 91 28 and and CCONJ iajs-3013 91 29 hence hence ADV iajs-3013 91 30 π‘₯ π‘₯ PRON iajs-3013 91 31 1 1 NUM iajs-3013 91 32 ∈ ∈ PROPN iajs-3013 91 33 𝐹 𝐹 PROPN iajs-3013 91 34 , , PUNCT iajs-3013 91 35 implies imply VERB iajs-3013 91 36 that that SCONJ iajs-3013 91 37 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3013 91 38 ) ) PUNCT iajs-3013 91 39 ∈ ∈ PROPN iajs-3013 91 40 𝐹 𝐹 PROPN iajs-3013 91 41 , , PUNCT iajs-3013 91 42 therefore therefore ADV iajs-3013 91 43 π‘₯ π‘₯ X iajs-3013 91 44 ∈ ∈ NOUN iajs-3013 91 45 π‘“βˆ’1(𝐹 π‘“βˆ’1(𝐹 NUM iajs-3013 91 46 ) ) PUNCT iajs-3013 91 47 and and CCONJ iajs-3013 91 48 π‘₯ π‘₯ X iajs-3013 91 49 𝑠 𝑠 X iajs-3013 91 50 ∈ ∈ PROPN iajs-3013 91 51 ( ( PUNCT iajs-3013 91 52 π‘“βˆ’1(𝐹))𝑆. π‘“βˆ’1(𝐹))𝑆. NOUN iajs-3013 91 53 thus thus ADV iajs-3013 91 54 , , PUNCT iajs-3013 91 55 𝐹 𝐹 PROPN iajs-3013 91 56 βŠ† βŠ† NUM iajs-3013 91 57 ( ( PUNCT iajs-3013 91 58 π‘“βˆ’1(𝐹))𝑆. π‘“βˆ’1(𝐹))𝑆. NOUN iajs-3013 91 59 therefore therefore ADV iajs-3013 91 60 𝐹 𝐹 PROPN iajs-3013 91 61 = = PRON iajs-3013 91 62 ( ( PUNCT iajs-3013 91 63 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 NOUN iajs-3013 91 64 for for ADP iajs-3013 91 65 any any DET iajs-3013 91 66 submodule submodule NOUN iajs-3013 91 67 𝐹 𝐹 PROPN iajs-3013 91 68 of of ADP iajs-3013 91 69 𝑄𝑆. 𝑄𝑆. PUNCT iajs-3013 91 70 since since SCONJ iajs-3013 91 71 ( ( PUNCT iajs-3013 91 72 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 NOUN iajs-3013 91 73 βŠ† βŠ† NUM iajs-3013 91 74 ⋃ ⋃ PROPN iajs-3013 91 75 ( ( PUNCT iajs-3013 91 76 π‘“βˆ’1(𝑃𝛼𝑖 π‘“βˆ’1(𝑃𝛼𝑖 NOUN iajs-3013 91 77 ) ) PUNCT iajs-3013 91 78 ) ) PUNCT iajs-3013 91 79 , , PUNCT iajs-3013 91 80 𝑆 𝑆 PROPN iajs-3013 91 81 𝑛 𝑛 PART iajs-3013 91 82 𝑖=1 𝑖=1 PROPN iajs-3013 91 83 , , PUNCT iajs-3013 91 84 we we PRON iajs-3013 91 85 have have VERB iajs-3013 91 86 𝐹 𝐹 PROPN iajs-3013 91 87 βŠ† βŠ† NUM iajs-3013 91 88 ⋃ ⋃ ADP iajs-3013 91 89 𝑃𝛼𝑖 𝑃𝛼𝑖 PROPN iajs-3013 91 90 𝑛 𝑛 PRON iajs-3013 91 91 𝑖=1 𝑖=1 PROPN iajs-3013 91 92 . . PUNCT iajs-3013 92 1 hence hence ADV iajs-3013 92 2 , , PUNCT iajs-3013 92 3 𝐹 𝐹 PROPN iajs-3013 92 4 is be AUX iajs-3013 92 5 an an DET iajs-3013 92 6 alappnq alappnq NOUN iajs-3013 92 7 compactly compactly ADV iajs-3013 92 8 packed pack VERB iajs-3013 92 9 submodule submodule NOUN iajs-3013 92 10 of of ADP iajs-3013 92 11 𝑄𝑆. 𝑄𝑆. NOUN iajs-3013 92 12 thus thus ADV iajs-3013 92 13 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 92 14 is be AUX iajs-3013 92 15 an an DET iajs-3013 92 16 alappnq alappnq NOUN iajs-3013 92 17 compactly compactly ADV iajs-3013 92 18 packed pack VERB iajs-3013 92 19 module module NOUN iajs-3013 92 20 . . PUNCT iajs-3013 93 1 4 4 X iajs-3013 93 2 . . X iajs-3013 93 3 strongly strongly ADV iajs-3013 93 4 almost almost ADV iajs-3013 93 5 approximately approximately ADV iajs-3013 93 6 nearly nearly ADV iajs-3013 93 7 quasi quasi ADJ iajs-3013 93 8 compactly compactly ADV iajs-3013 93 9 packed pack VERB iajs-3013 93 10 modules module NOUN iajs-3013 93 11 in in ADP iajs-3013 93 12 this this DET iajs-3013 93 13 section section NOUN iajs-3013 93 14 , , PUNCT iajs-3013 93 15 we we PRON iajs-3013 93 16 introduce introduce VERB iajs-3013 93 17 the the DET iajs-3013 93 18 strongly strongly ADV iajs-3013 93 19 alappnq alappnq NOUN iajs-3013 93 20 compactly compactly ADV iajs-3013 93 21 packed pack VERB iajs-3013 93 22 modules module NOUN iajs-3013 93 23 and and CCONJ iajs-3013 93 24 comprehensively comprehensively ADV iajs-3013 93 25 study study VERB iajs-3013 93 26 this this DET iajs-3013 93 27 concept concept NOUN iajs-3013 93 28 . . PUNCT iajs-3013 94 1 first first ADV iajs-3013 94 2 , , PUNCT iajs-3013 94 3 we we PRON iajs-3013 94 4 must must AUX iajs-3013 94 5 introduce introduce VERB iajs-3013 94 6 the the DET iajs-3013 94 7 definitions definition NOUN iajs-3013 94 8 of of ADP iajs-3013 94 9 alappnq alappnq ADJ iajs-3013 94 10 - - PUNCT iajs-3013 94 11 prime prime NOUN iajs-3013 94 12 radical radical NOUN iajs-3013 94 13 of of ADP iajs-3013 94 14 submodules submodule NOUN iajs-3013 94 15 , , PUNCT iajs-3013 94 16 alappnq alappnq NOUN iajs-3013 94 17 - - PUNCT iajs-3013 94 18 prime prime ADJ iajs-3013 94 19 radical radical ADJ iajs-3013 94 20 submodules submodule NOUN iajs-3013 94 21 , , PUNCT iajs-3013 94 22 and and CCONJ iajs-3013 94 23 some some DET iajs-3013 94 24 propositions proposition NOUN iajs-3013 94 25 of of ADP iajs-3013 94 26 these these DET iajs-3013 94 27 concepts concept NOUN iajs-3013 94 28 needed need VERB iajs-3013 94 29 in in ADP iajs-3013 94 30 the the DET iajs-3013 94 31 sequel sequel NOUN iajs-3013 94 32 . . PUNCT iajs-3013 95 1 definition definition NOUN iajs-3013 95 2 4.1 4.1 NUM iajs-3013 95 3 let let VERB iajs-3013 95 4 𝐹 𝐹 PRON iajs-3013 95 5 be be AUX iajs-3013 95 6 proper proper ADJ iajs-3013 95 7 submodule submodule NOUN iajs-3013 95 8 of of ADP iajs-3013 95 9 an an DET iajs-3013 95 10 𝑅-module 𝑅-module PROPN iajs-3013 95 11 𝑄. 𝑄. NOUN iajs-3013 95 12 if if SCONJ iajs-3013 95 13 there there PRON iajs-3013 95 14 exist exist VERB iajs-3013 95 15 an an DET iajs-3013 95 16 alappn alappn NOUN iajs-3013 95 17 - - PUNCT iajs-3013 95 18 prime prime ADJ iajs-3013 95 19 submodules submodule NOUN iajs-3013 95 20 that that PRON iajs-3013 95 21 contain contain VERB iajs-3013 95 22 𝐹 𝐹 PROPN iajs-3013 95 23 , , PUNCT iajs-3013 95 24 then then ADV iajs-3013 95 25 , , PUNCT iajs-3013 95 26 the the DET iajs-3013 95 27 intersection intersection NOUN iajs-3013 95 28 of of ADP iajs-3013 95 29 each each DET iajs-3013 95 30 alappn alappn NOUN iajs-3013 95 31 - - PUNCT iajs-3013 95 32 prime prime ADJ iajs-3013 95 33 submodules submodule NOUN iajs-3013 95 34 containing contain VERB iajs-3013 95 35 𝐹 𝐹 PROPN iajs-3013 95 36 is be AUX iajs-3013 95 37 called call VERB iajs-3013 95 38 alappnqprime alappnqprime NOUN iajs-3013 95 39 radical radical NOUN iajs-3013 95 40 of of ADP iajs-3013 95 41 𝐹 𝐹 PROPN iajs-3013 95 42 and and CCONJ iajs-3013 95 43 denoted denote VERB iajs-3013 95 44 by by ADP iajs-3013 95 45 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 95 46 ) ) PUNCT iajs-3013 95 47 . . PUNCT iajs-3013 96 1 if if SCONJ iajs-3013 96 2 there there PRON iajs-3013 96 3 exists exist VERB iajs-3013 96 4 no no DET iajs-3013 96 5 an an DET iajs-3013 96 6 alappnq alappnq ADJ iajs-3013 96 7 - - PUNCT iajs-3013 96 8 prime prime NOUN iajs-3013 96 9 submodule submodule NOUN iajs-3013 96 10 containing contain VERB iajs-3013 96 11 𝐹 𝐹 PROPN iajs-3013 96 12 , , PUNCT iajs-3013 96 13 we we PRON iajs-3013 96 14 put put VERB iajs-3013 96 15 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 96 16 ) ) PUNCT iajs-3013 97 1 = = SYM iajs-3013 97 2 𝑄. 𝑄. NOUN iajs-3013 97 3 definition definition NOUN iajs-3013 97 4 4.2 4.2 NUM iajs-3013 97 5 we we PRON iajs-3013 97 6 say say VERB iajs-3013 97 7 that that SCONJ iajs-3013 97 8 a a DET iajs-3013 97 9 submodule submodule NOUN iajs-3013 97 10 𝐹 𝐹 PROPN iajs-3013 97 11 of of ADP iajs-3013 97 12 𝑄 𝑄 PROPN iajs-3013 97 13 is be AUX iajs-3013 97 14 alappnq alappnq ADJ iajs-3013 97 15 - - PUNCT iajs-3013 97 16 prime prime NOUN iajs-3013 97 17 radical radical NOUN iajs-3013 97 18 , , PUNCT iajs-3013 97 19 if if SCONJ iajs-3013 97 20 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 97 21 ) ) PUNCT iajs-3013 98 1 = = SYM iajs-3013 98 2 𝐹. 𝐹. PROPN iajs-3013 98 3 proposition proposition NOUN iajs-3013 98 4 4.3 4.3 NUM iajs-3013 98 5 let let VERB iajs-3013 98 6 𝑄 𝑄 PRON iajs-3013 98 7 be be AUX iajs-3013 98 8 an an DET iajs-3013 98 9 𝑅-module 𝑅-module PROPN iajs-3013 98 10 and and CCONJ iajs-3013 98 11 𝐹 𝐹 PROPN iajs-3013 98 12 , , PUNCT iajs-3013 98 13 𝐿 𝐿 PROPN iajs-3013 98 14 are be AUX iajs-3013 98 15 submodules submodule NOUN iajs-3013 98 16 of of ADP iajs-3013 98 17 𝑄. 𝑄. NOUN iajs-3013 98 18 then then ADV iajs-3013 98 19 : : PUNCT iajs-3013 98 20 1 1 X iajs-3013 98 21 . . X iajs-3013 98 22 𝐹 𝐹 PROPN iajs-3013 98 23 βŠ† βŠ† NUM iajs-3013 98 24 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 98 25 ) ) PUNCT iajs-3013 98 26 . . PUNCT iajs-3013 99 1 2 2 X iajs-3013 99 2 . . X iajs-3013 100 1 if if SCONJ iajs-3013 100 2 𝐹 𝐹 PROPN iajs-3013 100 3 βŠ† βŠ† NUM iajs-3013 100 4 𝐿 𝐿 PROPN iajs-3013 100 5 , , PUNCT iajs-3013 100 6 then then ADV iajs-3013 100 7 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 100 8 ) ) PUNCT iajs-3013 100 9 βŠ† βŠ† NUM iajs-3013 100 10 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐿 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐿 NOUN iajs-3013 100 11 ) ) PUNCT iajs-3013 100 12 . . PUNCT iajs-3013 101 1 3 3 X iajs-3013 101 2 . . X iajs-3013 102 1 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ PROPN iajs-3013 102 2 ( ( PUNCT iajs-3013 102 3 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 102 4 ) ) PUNCT iajs-3013 102 5 ) ) PUNCT iajs-3013 103 1 = = SYM iajs-3013 103 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 103 3 ) ) PUNCT iajs-3013 103 4 . . PUNCT iajs-3013 104 1 proof proof NOUN iajs-3013 104 2 ( ( PUNCT iajs-3013 104 3 1 1 NUM iajs-3013 104 4 ) ) PUNCT iajs-3013 104 5 and and CCONJ iajs-3013 104 6 ( ( PUNCT iajs-3013 104 7 2 2 X iajs-3013 104 8 ) ) PUNCT iajs-3013 104 9 direct direct ADJ iajs-3013 104 10 from from ADP iajs-3013 104 11 definition definition NOUN iajs-3013 104 12 . . PUNCT iajs-3013 105 1 ( ( PUNCT iajs-3013 105 2 3 3 X iajs-3013 105 3 ) ) PUNCT iajs-3013 105 4 by by ADP iajs-3013 105 5 part part NOUN iajs-3013 105 6 ( ( PUNCT iajs-3013 105 7 1 1 X iajs-3013 105 8 ) ) PUNCT iajs-3013 105 9 we we PRON iajs-3013 105 10 have have VERB iajs-3013 105 11 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 105 12 ) ) PUNCT iajs-3013 106 1 βŠ† βŠ† X iajs-3013 106 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ PROPN iajs-3013 106 3 ( ( PUNCT iajs-3013 106 4 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 106 5 ) ) PUNCT iajs-3013 106 6 ) ) PUNCT iajs-3013 106 7 . . PUNCT iajs-3013 107 1 now now ADV iajs-3013 107 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 107 3 ) ) PUNCT iajs-3013 108 1 = = NOUN iajs-3013 108 2 ∩ ∩ NOUN iajs-3013 108 3 𝐾 𝐾 PROPN iajs-3013 108 4 , , PUNCT iajs-3013 108 5 where where SCONJ iajs-3013 108 6 the the DET iajs-3013 108 7 intersection intersection NOUN iajs-3013 108 8 runs run VERB iajs-3013 108 9 over over ADP iajs-3013 108 10 all all DET iajs-3013 108 11 alappnq alappnq ADJ iajs-3013 108 12 - - PUNCT iajs-3013 108 13 prime prime NOUN iajs-3013 108 14 submodules submodule NOUN iajs-3013 108 15 𝐾 𝐾 PROPN iajs-3013 108 16 of of ADP iajs-3013 108 17 𝑄 𝑄 PROPN iajs-3013 108 18 with with ADP iajs-3013 108 19 𝐹 𝐹 PROPN iajs-3013 109 1 βŠ† βŠ† NUM iajs-3013 109 2 𝐾. 𝐾. SYM iajs-3013 109 3 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ PROPN iajs-3013 109 4 ( ( PUNCT iajs-3013 109 5 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 109 6 ) ) PUNCT iajs-3013 109 7 ) ) PUNCT iajs-3013 110 1 = = PUNCT iajs-3013 110 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(∩ π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(∩ PROPN iajs-3013 110 3 𝐾 𝐾 PROPN iajs-3013 110 4 ) ) PUNCT iajs-3013 110 5 βŠ†βˆ© βŠ†βˆ© ADJ iajs-3013 110 6 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐾 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐾 NOUN iajs-3013 110 7 ) ) PUNCT iajs-3013 110 8 = = NOUN iajs-3013 110 9 ∩ ∩ X iajs-3013 110 10 𝐾. 𝐾. NOUN iajs-3013 110 11 hence hence ADV iajs-3013 110 12 , , PUNCT iajs-3013 110 13 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ PROPN iajs-3013 110 14 ( ( PUNCT iajs-3013 110 15 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 110 16 ) ) PUNCT iajs-3013 110 17 ) ) PUNCT iajs-3013 110 18 βŠ† βŠ† NUM iajs-3013 110 19 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 NOUN iajs-3013 110 20 ) ) PUNCT iajs-3013 110 21 . . PUNCT iajs-3013 111 1 thus thus ADV iajs-3013 111 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ X iajs-3013 111 3 ( ( PUNCT iajs-3013 111 4 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 111 5 ) ) PUNCT iajs-3013 111 6 ) ) PUNCT iajs-3013 112 1 = = SYM iajs-3013 112 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 112 3 ) ) PUNCT iajs-3013 112 4 . . PUNCT iajs-3013 113 1 ihjpas ihjpas PROPN iajs-3013 113 2 . . PUNCT iajs-3013 114 1 36(1)2023 36(1)2023 NUM iajs-3013 114 2 305 305 NUM iajs-3013 114 3 proposition proposition NOUN iajs-3013 114 4 4.4 4.4 NUM iajs-3013 114 5 let let VERB iajs-3013 114 6 𝑄 𝑄 PRON iajs-3013 114 7 be be AUX iajs-3013 114 8 an an DET iajs-3013 114 9 𝑅-module 𝑅-module PROPN iajs-3013 114 10 . . PUNCT iajs-3013 115 1 if if SCONJ iajs-3013 115 2 𝑄 𝑄 PRON iajs-3013 115 3 is be AUX iajs-3013 115 4 𝑍-regular 𝑍-regular PROPN iajs-3013 115 5 , , PUNCT iajs-3013 115 6 then then ADV iajs-3013 115 7 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 115 8 ) ) PUNCT iajs-3013 116 1 = = SYM iajs-3013 116 2 𝐹 𝐹 PROPN iajs-3013 116 3 for for ADP iajs-3013 116 4 all all DET iajs-3013 116 5 submodule submodule NOUN iajs-3013 116 6 𝐹 𝐹 PROPN iajs-3013 116 7 of of ADP iajs-3013 116 8 𝑄. 𝑄. PROPN iajs-3013 116 9 proof proof NOUN iajs-3013 116 10 let let VERB iajs-3013 116 11 𝐹 𝐹 PROPN iajs-3013 116 12 βŠ‚ βŠ‚ PROPN iajs-3013 116 13 𝑄. 𝑄. PROPN iajs-3013 116 14 then then ADV iajs-3013 116 15 by by ADP iajs-3013 116 16 proposition proposition NOUN iajs-3013 116 17 4.3(1 4.3(1 X iajs-3013 116 18 ) ) PUNCT iajs-3013 116 19 we we PRON iajs-3013 116 20 have have VERB iajs-3013 116 21 𝐹 𝐹 PROPN iajs-3013 116 22 βŠ† βŠ† NUM iajs-3013 116 23 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 116 24 ) ) PUNCT iajs-3013 116 25 . . PUNCT iajs-3013 117 1 since since SCONJ iajs-3013 117 2 𝑄 𝑄 PRON iajs-3013 117 3 is be AUX iajs-3013 117 4 𝑍-regular 𝑍-regular PROPN iajs-3013 117 5 and and CCONJ iajs-3013 117 6 𝐹 𝐹 PROPN iajs-3013 117 7 be be VERB iajs-3013 117 8 aproper aproper NOUN iajs-3013 117 9 submodule submodule NOUN iajs-3013 117 10 of of ADP iajs-3013 117 11 𝑄 𝑄 PRON iajs-3013 117 12 then then ADV iajs-3013 117 13 by by ADP iajs-3013 117 14 corollary corollary NOUN iajs-3013 117 15 2.11 2.11 NUM iajs-3013 117 16 we we PRON iajs-3013 117 17 have have VERB iajs-3013 117 18 𝐹 𝐹 PROPN iajs-3013 117 19 is be AUX iajs-3013 117 20 the the DET iajs-3013 117 21 intersection intersection NOUN iajs-3013 117 22 of of ADP iajs-3013 117 23 prime prime ADJ iajs-3013 117 24 submodules submodule NOUN iajs-3013 117 25 . . PUNCT iajs-3013 118 1 hence hence ADV iajs-3013 118 2 𝐹 𝐹 PROPN iajs-3013 118 3 = = SYM iajs-3013 118 4 β‹‚ β‹‚ PROPN iajs-3013 118 5 π‘ƒπ›Όπ›ΌβˆˆΚŒ π‘ƒπ›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 118 6 where where SCONJ iajs-3013 118 7 𝑃𝛼 𝑃𝛼 PROPN iajs-3013 118 8 is be AUX iajs-3013 118 9 a a DET iajs-3013 118 10 prime prime ADJ iajs-3013 118 11 submodule submodule NOUN iajs-3013 118 12 of of ADP iajs-3013 118 13 𝑄 𝑄 PROPN iajs-3013 118 14 for for ADP iajs-3013 118 15 each each DET iajs-3013 118 16 𝛼 𝛼 PROPN iajs-3013 118 17 ∈ ∈ PROPN iajs-3013 118 18 ʌ ʌ X iajs-3013 118 19 . . PUNCT iajs-3013 119 1 therefore therefore ADV iajs-3013 119 2 β‹‚ β‹‚ PROPN iajs-3013 119 3 π‘ƒπ›Όπ›ΌβˆˆΚŒ π‘ƒπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 119 4 βŠ† βŠ† NUM iajs-3013 119 5 𝐹 𝐹 PROPN iajs-3013 119 6 , , PUNCT iajs-3013 119 7 where where SCONJ iajs-3013 119 8 𝑃𝛼 𝑃𝛼 PROPN iajs-3013 119 9 is be AUX iajs-3013 119 10 a a DET iajs-3013 119 11 prime prime ADJ iajs-3013 119 12 submodule submodule NOUN iajs-3013 119 13 of of ADP iajs-3013 119 14 𝑄 𝑄 PRON iajs-3013 119 15 such such ADJ iajs-3013 119 16 that that SCONJ iajs-3013 119 17 𝐹 𝐹 PROPN iajs-3013 119 18 βŠ† βŠ† NUM iajs-3013 119 19 𝑃𝛼. 𝑃𝛼. PROPN iajs-3013 119 20 since since SCONJ iajs-3013 119 21 by by ADP iajs-3013 119 22 remark remark NOUN iajs-3013 119 23 2.12 2.12 NUM iajs-3013 119 24 every every DET iajs-3013 119 25 prime prime ADJ iajs-3013 119 26 submodule submodule NOUN iajs-3013 119 27 of of ADP iajs-3013 119 28 𝑄 𝑄 PROPN iajs-3013 119 29 is be AUX iajs-3013 119 30 an an DET iajs-3013 119 31 alappnq alappnq NOUN iajs-3013 119 32 - - PUNCT iajs-3013 119 33 prime prime NOUN iajs-3013 119 34 then then ADV iajs-3013 119 35 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 119 36 ) ) PUNCT iajs-3013 120 1 βŠ† βŠ† NUM iajs-3013 120 2 β‹‚ β‹‚ PROPN iajs-3013 120 3 π‘ƒπ›Όπ›ΌβˆˆΚŒ π‘ƒπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 120 4 implies imply VERB iajs-3013 120 5 that that SCONJ iajs-3013 120 6 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 120 7 ) ) PUNCT iajs-3013 121 1 = = SYM iajs-3013 121 2 𝐹. 𝐹. PROPN iajs-3013 121 3 proposition proposition NOUN iajs-3013 121 4 4.5 4.5 NUM iajs-3013 121 5 let let VERB iajs-3013 121 6 𝑄 𝑄 PRON iajs-3013 121 7 be be AUX iajs-3013 121 8 an an DET iajs-3013 121 9 𝑅-module 𝑅-module PROPN iajs-3013 121 10 . . PUNCT iajs-3013 122 1 if if SCONJ iajs-3013 122 2 𝑄 𝑄 PRON iajs-3013 122 3 satisfies satisfy VERB iajs-3013 122 4 the the DET iajs-3013 122 5 ascending ascend VERB iajs-3013 122 6 chain chain NOUN iajs-3013 122 7 condition condition NOUN iajs-3013 122 8 for for ADP iajs-3013 122 9 alappnq alappnq ADJ iajs-3013 122 10 - - PUNCT iajs-3013 122 11 prime prime ADJ iajs-3013 122 12 radical radical ADJ iajs-3013 122 13 submodules submodule NOUN iajs-3013 122 14 , , PUNCT iajs-3013 122 15 then then ADV iajs-3013 122 16 every every DET iajs-3013 122 17 proper proper ADJ iajs-3013 122 18 submodule submodule NOUN iajs-3013 122 19 of of ADP iajs-3013 122 20 𝑄 𝑄 PROPN iajs-3013 122 21 is be AUX iajs-3013 122 22 an an DET iajs-3013 122 23 alappnq alappnq ADJ iajs-3013 122 24 - - PUNCT iajs-3013 122 25 prime prime NOUN iajs-3013 122 26 radical radical NOUN iajs-3013 122 27 of of ADP iajs-3013 122 28 a a DET iajs-3013 122 29 finitely finitely ADV iajs-3013 122 30 generated generate VERB iajs-3013 122 31 submodule submodule NOUN iajs-3013 122 32 of of ADP iajs-3013 122 33 it it PRON iajs-3013 122 34 . . PUNCT iajs-3013 123 1 proof proof NOUN iajs-3013 123 2 assume assume VERB iajs-3013 123 3 that that SCONJ iajs-3013 123 4 there there PRON iajs-3013 123 5 exists exist VERB iajs-3013 123 6 a a DET iajs-3013 123 7 proper proper ADJ iajs-3013 123 8 submodule submodule NOUN iajs-3013 123 9 𝐹 𝐹 PROPN iajs-3013 123 10 of of ADP iajs-3013 123 11 𝑄 𝑄 PRON iajs-3013 123 12 which which PRON iajs-3013 123 13 is be AUX iajs-3013 123 14 not not PART iajs-3013 123 15 the the DET iajs-3013 123 16 alappnq alappnq ADJ iajs-3013 123 17 - - PUNCT iajs-3013 123 18 prime prime NOUN iajs-3013 123 19 radical radical NOUN iajs-3013 123 20 of of ADP iajs-3013 123 21 a a DET iajs-3013 123 22 finitely finitely ADV iajs-3013 123 23 generated generate VERB iajs-3013 123 24 submodule submodule NOUN iajs-3013 123 25 of of ADP iajs-3013 123 26 it it PRON iajs-3013 123 27 . . PUNCT iajs-3013 124 1 let let VERB iajs-3013 124 2 π‘ž1 π‘ž1 PROPN iajs-3013 124 3 ∈ ∈ PROPN iajs-3013 124 4 𝐹 𝐹 PROPN iajs-3013 124 5 and and CCONJ iajs-3013 124 6 𝐹1 𝐹1 NOUN iajs-3013 124 7 = = SYM iajs-3013 124 8 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž1 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž1 NOUN iajs-3013 124 9 ) ) PUNCT iajs-3013 124 10 , , PUNCT iajs-3013 124 11 so so ADV iajs-3013 124 12 𝐹1 𝐹1 PROPN iajs-3013 124 13 βŠ‚ βŠ‚ PROPN iajs-3013 124 14 𝐹. 𝐹. PROPN iajs-3013 124 15 thus thus ADV iajs-3013 124 16 , , PUNCT iajs-3013 124 17 there there PRON iajs-3013 124 18 exists exist VERB iajs-3013 124 19 π‘ž2 π‘ž2 PROPN iajs-3013 124 20 ∈ ∈ PROPN iajs-3013 124 21 𝐹 𝐹 PROPN iajs-3013 125 1 βˆ’ βˆ’ PROPN iajs-3013 125 2 𝐹1 𝐹1 PROPN iajs-3013 125 3 . . PUNCT iajs-3013 126 1 let let VERB iajs-3013 126 2 𝐹2 𝐹2 NOUN iajs-3013 126 3 = = PUNCT iajs-3013 127 1 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž1 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž1 PROPN iajs-3013 127 2 + + CCONJ iajs-3013 127 3 π‘…π‘ž2 π‘…π‘ž2 NOUN iajs-3013 127 4 ) ) PUNCT iajs-3013 128 1 , , PUNCT iajs-3013 128 2 then then ADV iajs-3013 128 3 , , PUNCT iajs-3013 128 4 𝐹1 𝐹1 PROPN iajs-3013 128 5 βŠ‚ βŠ‚ PROPN iajs-3013 128 6 𝐹2 𝐹2 PROPN iajs-3013 129 1 βŠ‚ βŠ‚ PROPN iajs-3013 129 2 𝐹 𝐹 PROPN iajs-3013 129 3 , , PUNCT iajs-3013 129 4 hence hence ADV iajs-3013 129 5 there there PRON iajs-3013 129 6 exists exist VERB iajs-3013 129 7 π‘ž3 π‘ž3 PROPN iajs-3013 129 8 ∈ ∈ PROPN iajs-3013 130 1 𝐹 𝐹 PROPN iajs-3013 131 1 βˆ’ βˆ’ PROPN iajs-3013 131 2 𝐹3 𝐹3 PROPN iajs-3013 131 3 . . PUNCT iajs-3013 132 1 this this PRON iajs-3013 132 2 implies imply VERB iajs-3013 132 3 an an DET iajs-3013 132 4 ascending ascend VERB iajs-3013 132 5 chain chain NOUN iajs-3013 132 6 of of ADP iajs-3013 132 7 alappnq alappnq ADJ iajs-3013 132 8 - - PUNCT iajs-3013 132 9 prime prime ADJ iajs-3013 132 10 radical radical ADJ iajs-3013 132 11 submodules submodule NOUN iajs-3013 132 12 𝐹1 𝐹1 PROPN iajs-3013 132 13 βŠ† βŠ† NUM iajs-3013 132 14 𝐹2 𝐹2 NOUN iajs-3013 132 15 βŠ† βŠ† NUM iajs-3013 132 16 𝐹3 𝐹3 PROPN iajs-3013 132 17 βŠ† βŠ† NUM iajs-3013 132 18 β‹― β‹― PROPN iajs-3013 132 19 , , PUNCT iajs-3013 132 20 which which PRON iajs-3013 132 21 does do AUX iajs-3013 132 22 not not PART iajs-3013 132 23 terminate terminate VERB iajs-3013 132 24 and and CCONJ iajs-3013 132 25 this this PRON iajs-3013 132 26 contradicts contradict VERB iajs-3013 132 27 with with ADP iajs-3013 132 28 hypothesis hypothesis NOUN iajs-3013 132 29 . . PUNCT iajs-3013 133 1 now now ADV iajs-3013 133 2 , , PUNCT iajs-3013 133 3 we we PRON iajs-3013 133 4 introduce introduce VERB iajs-3013 133 5 the the DET iajs-3013 133 6 concept concept NOUN iajs-3013 133 7 of of ADP iajs-3013 133 8 strongly strongly ADV iajs-3013 133 9 alappnq alappnq ADJ iajs-3013 133 10 compactly compactly ADV iajs-3013 133 11 packed pack VERB iajs-3013 133 12 modules module NOUN iajs-3013 133 13 , , PUNCT iajs-3013 133 14 and and CCONJ iajs-3013 133 15 study study VERB iajs-3013 133 16 some some DET iajs-3013 133 17 properties property NOUN iajs-3013 133 18 . . PUNCT iajs-3013 134 1 definition definition NOUN iajs-3013 134 2 4.6 4.6 NUM iajs-3013 134 3 a a DET iajs-3013 134 4 proper proper ADJ iajs-3013 134 5 submodule submodule NOUN iajs-3013 134 6 𝐹 𝐹 PROPN iajs-3013 134 7 of of ADP iajs-3013 134 8 an an DET iajs-3013 134 9 𝑅-module 𝑅-module PROPN iajs-3013 134 10 𝑄 𝑄 PROPN iajs-3013 134 11 is be AUX iajs-3013 134 12 called call VERB iajs-3013 134 13 strongly strongly ADV iajs-3013 134 14 alappnq alappnq NOUN iajs-3013 134 15 compactly compactly ADV iajs-3013 134 16 packed pack VERB iajs-3013 134 17 if if SCONJ iajs-3013 134 18 for for ADP iajs-3013 134 19 each each DET iajs-3013 134 20 family family NOUN iajs-3013 134 21 { { PUNCT iajs-3013 134 22 𝐹𝛼}π›ΌβˆˆΚŒ 𝐹𝛼}π›ΌβˆˆΚŒ PUNCT iajs-3013 134 23 of of ADP iajs-3013 134 24 alappnq alappnq ADJ iajs-3013 134 25 - - PUNCT iajs-3013 134 26 prime prime NOUN iajs-3013 134 27 submodules submodule NOUN iajs-3013 134 28 of of ADP iajs-3013 134 29 𝑄 𝑄 PRON iajs-3013 134 30 with with ADP iajs-3013 134 31 𝐹 𝐹 PROPN iajs-3013 134 32 βŠ† βŠ† NUM iajs-3013 134 33 ⋃ ⋃ ADP iajs-3013 134 34 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 134 35 there there PRON iajs-3013 134 36 exists exist VERB iajs-3013 134 37 𝛽 𝛽 PROPN iajs-3013 134 38 ∈ ∈ PROPN iajs-3013 134 39 ʌ ʌ NOUN iajs-3013 134 40 such such ADJ iajs-3013 134 41 that that SCONJ iajs-3013 134 42 𝐹 𝐹 PROPN iajs-3013 134 43 βŠ† βŠ† NUM iajs-3013 134 44 𝐹𝛽. 𝐹𝛽. NUM iajs-3013 134 45 definition definition NOUN iajs-3013 134 46 4.7 4.7 NUM iajs-3013 134 47 an an DET iajs-3013 134 48 𝑅-module 𝑅-module PROPN iajs-3013 134 49 𝑄 𝑄 PROPN iajs-3013 134 50 is be AUX iajs-3013 134 51 called call VERB iajs-3013 134 52 strongly strongly ADV iajs-3013 134 53 alappnq alappnq NOUN iajs-3013 134 54 compactly compactly ADV iajs-3013 134 55 packed pack VERB iajs-3013 134 56 if if SCONJ iajs-3013 134 57 every every DET iajs-3013 134 58 proper proper ADJ iajs-3013 134 59 submodule submodule NOUN iajs-3013 134 60 of of ADP iajs-3013 134 61 𝑄 𝑄 PRON iajs-3013 134 62 is be AUX iajs-3013 134 63 strongly strongly ADV iajs-3013 134 64 alappnq alappnq ADJ iajs-3013 134 65 compactly compactly ADV iajs-3013 134 66 packed pack VERB iajs-3013 134 67 . . PUNCT iajs-3013 135 1 remark remark VERB iajs-3013 135 2 4.8 4.8 NUM iajs-3013 135 3 every every DET iajs-3013 135 4 strongly strongly ADV iajs-3013 135 5 alappnq alappnq NOUN iajs-3013 135 6 compactly compactly ADV iajs-3013 135 7 packed pack VERB iajs-3013 135 8 submodule submodule NOUN iajs-3013 135 9 is be AUX iajs-3013 135 10 alappnq alappnq NOUN iajs-3013 135 11 compactly compactly ADV iajs-3013 135 12 packed pack VERB iajs-3013 135 13 , , PUNCT iajs-3013 135 14 but but CCONJ iajs-3013 135 15 the the DET iajs-3013 135 16 convers conver NOUN iajs-3013 135 17 is be AUX iajs-3013 135 18 not not PART iajs-3013 135 19 true true ADJ iajs-3013 135 20 as as SCONJ iajs-3013 135 21 explain explain VERB iajs-3013 135 22 in in ADP iajs-3013 135 23 the the DET iajs-3013 135 24 following follow VERB iajs-3013 135 25 example example NOUN iajs-3013 135 26 : : PUNCT iajs-3013 135 27 let let VERB iajs-3013 135 28 𝑄 𝑄 PRON iajs-3013 135 29 = = SYM iajs-3013 135 30 𝑍2[π‘₯ 𝑍2[π‘₯ X iajs-3013 135 31 ] ] PUNCT iajs-3013 135 32 be be AUX iajs-3013 135 33 a a DET iajs-3013 135 34 module module NOUN iajs-3013 135 35 over over ADP iajs-3013 135 36 𝑍2 𝑍2 PROPN iajs-3013 135 37 . . PUNCT iajs-3013 136 1 let let VERB iajs-3013 136 2 𝐿 𝐿 PROPN iajs-3013 136 3 = = SYM iajs-3013 136 4 { { PUNCT iajs-3013 136 5 0Μ… 0Μ… PROPN iajs-3013 136 6 , , PUNCT iajs-3013 136 7 1Μ… 1Μ… NUM iajs-3013 136 8 , , PUNCT iajs-3013 136 9 π‘₯ π‘₯ PRON iajs-3013 136 10 , , PUNCT iajs-3013 136 11 1Μ… 1Μ… PROPN iajs-3013 136 12 + + NUM iajs-3013 136 13 π‘₯ π‘₯ X iajs-3013 136 14 } } PUNCT iajs-3013 136 15 is be AUX iajs-3013 136 16 a a DET iajs-3013 136 17 𝑍2 𝑍2 ADJ iajs-3013 136 18 - - PUNCT iajs-3013 136 19 submodule submodule NOUN iajs-3013 136 20 of of ADP iajs-3013 136 21 𝑄. 𝑄. PROPN iajs-3013 136 22 𝐿 𝐿 PROPN iajs-3013 136 23 βŠ† βŠ† PROPN iajs-3013 136 24 1̅𝑍2 1̅𝑍2 NUM iajs-3013 136 25 βˆͺ βˆͺ ADP iajs-3013 136 26 π‘₯𝑍2 π‘₯𝑍2 PROPN iajs-3013 136 27 βˆͺ βˆͺ ADP iajs-3013 136 28 π‘₯ π‘₯ PROPN iajs-3013 136 29 2𝑍2 2𝑍2 NUM iajs-3013 136 30 , , PUNCT iajs-3013 136 31 where where SCONJ iajs-3013 136 32 1̅𝑍2 1̅𝑍2 NUM iajs-3013 136 33 , , PUNCT iajs-3013 136 34 π‘₯𝑍2 π‘₯𝑍2 PROPN iajs-3013 136 35 , , PUNCT iajs-3013 136 36 π‘₯ π‘₯ PROPN iajs-3013 136 37 2𝑍2 2𝑍2 NUM iajs-3013 136 38 are be AUX iajs-3013 136 39 prime prime ADJ iajs-3013 136 40 submodules submodule NOUN iajs-3013 136 41 of of ADP iajs-3013 136 42 𝑄. 𝑄. PROPN iajs-3013 136 43 but but CCONJ iajs-3013 136 44 𝐿 𝐿 PROPN iajs-3013 136 45 βŠ„ βŠ„ NOUN iajs-3013 136 46 1̅𝑍2 1̅𝑍2 NUM iajs-3013 136 47 βˆͺ βˆͺ ADP iajs-3013 136 48 π‘₯𝑍2 π‘₯𝑍2 PROPN iajs-3013 136 49 βˆͺ βˆͺ ADP iajs-3013 136 50 π‘₯ π‘₯ PROPN iajs-3013 136 51 2𝑍2 2𝑍2 NUM iajs-3013 136 52 , , PUNCT iajs-3013 136 53 that that SCONJ iajs-3013 136 54 𝐿 𝐿 PROPN iajs-3013 136 55 is be AUX iajs-3013 136 56 alappnq alappnq NOUN iajs-3013 136 57 compactly compactly ADV iajs-3013 136 58 packed pack VERB iajs-3013 136 59 , , PUNCT iajs-3013 136 60 but but CCONJ iajs-3013 136 61 it it PRON iajs-3013 136 62 is be AUX iajs-3013 136 63 not not PART iajs-3013 136 64 strongly strongly ADV iajs-3013 136 65 alappnq alappnq ADJ iajs-3013 136 66 compactly compactly ADV iajs-3013 136 67 packed pack VERB iajs-3013 136 68 . . PUNCT iajs-3013 137 1 the the DET iajs-3013 137 2 following follow VERB iajs-3013 137 3 proposition proposition NOUN iajs-3013 137 4 gives give VERB iajs-3013 137 5 a a DET iajs-3013 137 6 characterization characterization NOUN iajs-3013 137 7 of of ADP iajs-3013 137 8 strongly strongly ADV iajs-3013 137 9 alappnq alappnq ADJ iajs-3013 137 10 compactly compactly ADV iajs-3013 137 11 packed pack VERB iajs-3013 137 12 modules module NOUN iajs-3013 137 13 . . PUNCT iajs-3013 138 1 proposition proposition NOUN iajs-3013 138 2 4.9 4.9 NUM iajs-3013 138 3 let let VERB iajs-3013 138 4 𝑄 𝑄 PRON iajs-3013 138 5 be be AUX iajs-3013 138 6 an an DET iajs-3013 138 7 𝑅-module 𝑅-module PROPN iajs-3013 138 8 . . PUNCT iajs-3013 139 1 then then ADV iajs-3013 139 2 𝑄 𝑄 PRON iajs-3013 139 3 is be AUX iajs-3013 139 4 strongly strongly ADV iajs-3013 139 5 alappnq alappnq ADJ iajs-3013 139 6 compactly compactly ADV iajs-3013 139 7 packed pack VERB iajs-3013 139 8 if if SCONJ iajs-3013 139 9 and and CCONJ iajs-3013 139 10 only only ADV iajs-3013 139 11 if if SCONJ iajs-3013 139 12 every every DET iajs-3013 139 13 proper proper ADJ iajs-3013 139 14 submodule submodule NOUN iajs-3013 139 15 of of ADP iajs-3013 139 16 𝑄 𝑄 PROPN iajs-3013 139 17 is be AUX iajs-3013 139 18 alappnq alappnq ADJ iajs-3013 139 19 - - PUNCT iajs-3013 139 20 prime prime NOUN iajs-3013 139 21 radical radical NOUN iajs-3013 139 22 of of ADP iajs-3013 139 23 a a DET iajs-3013 139 24 cyclic cyclic ADJ iajs-3013 139 25 submodule submodule NOUN iajs-3013 139 26 of of ADP iajs-3013 139 27 it it PRON iajs-3013 139 28 . . PUNCT iajs-3013 140 1 proof proof NOUN iajs-3013 140 2 ( ( PUNCT iajs-3013 140 3 ⟾ ⟾ ADJ iajs-3013 140 4 ) ) PUNCT iajs-3013 140 5 let let VERB iajs-3013 140 6 𝐹 𝐹 PRON iajs-3013 140 7 be be AUX iajs-3013 140 8 a a DET iajs-3013 140 9 proper proper ADJ iajs-3013 140 10 submodule submodule NOUN iajs-3013 140 11 of of ADP iajs-3013 140 12 𝑄 𝑄 PRON iajs-3013 140 13 such such ADJ iajs-3013 140 14 that that SCONJ iajs-3013 140 15 𝐹 𝐹 PROPN iajs-3013 140 16 is be AUX iajs-3013 140 17 not not PART iajs-3013 140 18 alappnq alappnq ADJ iajs-3013 140 19 - - PUNCT iajs-3013 140 20 prime prime NOUN iajs-3013 140 21 radical radical NOUN iajs-3013 140 22 of of ADP iajs-3013 140 23 a a DET iajs-3013 140 24 cyclic cyclic ADJ iajs-3013 140 25 ihjpas ihjpa NOUN iajs-3013 140 26 . . PUNCT iajs-3013 141 1 36(1)2023 36(1)2023 NUM iajs-3013 141 2 306 306 NUM iajs-3013 141 3 submodule submodule NOUN iajs-3013 141 4 of of ADP iajs-3013 141 5 it it PRON iajs-3013 141 6 , , PUNCT iajs-3013 141 7 thus thus ADV iajs-3013 141 8 for for ADP iajs-3013 141 9 each each DET iajs-3013 141 10 π‘ž π‘ž PROPN iajs-3013 141 11 ∈ ∈ PROPN iajs-3013 141 12 𝐹 𝐹 PROPN iajs-3013 141 13 , , PUNCT iajs-3013 141 14 𝐹 𝐹 PROPN iajs-3013 141 15 β‰  β‰  PROPN iajs-3013 141 16 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„((π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„((π‘ž PROPN iajs-3013 141 17 ) ) PUNCT iajs-3013 141 18 ) ) PUNCT iajs-3013 141 19 . . PUNCT iajs-3013 142 1 so so ADV iajs-3013 142 2 there there PRON iajs-3013 142 3 exists exist VERB iajs-3013 142 4 an an DET iajs-3013 142 5 alappnq alappnq ADJ iajs-3013 142 6 - - PUNCT iajs-3013 142 7 prime prime NOUN iajs-3013 142 8 submodule submodule NOUN iajs-3013 142 9 πΏπ‘ž πΏπ‘ž PROPN iajs-3013 142 10 βŠ‡ βŠ‡ NOUN iajs-3013 142 11 ( ( PUNCT iajs-3013 142 12 π‘ž π‘ž NOUN iajs-3013 142 13 ) ) PUNCT iajs-3013 142 14 for for ADP iajs-3013 142 15 each each DET iajs-3013 142 16 π‘ž π‘ž PROPN iajs-3013 142 17 ∈ ∈ PROPN iajs-3013 142 18 𝐹 𝐹 PROPN iajs-3013 142 19 and and CCONJ iajs-3013 142 20 𝐹 𝐹 PROPN iajs-3013 142 21 βŠ„ βŠ„ NOUN iajs-3013 142 22 πΏπ‘ž. πΏπ‘ž. PROPN iajs-3013 142 23 thus thus ADV iajs-3013 142 24 𝐹 𝐹 PROPN iajs-3013 142 25 = = SYM iajs-3013 142 26 ⋃ ⋃ PROPN iajs-3013 142 27 ( ( PUNCT iajs-3013 142 28 π‘ž)π‘žβˆˆπΉ π‘ž)π‘žβˆˆπΉ ADP iajs-3013 142 29 βŠ† βŠ† NUM iajs-3013 142 30 ⋃ ⋃ PUNCT iajs-3013 142 31 πΏπ‘žπ‘žβˆˆπΉ πΏπ‘žπ‘žβˆˆπΉ PROPN iajs-3013 142 32 . . PUNCT iajs-3013 143 1 since since SCONJ iajs-3013 143 2 𝑄 𝑄 PRON iajs-3013 143 3 is be AUX iajs-3013 143 4 strongly strongly ADV iajs-3013 143 5 alappnq alappnq ADJ iajs-3013 143 6 compactly compactly ADV iajs-3013 143 7 packed pack VERB iajs-3013 143 8 , , PUNCT iajs-3013 143 9 then then ADV iajs-3013 143 10 there there PRON iajs-3013 143 11 exists exist VERB iajs-3013 143 12 π‘ž0 π‘ž0 PROPN iajs-3013 143 13 ∈ ∈ PROPN iajs-3013 143 14 𝐹 𝐹 PROPN iajs-3013 143 15 such such ADJ iajs-3013 143 16 that that SCONJ iajs-3013 143 17 𝐹 𝐹 PROPN iajs-3013 143 18 βŠ† βŠ† NUM iajs-3013 143 19 πΏπ‘ž0 πΏπ‘ž0 NOUN iajs-3013 143 20 which which PRON iajs-3013 143 21 is be AUX iajs-3013 143 22 a a DET iajs-3013 143 23 contradiction contradiction NOUN iajs-3013 143 24 . . PUNCT iajs-3013 144 1 hence hence ADV iajs-3013 144 2 𝐹 𝐹 PROPN iajs-3013 144 3 is be AUX iajs-3013 144 4 alappnq alappnq ADJ iajs-3013 144 5 - - PUNCT iajs-3013 144 6 prime prime NOUN iajs-3013 144 7 radical radical NOUN iajs-3013 144 8 of of ADP iajs-3013 144 9 a a DET iajs-3013 144 10 cyclic cyclic ADJ iajs-3013 144 11 submodule submodule NOUN iajs-3013 144 12 of of ADP iajs-3013 144 13 it it PRON iajs-3013 144 14 . . PUNCT iajs-3013 145 1 ( ( PUNCT iajs-3013 145 2 ⟽ ⟽ X iajs-3013 145 3 ) ) PUNCT iajs-3013 145 4 let let VERB iajs-3013 145 5 𝐹 𝐹 PROPN iajs-3013 145 6 βŠ† βŠ† NUM iajs-3013 145 7 ⋃ ⋃ ADP iajs-3013 145 8 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 145 9 , , PUNCT iajs-3013 145 10 where where SCONJ iajs-3013 145 11 𝐹𝛼 𝐹𝛼 PROPN iajs-3013 145 12 is be AUX iajs-3013 145 13 an an DET iajs-3013 145 14 alappnq alappnq ADJ iajs-3013 145 15 - - PUNCT iajs-3013 145 16 prime prime NOUN iajs-3013 145 17 submodule submodule NOUN iajs-3013 145 18 of of ADP iajs-3013 145 19 𝑄 𝑄 PROPN iajs-3013 145 20 for for ADP iajs-3013 145 21 all all DET iajs-3013 145 22 𝛼 𝛼 PRON iajs-3013 145 23 ∈ ∈ NOUN iajs-3013 145 24 ʌ ʌ X iajs-3013 145 25 and and CCONJ iajs-3013 145 26 𝐹 𝐹 PROPN iajs-3013 145 27 = = PUNCT iajs-3013 145 28 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„((π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„((π‘ž PROPN iajs-3013 145 29 ) ) PUNCT iajs-3013 145 30 ) ) PUNCT iajs-3013 145 31 for for ADP iajs-3013 145 32 some some DET iajs-3013 145 33 π‘ž π‘ž PROPN iajs-3013 145 34 ∈ ∈ PROPN iajs-3013 145 35 𝐹. 𝐹. PROPN iajs-3013 145 36 since since SCONJ iajs-3013 145 37 π‘ž π‘ž PROPN iajs-3013 145 38 ∈ ∈ PROPN iajs-3013 145 39 𝐹 𝐹 PROPN iajs-3013 145 40 , , PUNCT iajs-3013 145 41 thus thus ADV iajs-3013 145 42 π‘ž π‘ž X iajs-3013 145 43 ∈ ∈ PROPN iajs-3013 145 44 ⋃ ⋃ PROPN iajs-3013 145 45 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 145 46 . . PUNCT iajs-3013 146 1 hence hence ADV iajs-3013 146 2 there there PRON iajs-3013 146 3 exists exist VERB iajs-3013 146 4 𝛽 𝛽 PROPN iajs-3013 146 5 ∈ ∈ PROPN iajs-3013 146 6 ʌ ʌ NOUN iajs-3013 146 7 such such ADJ iajs-3013 146 8 that that SCONJ iajs-3013 146 9 π‘ž π‘ž PROPN iajs-3013 146 10 ∈ ∈ PROPN iajs-3013 146 11 𝐹𝛽. 𝐹𝛽. PROPN iajs-3013 146 12 thus thus ADV iajs-3013 146 13 implies imply VERB iajs-3013 146 14 that that SCONJ iajs-3013 146 15 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„((π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„((π‘ž NOUN iajs-3013 146 16 ) ) PUNCT iajs-3013 146 17 ) ) PUNCT iajs-3013 146 18 βŠ† βŠ† NUM iajs-3013 146 19 𝐹𝛽 𝐹𝛽 PROPN iajs-3013 146 20 and and CCONJ iajs-3013 146 21 consequently consequently ADV iajs-3013 146 22 𝐹 𝐹 PROPN iajs-3013 146 23 βŠ† βŠ† NUM iajs-3013 146 24 𝐹𝛽 𝐹𝛽 PROPN iajs-3013 146 25 which which PRON iajs-3013 146 26 prove prove VERB iajs-3013 146 27 that that SCONJ iajs-3013 146 28 𝑄 𝑄 PRON iajs-3013 146 29 is be AUX iajs-3013 146 30 strongly strongly ADV iajs-3013 146 31 alappnq alappnq ADJ iajs-3013 146 32 compactly compactly ADV iajs-3013 146 33 packed pack VERB iajs-3013 146 34 . . PUNCT iajs-3013 147 1 the the DET iajs-3013 147 2 following follow VERB iajs-3013 147 3 theorem theorem NOUN iajs-3013 147 4 gives give VERB iajs-3013 147 5 characterizations characterization NOUN iajs-3013 147 6 of of ADP iajs-3013 147 7 strongly strongly ADV iajs-3013 147 8 alappnq alappnq ADJ iajs-3013 147 9 compactly compactly ADV iajs-3013 147 10 packed pack VERB iajs-3013 147 11 modules module NOUN iajs-3013 147 12 . . PUNCT iajs-3013 148 1 theorem theorem VERB iajs-3013 148 2 4.10 4.10 NUM iajs-3013 148 3 let let VERB iajs-3013 148 4 𝑄 𝑄 PRON iajs-3013 148 5 be be AUX iajs-3013 148 6 an an DET iajs-3013 148 7 𝑅-module 𝑅-module PROPN iajs-3013 148 8 . . PUNCT iajs-3013 149 1 then then ADV iajs-3013 149 2 the the DET iajs-3013 149 3 following follow VERB iajs-3013 149 4 statements statement NOUN iajs-3013 149 5 are be AUX iajs-3013 149 6 equivalent equivalent ADJ iajs-3013 149 7 : : PUNCT iajs-3013 149 8 1 1 X iajs-3013 149 9 . . X iajs-3013 150 1 𝑄 𝑄 PRON iajs-3013 150 2 is be AUX iajs-3013 150 3 strongly strongly ADV iajs-3013 150 4 alappnq alappnq ADJ iajs-3013 150 5 compactly compactly ADV iajs-3013 150 6 packed pack VERB iajs-3013 150 7 module module NOUN iajs-3013 150 8 . . PUNCT iajs-3013 151 1 2 2 X iajs-3013 151 2 . . X iajs-3013 151 3 for for ADP iajs-3013 151 4 each each DET iajs-3013 151 5 𝐹 𝐹 PROPN iajs-3013 151 6 βŠ‚ βŠ‚ PROPN iajs-3013 151 7 𝑄 𝑄 PROPN iajs-3013 151 8 , , PUNCT iajs-3013 151 9 there there PRON iajs-3013 151 10 exists exist VERB iajs-3013 151 11 π‘ž π‘ž PROPN iajs-3013 151 12 ∈ ∈ PROPN iajs-3013 151 13 𝐹 𝐹 PROPN iajs-3013 151 14 such such ADJ iajs-3013 151 15 that that PRON iajs-3013 151 16 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 151 17 ) ) PUNCT iajs-3013 152 1 = = PUNCT iajs-3013 152 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž NOUN iajs-3013 152 3 ) ) PUNCT iajs-3013 152 4 . . PUNCT iajs-3013 153 1 3 3 X iajs-3013 153 2 . . X iajs-3013 153 3 for for ADP iajs-3013 153 4 each each DET iajs-3013 153 5 𝐹 𝐹 PROPN iajs-3013 153 6 βŠ‚ βŠ‚ PROPN iajs-3013 153 7 𝑄 𝑄 PROPN iajs-3013 153 8 , , PUNCT iajs-3013 153 9 if if SCONJ iajs-3013 153 10 { { PUNCT iajs-3013 153 11 𝐹𝛼}π›ΌβˆˆΚŒ 𝐹𝛼}π›ΌβˆˆΚŒ PUNCT iajs-3013 153 12 is be AUX iajs-3013 153 13 a a DET iajs-3013 153 14 family family NOUN iajs-3013 153 15 of of ADP iajs-3013 153 16 submodules submodule NOUN iajs-3013 153 17 of of ADP iajs-3013 153 18 𝑄 𝑄 PROPN iajs-3013 153 19 , , PUNCT iajs-3013 153 20 such such ADJ iajs-3013 153 21 that that SCONJ iajs-3013 153 22 𝐹 𝐹 PROPN iajs-3013 153 23 βŠ† βŠ† NUM iajs-3013 153 24 ⋃ ⋃ ADP iajs-3013 153 25 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 153 26 , , PUNCT iajs-3013 153 27 then then ADV iajs-3013 153 28 there there PRON iajs-3013 153 29 exists exist VERB iajs-3013 153 30 𝛽 𝛽 PROPN iajs-3013 153 31 ∈ ∈ PROPN iajs-3013 153 32 ʌ ʌ NOUN iajs-3013 153 33 such such ADJ iajs-3013 153 34 that that SCONJ iajs-3013 153 35 𝐹 𝐹 PROPN iajs-3013 153 36 βŠ† βŠ† NUM iajs-3013 153 37 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛽 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛽 NOUN iajs-3013 153 38 ) ) PUNCT iajs-3013 153 39 . . PUNCT iajs-3013 154 1 4 4 X iajs-3013 154 2 . . X iajs-3013 154 3 for for ADP iajs-3013 154 4 each each DET iajs-3013 154 5 𝐹 𝐹 PROPN iajs-3013 154 6 βŠ‚ βŠ‚ PROPN iajs-3013 154 7 𝑄 𝑄 PROPN iajs-3013 154 8 , , PUNCT iajs-3013 154 9 if if SCONJ iajs-3013 154 10 { { PUNCT iajs-3013 154 11 𝐹𝛼}π›ΌβˆˆΚŒ 𝐹𝛼}π›ΌβˆˆΚŒ PUNCT iajs-3013 154 12 is be AUX iajs-3013 154 13 a a DET iajs-3013 154 14 family family NOUN iajs-3013 154 15 of of ADP iajs-3013 154 16 alappnq alappnq ADJ iajs-3013 154 17 - - PUNCT iajs-3013 154 18 prime prime ADJ iajs-3013 154 19 radical radical ADJ iajs-3013 154 20 submodule submodule NOUN iajs-3013 154 21 of of ADP iajs-3013 154 22 𝑄 𝑄 PROPN iajs-3013 154 23 , , PUNCT iajs-3013 154 24 with with ADP iajs-3013 154 25 𝐹 𝐹 PROPN iajs-3013 154 26 βŠ† βŠ† NUM iajs-3013 154 27 ⋃ ⋃ ADP iajs-3013 154 28 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 154 29 , , PUNCT iajs-3013 154 30 then then ADV iajs-3013 154 31 there there PRON iajs-3013 154 32 exists exist VERB iajs-3013 154 33 𝛽 𝛽 PROPN iajs-3013 154 34 ∈ ∈ PROPN iajs-3013 154 35 ʌ ʌ NOUN iajs-3013 154 36 such such ADJ iajs-3013 154 37 that that SCONJ iajs-3013 154 38 𝐹 𝐹 PROPN iajs-3013 154 39 βŠ† βŠ† NUM iajs-3013 154 40 𝐹𝛽. 𝐹𝛽. ADJ iajs-3013 154 41 proof proof NOUN iajs-3013 154 42 ( ( PUNCT iajs-3013 154 43 1 1 X iajs-3013 154 44 ) ) PUNCT iajs-3013 154 45 β‡’ β‡’ NOUN iajs-3013 154 46 ( ( PUNCT iajs-3013 154 47 2 2 X iajs-3013 154 48 ) ) PUNCT iajs-3013 154 49 let let VERB iajs-3013 154 50 𝐹 𝐹 PROPN iajs-3013 154 51 βŠ‚ βŠ‚ PROPN iajs-3013 154 52 𝑄. 𝑄. PROPN iajs-3013 154 53 suppose suppose VERB iajs-3013 154 54 that that SCONJ iajs-3013 154 55 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 154 56 ) ) PUNCT iajs-3013 155 1 β‰  β‰  PROPN iajs-3013 155 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž PROPN iajs-3013 155 3 ) ) PUNCT iajs-3013 155 4 for for ADP iajs-3013 155 5 all all DET iajs-3013 155 6 π‘ž π‘ž PROPN iajs-3013 155 7 ∈ ∈ PROPN iajs-3013 155 8 𝐹. 𝐹. PROPN iajs-3013 155 9 implies imply VERB iajs-3013 155 10 that that SCONJ iajs-3013 155 11 for for ADP iajs-3013 155 12 all all PRON iajs-3013 155 13 π‘ž π‘ž PROPN iajs-3013 155 14 ∈ ∈ PROPN iajs-3013 155 15 𝐹 𝐹 PROPN iajs-3013 155 16 there there PRON iajs-3013 155 17 exists exist VERB iajs-3013 155 18 an an DET iajs-3013 155 19 alappnq alappnq ADJ iajs-3013 155 20 - - PUNCT iajs-3013 155 21 prime prime NOUN iajs-3013 155 22 submodule submodule NOUN iajs-3013 156 1 πΎπ‘ž πΎπ‘ž PROPN iajs-3013 156 2 containing contain VERB iajs-3013 156 3 π‘…π‘ž π‘…π‘ž NOUN iajs-3013 156 4 and and CCONJ iajs-3013 156 5 𝐹 𝐹 PROPN iajs-3013 156 6 βŠ„ βŠ„ VERB iajs-3013 156 7 πΎπ‘ž. πΎπ‘ž. PROPN iajs-3013 157 1 but but CCONJ iajs-3013 157 2 𝐹 𝐹 PROPN iajs-3013 157 3 = = PUNCT iajs-3013 157 4 ⋃ ⋃ ADP iajs-3013 157 5 π‘…π‘ž π‘…π‘ž NOUN iajs-3013 157 6 βŠ† βŠ† NUM iajs-3013 157 7 ⋃ ⋃ NOUN iajs-3013 157 8 πΎπ‘žπ‘žβˆˆπΉπ‘žβˆˆπΉ πΎπ‘žπ‘žβˆˆπΉπ‘žβˆˆπΉ NOUN iajs-3013 157 9 and and CCONJ iajs-3013 157 10 since since SCONJ iajs-3013 157 11 𝑄 𝑄 PRON iajs-3013 157 12 is be AUX iajs-3013 157 13 strongly strongly ADV iajs-3013 157 14 alappnq alappnq ADJ iajs-3013 157 15 compactly compactly ADV iajs-3013 157 16 packed pack VERB iajs-3013 157 17 , , PUNCT iajs-3013 157 18 then then ADV iajs-3013 157 19 there there PRON iajs-3013 157 20 exists exist VERB iajs-3013 157 21 π‘ž π‘ž PROPN iajs-3013 157 22 ∈ ∈ PROPN iajs-3013 157 23 𝐹 𝐹 PROPN iajs-3013 157 24 such such ADJ iajs-3013 157 25 that that SCONJ iajs-3013 157 26 𝐹 𝐹 PROPN iajs-3013 157 27 βŠ† βŠ† NUM iajs-3013 157 28 πΎπ‘ž πΎπ‘ž NOUN iajs-3013 157 29 which which PRON iajs-3013 157 30 is be AUX iajs-3013 157 31 a a DET iajs-3013 157 32 contradiction contradiction NOUN iajs-3013 157 33 . . PUNCT iajs-3013 158 1 hence hence ADV iajs-3013 158 2 there there PRON iajs-3013 158 3 exists exist VERB iajs-3013 158 4 π‘ž π‘ž PROPN iajs-3013 158 5 ∈ ∈ PROPN iajs-3013 158 6 𝐹 𝐹 PROPN iajs-3013 158 7 such such ADJ iajs-3013 158 8 that that PRON iajs-3013 158 9 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 158 10 ) ) PUNCT iajs-3013 159 1 = = PUNCT iajs-3013 159 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž NOUN iajs-3013 159 3 ) ) PUNCT iajs-3013 159 4 . . PUNCT iajs-3013 160 1 ( ( PUNCT iajs-3013 160 2 2 2 X iajs-3013 160 3 ) ) PUNCT iajs-3013 160 4 β‡’ β‡’ NOUN iajs-3013 160 5 ( ( PUNCT iajs-3013 160 6 3 3 X iajs-3013 160 7 ) ) PUNCT iajs-3013 160 8 let let VERB iajs-3013 160 9 𝐹 𝐹 PROPN iajs-3013 160 10 βŠ‚ βŠ‚ PROPN iajs-3013 160 11 𝑄 𝑄 PROPN iajs-3013 160 12 , , PUNCT iajs-3013 160 13 and and CCONJ iajs-3013 160 14 { { PUNCT iajs-3013 160 15 𝐹𝛼}π›ΌβˆˆΚŒ 𝐹𝛼}π›ΌβˆˆΚŒ PUNCT iajs-3013 160 16 be be AUX iajs-3013 160 17 a a DET iajs-3013 160 18 family family NOUN iajs-3013 160 19 of of ADP iajs-3013 160 20 submodules submodule NOUN iajs-3013 160 21 of of ADP iajs-3013 160 22 𝑄 𝑄 PROPN iajs-3013 160 23 , , PUNCT iajs-3013 160 24 such such ADJ iajs-3013 160 25 that that SCONJ iajs-3013 160 26 𝐹 𝐹 PROPN iajs-3013 160 27 βŠ† βŠ† NUM iajs-3013 160 28 ⋃ ⋃ ADP iajs-3013 160 29 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 160 30 . . PUNCT iajs-3013 161 1 hence hence ADV iajs-3013 161 2 , , PUNCT iajs-3013 161 3 by by ADP iajs-3013 161 4 hypothesis hypothesis NOUN iajs-3013 161 5 there there PRON iajs-3013 161 6 exists exist VERB iajs-3013 161 7 π‘ž π‘ž PROPN iajs-3013 161 8 ∈ ∈ PROPN iajs-3013 161 9 𝐹 𝐹 PROPN iajs-3013 161 10 such such ADJ iajs-3013 161 11 that that PRON iajs-3013 161 12 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 161 13 ) ) PUNCT iajs-3013 162 1 = = PUNCT iajs-3013 162 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž NOUN iajs-3013 162 3 ) ) PUNCT iajs-3013 162 4 . . PUNCT iajs-3013 163 1 since since SCONJ iajs-3013 163 2 π‘ž π‘ž PROPN iajs-3013 163 3 ∈ ∈ PROPN iajs-3013 163 4 𝐹 𝐹 PROPN iajs-3013 163 5 , , PUNCT iajs-3013 163 6 then then ADV iajs-3013 163 7 π‘ž π‘ž X iajs-3013 163 8 ∈ ∈ PROPN iajs-3013 163 9 ⋃ ⋃ ADP iajs-3013 163 10 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 163 11 implies imply VERB iajs-3013 163 12 that that SCONJ iajs-3013 163 13 π‘ž π‘ž PROPN iajs-3013 163 14 ∈ ∈ PROPN iajs-3013 163 15 𝐹𝛽 𝐹𝛽 PROPN iajs-3013 163 16 for for ADP iajs-3013 163 17 some some DET iajs-3013 163 18 𝛽 𝛽 NOUN iajs-3013 163 19 ∈ ∈ PROPN iajs-3013 163 20 ʌ ʌ PROPN iajs-3013 163 21 . . PUNCT iajs-3013 164 1 hence hence ADV iajs-3013 164 2 , , PUNCT iajs-3013 164 3 π‘…π‘ž π‘…π‘ž PROPN iajs-3013 164 4 βŠ† βŠ† NUM iajs-3013 164 5 𝐹𝛽 𝐹𝛽 PROPN iajs-3013 164 6 and and CCONJ iajs-3013 164 7 𝐹 𝐹 PROPN iajs-3013 164 8 βŠ† βŠ† NUM iajs-3013 164 9 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 164 10 ) ) PUNCT iajs-3013 165 1 = = PUNCT iajs-3013 165 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž NOUN iajs-3013 165 3 ) ) PUNCT iajs-3013 165 4 βŠ† βŠ† NUM iajs-3013 165 5 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛽 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛽 NOUN iajs-3013 165 6 ) ) PUNCT iajs-3013 165 7 . . PUNCT iajs-3013 166 1 that that PRON iajs-3013 166 2 is be AUX iajs-3013 166 3 𝐹 𝐹 PROPN iajs-3013 166 4 βŠ† βŠ† NUM iajs-3013 166 5 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛼 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛼 NOUN iajs-3013 166 6 ) ) PUNCT iajs-3013 166 7 . . PUNCT iajs-3013 167 1 ( ( PUNCT iajs-3013 167 2 3 3 X iajs-3013 167 3 ) ) PUNCT iajs-3013 167 4 β‡’ β‡’ NOUN iajs-3013 167 5 ( ( PUNCT iajs-3013 167 6 4 4 X iajs-3013 167 7 ) ) PUNCT iajs-3013 167 8 let let VERB iajs-3013 167 9 𝐹 𝐹 PROPN iajs-3013 167 10 βŠ‚ βŠ‚ PROPN iajs-3013 167 11 𝑄 𝑄 PROPN iajs-3013 167 12 , , PUNCT iajs-3013 167 13 and and CCONJ iajs-3013 167 14 { { PUNCT iajs-3013 167 15 𝐹𝛼}π›ΌβˆˆΚŒ 𝐹𝛼}π›ΌβˆˆΚŒ PUNCT iajs-3013 167 16 be be AUX iajs-3013 167 17 a a DET iajs-3013 167 18 family family NOUN iajs-3013 167 19 of of ADP iajs-3013 167 20 alappnq alappnq ADJ iajs-3013 167 21 - - PUNCT iajs-3013 167 22 prime prime ADJ iajs-3013 167 23 radical radical ADJ iajs-3013 167 24 submodules submodule NOUN iajs-3013 167 25 of of ADP iajs-3013 167 26 𝑄 𝑄 PROPN iajs-3013 167 27 , , PUNCT iajs-3013 167 28 such such ADJ iajs-3013 167 29 that that SCONJ iajs-3013 167 30 𝐹 𝐹 PROPN iajs-3013 167 31 βŠ† βŠ† NUM iajs-3013 167 32 ⋃ ⋃ ADP iajs-3013 167 33 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 167 34 , , PUNCT iajs-3013 167 35 then then ADV iajs-3013 167 36 by by ADP iajs-3013 167 37 hypothesis hypothesis NOUN iajs-3013 167 38 there there PRON iajs-3013 167 39 exists exist VERB iajs-3013 167 40 𝛽 𝛽 PROPN iajs-3013 167 41 ∈ ∈ PROPN iajs-3013 167 42 ʌ ʌ NOUN iajs-3013 167 43 such such ADJ iajs-3013 167 44 that that SCONJ iajs-3013 167 45 𝐹 𝐹 PROPN iajs-3013 167 46 βŠ† βŠ† NUM iajs-3013 167 47 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛽 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛽 NOUN iajs-3013 167 48 ) ) PUNCT iajs-3013 167 49 = = SYM iajs-3013 167 50 𝐹𝛽. 𝐹𝛽. PROPN iajs-3013 167 51 since since SCONJ iajs-3013 167 52 𝐹𝛽is 𝐹𝛽is PROPN iajs-3013 167 53 alappnq alappnq ADJ iajs-3013 167 54 - - PUNCT iajs-3013 167 55 prime prime ADJ iajs-3013 167 56 radical radical ADJ iajs-3013 167 57 submodules submodule NOUN iajs-3013 167 58 of of ADP iajs-3013 167 59 𝑄. 𝑄. NOUN iajs-3013 167 60 ( ( PUNCT iajs-3013 167 61 4 4 NUM iajs-3013 167 62 ) ) PUNCT iajs-3013 167 63 β‡’ β‡’ NOUN iajs-3013 167 64 ( ( PUNCT iajs-3013 167 65 1 1 X iajs-3013 167 66 ) ) PUNCT iajs-3013 167 67 let let VERB iajs-3013 167 68 𝐹 𝐹 PROPN iajs-3013 167 69 βŠ‚ βŠ‚ PROPN iajs-3013 167 70 𝑄 𝑄 PROPN iajs-3013 167 71 , , PUNCT iajs-3013 167 72 and and CCONJ iajs-3013 167 73 suppose suppose VERB iajs-3013 167 74 { { PUNCT iajs-3013 167 75 𝐹𝛼}π›ΌβˆˆΚŒ 𝐹𝛼}π›ΌβˆˆΚŒ PUNCT iajs-3013 167 76 is be AUX iajs-3013 167 77 a a DET iajs-3013 167 78 family family NOUN iajs-3013 167 79 of of ADP iajs-3013 167 80 alappnq alappnq ADJ iajs-3013 167 81 - - PUNCT iajs-3013 167 82 prime prime NOUN iajs-3013 167 83 submodules submodule NOUN iajs-3013 167 84 of of ADP iajs-3013 167 85 𝑄 𝑄 PROPN iajs-3013 167 86 , , PUNCT iajs-3013 167 87 such such ADJ iajs-3013 167 88 that that SCONJ iajs-3013 167 89 𝐹 𝐹 PROPN iajs-3013 167 90 βŠ† βŠ† NUM iajs-3013 167 91 ⋃ ⋃ ADP iajs-3013 167 92 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 167 93 . . PUNCT iajs-3013 168 1 since since SCONJ iajs-3013 168 2 𝐹𝛼 𝐹𝛼 PROPN iajs-3013 168 3 is be AUX iajs-3013 168 4 alappnq alappnq ADJ iajs-3013 168 5 - - PUNCT iajs-3013 168 6 prime prime NOUN iajs-3013 168 7 submodules submodule NOUN iajs-3013 168 8 for for ADP iajs-3013 168 9 each each DET iajs-3013 168 10 𝛼 𝛼 PROPN iajs-3013 168 11 ∈ ∈ NOUN iajs-3013 168 12 ʌ ʌ NOUN iajs-3013 168 13 then then ADV iajs-3013 168 14 𝐹𝛼 𝐹𝛼 PROPN iajs-3013 168 15 = = SYM iajs-3013 168 16 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛼 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛼 X iajs-3013 168 17 ) ) PUNCT iajs-3013 168 18 . . PUNCT iajs-3013 169 1 thus thus ADV iajs-3013 169 2 𝐹 𝐹 PROPN iajs-3013 169 3 βŠ† βŠ† NUM iajs-3013 169 4 ⋃ ⋃ ADP iajs-3013 169 5 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 169 6 = = PUNCT iajs-3013 169 7 ⋃ ⋃ VERB iajs-3013 169 8 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛼)π›ΌβˆˆΚŒ π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛼)π›ΌβˆˆΚŒ PROPN iajs-3013 169 9 . . PUNCT iajs-3013 170 1 then then ADV iajs-3013 170 2 by by ADP iajs-3013 170 3 hypothesis hypothesis NOUN iajs-3013 170 4 , , PUNCT iajs-3013 170 5 there there PRON iajs-3013 170 6 exists exist VERB iajs-3013 170 7 𝛽 𝛽 PROPN iajs-3013 170 8 ∈ ∈ PROPN iajs-3013 170 9 ʌ ʌ NOUN iajs-3013 170 10 such such ADJ iajs-3013 170 11 that that SCONJ iajs-3013 170 12 𝐹 𝐹 PROPN iajs-3013 170 13 βŠ† βŠ† NUM iajs-3013 170 14 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛼 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛼 X iajs-3013 170 15 ) ) PUNCT iajs-3013 171 1 = = SYM iajs-3013 171 2 𝐹𝛼. 𝐹𝛼. PROPN iajs-3013 171 3 thus thus ADV iajs-3013 171 4 𝑄 𝑄 PRON iajs-3013 171 5 is be AUX iajs-3013 171 6 strongly strongly ADV iajs-3013 171 7 alappnq alappnq ADJ iajs-3013 171 8 compactly compactly ADV iajs-3013 171 9 packed pack VERB iajs-3013 171 10 . . PUNCT iajs-3013 172 1 proposition proposition NOUN iajs-3013 172 2 4.11 4.11 NUM iajs-3013 172 3 let let VERB iajs-3013 172 4 𝑅 𝑅 PROPN iajs-3013 172 5 be be AUX iajs-3013 172 6 alappnq alappnq NOUN iajs-3013 172 7 compactly compactly ADV iajs-3013 172 8 packed pack VERB iajs-3013 172 9 ring ring NOUN iajs-3013 172 10 . . PUNCT iajs-3013 173 1 let let VERB iajs-3013 173 2 𝑄 𝑄 PRON iajs-3013 173 3 be be AUX iajs-3013 173 4 a a DET iajs-3013 173 5 faithful faithful ADJ iajs-3013 173 6 cyclic cyclic NOUN iajs-3013 173 7 𝑅-module 𝑅-module PROPN iajs-3013 173 8 such such ADJ iajs-3013 173 9 that that PRON iajs-3013 173 10 for for ADP iajs-3013 173 11 every every DET iajs-3013 173 12 submodule submodule NOUN iajs-3013 173 13 𝐹1 𝐹1 NOUN iajs-3013 173 14 , , PUNCT iajs-3013 173 15 𝐹2 𝐹2 NOUN iajs-3013 173 16 of of ADP iajs-3013 173 17 𝑄 𝑄 PROPN iajs-3013 173 18 with with ADP iajs-3013 173 19 𝐹1 𝐹1 PROPN iajs-3013 173 20 βŠ† βŠ† NUM iajs-3013 173 21 𝐹2 𝐹2 NOUN iajs-3013 173 22 , , PUNCT iajs-3013 173 23 whenever whenever SCONJ iajs-3013 173 24 [ [ X iajs-3013 173 25 𝐹1:𝑅 𝐹1:𝑅 NOUN iajs-3013 173 26 𝑄 𝑄 X iajs-3013 173 27 ] ] PUNCT iajs-3013 173 28 βŠ† βŠ† NUM iajs-3013 173 29 [ [ X iajs-3013 173 30 𝐹2:𝑅 𝐹2:𝑅 NOUN iajs-3013 173 31 𝑄 𝑄 PRON iajs-3013 173 32 ] ] PUNCT iajs-3013 173 33 . . PUNCT iajs-3013 174 1 then then ADV iajs-3013 174 2 𝑄 𝑄 PRON iajs-3013 174 3 is be AUX iajs-3013 174 4 strongly strongly ADV iajs-3013 174 5 alappnq alappnq ADJ iajs-3013 174 6 compactly compactly ADV iajs-3013 174 7 packed pack VERB iajs-3013 174 8 . . PUNCT iajs-3013 175 1 proof proof NOUN iajs-3013 175 2 let let VERB iajs-3013 175 3 𝐹 𝐹 PRON iajs-3013 175 4 be be AUX iajs-3013 175 5 a a DET iajs-3013 175 6 proper proper ADJ iajs-3013 175 7 submodule submodule NOUN iajs-3013 175 8 of of ADP iajs-3013 175 9 𝑄 𝑄 PROPN iajs-3013 175 10 , , PUNCT iajs-3013 175 11 and and CCONJ iajs-3013 175 12 { { PUNCT iajs-3013 175 13 𝐹𝛼}π›ΌβˆˆΚŒ 𝐹𝛼}π›ΌβˆˆΚŒ PUNCT iajs-3013 175 14 is be AUX iajs-3013 175 15 a a DET iajs-3013 175 16 family family NOUN iajs-3013 175 17 of of ADP iajs-3013 175 18 alappnq alappnq ADJ iajs-3013 175 19 - - PUNCT iajs-3013 175 20 prime prime NOUN iajs-3013 175 21 submodules submodule NOUN iajs-3013 175 22 of of ADP iajs-3013 175 23 𝑄 𝑄 PROPN iajs-3013 175 24 , , PUNCT iajs-3013 175 25 ihjpas ihjpa VERB iajs-3013 175 26 . . PUNCT iajs-3013 176 1 36(1)2023 36(1)2023 NUM iajs-3013 176 2 307 307 NUM iajs-3013 176 3 such such ADJ iajs-3013 176 4 that that SCONJ iajs-3013 176 5 𝐹 𝐹 PROPN iajs-3013 176 6 βŠ† βŠ† NUM iajs-3013 176 7 ⋃ ⋃ ADP iajs-3013 176 8 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 176 9 . . PUNCT iajs-3013 177 1 since since SCONJ iajs-3013 177 2 𝑄 𝑄 PRON iajs-3013 177 3 is be AUX iajs-3013 177 4 cyclic cyclic ADJ iajs-3013 177 5 , , PUNCT iajs-3013 177 6 then then ADV iajs-3013 177 7 [ [ X iajs-3013 177 8 ⋃ ⋃ ADP iajs-3013 177 9 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ X iajs-3013 177 10 : : PUNCT iajs-3013 177 11 𝑅 𝑅 NOUN iajs-3013 177 12 𝑄 𝑄 PROPN iajs-3013 177 13 ] ] PUNCT iajs-3013 177 14 = = PUNCT iajs-3013 177 15 ⋃ ⋃ PROPN iajs-3013 178 1 [ [ X iajs-3013 178 2 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ X iajs-3013 178 3 : : PUNCT iajs-3013 178 4 𝑅 𝑅 PROPN iajs-3013 178 5 𝑄 𝑄 PROPN iajs-3013 178 6 ] ] PUNCT iajs-3013 178 7 implies imply VERB iajs-3013 178 8 that that SCONJ iajs-3013 178 9 [ [ X iajs-3013 178 10 𝐹:𝑅 𝐹:𝑅 ADP iajs-3013 178 11 𝑄 𝑄 NOUN iajs-3013 178 12 ] ] PUNCT iajs-3013 178 13 βŠ† βŠ† NUM iajs-3013 178 14 [ [ X iajs-3013 178 15 ⋃ ⋃ ADP iajs-3013 178 16 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ X iajs-3013 178 17 : : PUNCT iajs-3013 178 18 𝑅 𝑅 PROPN iajs-3013 178 19 𝑄 𝑄 PROPN iajs-3013 178 20 ] ] PUNCT iajs-3013 178 21 βŠ† βŠ† NUM iajs-3013 178 22 ⋃ ⋃ NOUN iajs-3013 178 23 [ [ X iajs-3013 178 24 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ X iajs-3013 178 25 : : PUNCT iajs-3013 178 26 𝑅 𝑅 NOUN iajs-3013 178 27 𝑄 𝑄 PROPN iajs-3013 178 28 ] ] PUNCT iajs-3013 178 29 . . PUNCT iajs-3013 179 1 since since SCONJ iajs-3013 179 2 𝑄 𝑄 PRON iajs-3013 179 3 is be AUX iajs-3013 179 4 cyclic cyclic ADJ iajs-3013 179 5 then then ADV iajs-3013 179 6 𝑄 𝑄 PROPN iajs-3013 179 7 is be AUX iajs-3013 179 8 multiplication multiplication NOUN iajs-3013 179 9 [ [ X iajs-3013 179 10 12 12 NUM iajs-3013 179 11 ] ] PUNCT iajs-3013 179 12 , , PUNCT iajs-3013 179 13 then then ADV iajs-3013 179 14 by by ADP iajs-3013 179 15 proposition proposition NOUN iajs-3013 179 16 2.14 2.14 NUM iajs-3013 179 17 [ [ X iajs-3013 179 18 𝐹𝛼:𝑅 𝐹𝛼:𝑅 NOUN iajs-3013 179 19 𝑄 𝑄 NOUN iajs-3013 179 20 ] ] PUNCT iajs-3013 179 21 is be AUX iajs-3013 179 22 alappnq alappnq ADJ iajs-3013 179 23 - - PUNCT iajs-3013 179 24 prime prime ADJ iajs-3013 179 25 ideal ideal NOUN iajs-3013 179 26 of of ADP iajs-3013 179 27 𝑅. 𝑅. NOUN iajs-3013 179 28 but but CCONJ iajs-3013 179 29 𝑅 𝑅 PROPN iajs-3013 179 30 is be AUX iajs-3013 179 31 alappnq alappnq NOUN iajs-3013 179 32 compactly compactly ADV iajs-3013 179 33 packed pack VERB iajs-3013 179 34 ring ring NOUN iajs-3013 179 35 then then ADV iajs-3013 179 36 there there PRON iajs-3013 179 37 exists exist VERB iajs-3013 179 38 𝛼𝑗 𝛼𝑗 PROPN iajs-3013 179 39 ∈ ∈ PROPN iajs-3013 179 40 ʌ ʌ NOUN iajs-3013 179 41 such such ADJ iajs-3013 179 42 that that SCONJ iajs-3013 180 1 [ [ X iajs-3013 180 2 𝐹:𝑅 𝐹:𝑅 ADP iajs-3013 180 3 𝑄 𝑄 NOUN iajs-3013 180 4 ] ] PUNCT iajs-3013 180 5 βŠ† βŠ† NUM iajs-3013 180 6 [ [ X iajs-3013 180 7 𝐹𝛼𝑗:𝑅 𝐹𝛼𝑗:𝑅 X iajs-3013 180 8 𝑄 𝑄 X iajs-3013 180 9 ] ] PUNCT iajs-3013 180 10 and and CCONJ iajs-3013 180 11 by by ADP iajs-3013 180 12 hypothesis hypothesis NOUN iajs-3013 180 13 , , PUNCT iajs-3013 180 14 hence hence ADV iajs-3013 180 15 𝐹 𝐹 PROPN iajs-3013 180 16 βŠ† βŠ† NUM iajs-3013 180 17 𝐹𝛼𝑗 𝐹𝛼𝑗 PROPN iajs-3013 180 18 . . PUNCT iajs-3013 181 1 thus thus ADV iajs-3013 181 2 𝑄 𝑄 PRON iajs-3013 181 3 is be AUX iajs-3013 181 4 strongly strongly ADV iajs-3013 181 5 alappnq alappnq ADJ iajs-3013 181 6 compactly compactly ADV iajs-3013 181 7 packed pack VERB iajs-3013 181 8 . . PUNCT iajs-3013 182 1 the the DET iajs-3013 182 2 following follow VERB iajs-3013 182 3 proposition proposition NOUN iajs-3013 182 4 gives give VERB iajs-3013 182 5 a a DET iajs-3013 182 6 necessary necessary ADJ iajs-3013 182 7 and and CCONJ iajs-3013 182 8 sufficient sufficient ADJ iajs-3013 182 9 condition condition NOUN iajs-3013 182 10 for for ADP iajs-3013 182 11 𝑍-regular 𝑍-regular ADJ iajs-3013 182 12 module module NOUN iajs-3013 182 13 to to PART iajs-3013 182 14 be be AUX iajs-3013 182 15 strongly strongly ADV iajs-3013 182 16 alappnq alappnq ADJ iajs-3013 182 17 compactly compactly ADV iajs-3013 182 18 packed pack VERB iajs-3013 182 19 . . PUNCT iajs-3013 183 1 proposition proposition NOUN iajs-3013 183 2 4.12 4.12 NUM iajs-3013 183 3 let let VERB iajs-3013 183 4 𝑄 𝑄 PRON iajs-3013 183 5 be be AUX iajs-3013 183 6 a a DET iajs-3013 183 7 𝑍-regular 𝑍-regular PROPN iajs-3013 183 8 𝑅-module 𝑅-module PROPN iajs-3013 183 9 , , PUNCT iajs-3013 183 10 then then ADV iajs-3013 183 11 𝑄 𝑄 PRON iajs-3013 183 12 is be AUX iajs-3013 183 13 strongly strongly ADV iajs-3013 183 14 alappnq alappnq ADJ iajs-3013 183 15 compactly compactly ADV iajs-3013 183 16 packed pack VERB iajs-3013 183 17 if if SCONJ iajs-3013 183 18 and and CCONJ iajs-3013 183 19 only only ADV iajs-3013 183 20 if if SCONJ iajs-3013 183 21 every every DET iajs-3013 183 22 proper proper ADJ iajs-3013 183 23 submodule submodule NOUN iajs-3013 183 24 of of ADP iajs-3013 183 25 𝑄 𝑄 PROPN iajs-3013 183 26 is be AUX iajs-3013 183 27 cyclic cyclic ADJ iajs-3013 183 28 . . PUNCT iajs-3013 184 1 proof proof NOUN iajs-3013 184 2 ( ( PUNCT iajs-3013 184 3 ⟾ ⟾ ADJ iajs-3013 184 4 ) ) PUNCT iajs-3013 184 5 suppose suppose VERB iajs-3013 184 6 that that SCONJ iajs-3013 184 7 𝑄 𝑄 PRON iajs-3013 184 8 is be AUX iajs-3013 184 9 a a DET iajs-3013 184 10 strongly strongly ADV iajs-3013 184 11 alappnq alappnq NOUN iajs-3013 184 12 compactly compactly ADV iajs-3013 184 13 packed pack VERB iajs-3013 184 14 𝑅-module 𝑅-module PROPN iajs-3013 184 15 and and CCONJ iajs-3013 184 16 let let VERB iajs-3013 184 17 𝐹 𝐹 PROPN iajs-3013 184 18 βŠ‚ βŠ‚ PROPN iajs-3013 184 19 𝑄. 𝑄. PROPN iajs-3013 184 20 since since SCONJ iajs-3013 184 21 𝑄 𝑄 PRON iajs-3013 184 22 is be AUX iajs-3013 184 23 a a DET iajs-3013 184 24 strongly strongly ADV iajs-3013 184 25 alappnq alappnq NOUN iajs-3013 184 26 compactly compactly ADV iajs-3013 184 27 packed pack VERB iajs-3013 184 28 , , PUNCT iajs-3013 184 29 then then ADV iajs-3013 184 30 by by ADP iajs-3013 184 31 theorem theorem NOUN iajs-3013 184 32 4.10 4.10 NUM iajs-3013 184 33 , , PUNCT iajs-3013 184 34 there there PRON iajs-3013 184 35 exists exist VERB iajs-3013 184 36 π‘ž π‘ž PROPN iajs-3013 184 37 ∈ ∈ PROPN iajs-3013 184 38 𝐹 𝐹 PROPN iajs-3013 184 39 such such ADJ iajs-3013 184 40 that that PRON iajs-3013 184 41 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 184 42 ) ) PUNCT iajs-3013 185 1 = = PUNCT iajs-3013 185 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž NOUN iajs-3013 185 3 ) ) PUNCT iajs-3013 185 4 . . PUNCT iajs-3013 186 1 but but CCONJ iajs-3013 186 2 𝑄 𝑄 PRON iajs-3013 186 3 is be AUX iajs-3013 186 4 𝑍-regular 𝑍-regular ADJ iajs-3013 186 5 module module NOUN iajs-3013 186 6 , , PUNCT iajs-3013 186 7 then then ADV iajs-3013 186 8 by by ADP iajs-3013 186 9 proposition proposition NOUN iajs-3013 186 10 4.4 4.4 NUM iajs-3013 186 11 , , PUNCT iajs-3013 186 12 we we PRON iajs-3013 186 13 have have VERB iajs-3013 186 14 𝐹 𝐹 PROPN iajs-3013 186 15 = = SYM iajs-3013 186 16 π‘…π‘ž π‘…π‘ž NOUN iajs-3013 186 17 , , PUNCT iajs-3013 186 18 thus thus ADV iajs-3013 186 19 𝐹 𝐹 PROPN iajs-3013 186 20 is be AUX iajs-3013 186 21 cyclic cyclic ADJ iajs-3013 186 22 . . PUNCT iajs-3013 187 1 ( ( PUNCT iajs-3013 187 2 ⟽ ⟽ X iajs-3013 187 3 ) ) PUNCT iajs-3013 187 4 suppose suppose VERB iajs-3013 187 5 that that SCONJ iajs-3013 187 6 every every DET iajs-3013 187 7 proper proper ADJ iajs-3013 187 8 submodule submodule NOUN iajs-3013 187 9 of of ADP iajs-3013 187 10 𝑄 𝑄 PROPN iajs-3013 187 11 is be AUX iajs-3013 187 12 cyclic cyclic ADJ iajs-3013 187 13 . . PUNCT iajs-3013 188 1 let let VERB iajs-3013 188 2 𝐹 𝐹 PRON iajs-3013 188 3 be be AUX iajs-3013 188 4 a a DET iajs-3013 188 5 proper proper ADJ iajs-3013 188 6 submodule submodule NOUN iajs-3013 188 7 of of ADP iajs-3013 188 8 𝑄 𝑄 PRON iajs-3013 188 9 then then ADV iajs-3013 188 10 , , PUNCT iajs-3013 188 11 𝐹 𝐹 PROPN iajs-3013 188 12 is be AUX iajs-3013 188 13 cyclic cyclic ADJ iajs-3013 188 14 , , PUNCT iajs-3013 188 15 thus thus ADV iajs-3013 188 16 there there PRON iajs-3013 188 17 exists exist VERB iajs-3013 188 18 π‘ž π‘ž PROPN iajs-3013 188 19 ∈ ∈ PROPN iajs-3013 188 20 𝐹 𝐹 PROPN iajs-3013 188 21 such such ADJ iajs-3013 188 22 that that SCONJ iajs-3013 188 23 𝐹 𝐹 PROPN iajs-3013 188 24 = = SYM iajs-3013 188 25 π‘…π‘ž π‘…π‘ž PROPN iajs-3013 188 26 , , PUNCT iajs-3013 188 27 so so SCONJ iajs-3013 188 28 we we PRON iajs-3013 188 29 have have VERB iajs-3013 188 30 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 188 31 ) ) PUNCT iajs-3013 189 1 = = PUNCT iajs-3013 189 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž NOUN iajs-3013 189 3 ) ) PUNCT iajs-3013 189 4 . . PUNCT iajs-3013 190 1 hence hence ADV iajs-3013 190 2 by by ADP iajs-3013 190 3 theorem theorem NOUN iajs-3013 190 4 4.10 4.10 NUM iajs-3013 190 5 , , PUNCT iajs-3013 190 6 𝑄 𝑄 PRON iajs-3013 190 7 is be AUX iajs-3013 190 8 strongly strongly ADV iajs-3013 190 9 alappnq alappnq ADJ iajs-3013 190 10 compactly compactly ADV iajs-3013 190 11 packed pack VERB iajs-3013 190 12 . . PUNCT iajs-3013 191 1 the the DET iajs-3013 191 2 following follow VERB iajs-3013 191 3 proposition proposition NOUN iajs-3013 191 4 gives give VERB iajs-3013 191 5 condition condition NOUN iajs-3013 191 6 under under ADP iajs-3013 191 7 which which PRON iajs-3013 191 8 strongly strongly ADV iajs-3013 191 9 alappnq alappnq ADJ iajs-3013 191 10 compactly compactly ADV iajs-3013 191 11 packed pack VERB iajs-3013 191 12 module module NOUN iajs-3013 191 13 satisfy satisfy NOUN iajs-3013 191 14 ascending ascend VERB iajs-3013 191 15 chain chain NOUN iajs-3013 191 16 condition condition NOUN iajs-3013 191 17 on on ADP iajs-3013 191 18 alappnq alappnq ADJ iajs-3013 191 19 - - PUNCT iajs-3013 191 20 prime prime ADJ iajs-3013 191 21 radical radical ADJ iajs-3013 191 22 submodules submodule NOUN iajs-3013 191 23 . . PUNCT iajs-3013 192 1 proposition proposition NOUN iajs-3013 192 2 4.13 4.13 NUM iajs-3013 192 3 let let VERB iajs-3013 192 4 𝑄 𝑄 PRON iajs-3013 192 5 be be AUX iajs-3013 192 6 strongly strongly ADV iajs-3013 192 7 alappnq alappnq ADJ iajs-3013 192 8 compactly compactly ADV iajs-3013 192 9 packed pack VERB iajs-3013 192 10 𝑅-module 𝑅-module PROPN iajs-3013 192 11 which which PRON iajs-3013 192 12 has have VERB iajs-3013 192 13 at at ADV iajs-3013 192 14 least least ADV iajs-3013 192 15 one one NUM iajs-3013 192 16 maximal maximal ADJ iajs-3013 192 17 submodule submodule NOUN iajs-3013 192 18 , , PUNCT iajs-3013 192 19 then then ADV iajs-3013 192 20 𝑄 𝑄 PRON iajs-3013 192 21 satisfies satisfy VERB iajs-3013 192 22 the the DET iajs-3013 192 23 ascending ascend VERB iajs-3013 192 24 chain chain NOUN iajs-3013 192 25 condition condition NOUN iajs-3013 192 26 on on ADP iajs-3013 192 27 alappnq alappnq ADJ iajs-3013 192 28 - - PUNCT iajs-3013 192 29 prime prime ADJ iajs-3013 192 30 radical radical ADJ iajs-3013 192 31 submodules submodule NOUN iajs-3013 192 32 . . PUNCT iajs-3013 193 1 proof proof NOUN iajs-3013 193 2 let let VERB iajs-3013 193 3 𝐹1 𝐹1 PROPN iajs-3013 193 4 βŠ† βŠ† NUM iajs-3013 193 5 𝐹2 𝐹2 NOUN iajs-3013 193 6 βŠ† βŠ† NUM iajs-3013 193 7 β‹― β‹― NOUN iajs-3013 193 8 be be AUX iajs-3013 193 9 an an DET iajs-3013 193 10 ascending ascend VERB iajs-3013 193 11 chain chain NOUN iajs-3013 193 12 condition condition NOUN iajs-3013 193 13 for for ADP iajs-3013 193 14 alappnq alappnq ADJ iajs-3013 193 15 - - PUNCT iajs-3013 193 16 prime prime ADJ iajs-3013 193 17 radical radical ADJ iajs-3013 193 18 submodules submodule NOUN iajs-3013 193 19 of of ADP iajs-3013 193 20 𝑄 𝑄 PROPN iajs-3013 193 21 , , PUNCT iajs-3013 193 22 let let VERB iajs-3013 193 23 𝐿 𝐿 PROPN iajs-3013 193 24 = = PRON iajs-3013 193 25 ⋃ ⋃ PROPN iajs-3013 193 26 𝐹𝑖𝑖 𝐹𝑖𝑖 PROPN iajs-3013 193 27 then then ADV iajs-3013 193 28 𝐿 𝐿 PROPN iajs-3013 193 29 is be AUX iajs-3013 193 30 a a DET iajs-3013 193 31 submodule submodule NOUN iajs-3013 193 32 of of ADP iajs-3013 193 33 𝑄. 𝑄. NOUN iajs-3013 193 34 we we PRON iajs-3013 193 35 claim claim VERB iajs-3013 193 36 that that SCONJ iajs-3013 193 37 𝐿 𝐿 PROPN iajs-3013 193 38 βŠ‚ βŠ‚ PROPN iajs-3013 193 39 𝑄. 𝑄. PROPN iajs-3013 193 40 in in ADP iajs-3013 193 41 fact fact NOUN iajs-3013 193 42 , , PUNCT iajs-3013 193 43 if if SCONJ iajs-3013 193 44 𝐿 𝐿 PROPN iajs-3013 193 45 = = SYM iajs-3013 193 46 𝑄 𝑄 PROPN iajs-3013 193 47 and and CCONJ iajs-3013 193 48 𝐻 𝐻 PROPN iajs-3013 193 49 is be AUX iajs-3013 193 50 a a DET iajs-3013 193 51 maximal maximal ADJ iajs-3013 193 52 submodule submodule NOUN iajs-3013 193 53 of of ADP iajs-3013 193 54 𝑄 𝑄 PROPN iajs-3013 193 55 , , PUNCT iajs-3013 193 56 then then ADV iajs-3013 193 57 𝐻 𝐻 PROPN iajs-3013 193 58 ⊊ ⊊ VERB iajs-3013 193 59 ⋃ ⋃ VERB iajs-3013 193 60 𝐹𝑖𝑖 𝐹𝑖𝑖 PROPN iajs-3013 193 61 . . PUNCT iajs-3013 194 1 since since SCONJ iajs-3013 194 2 𝑄 𝑄 PRON iajs-3013 194 3 is be AUX iajs-3013 194 4 strongly strongly ADV iajs-3013 194 5 alappnq alappnq ADJ iajs-3013 194 6 compactly compactly ADV iajs-3013 194 7 packed pack VERB iajs-3013 194 8 then then ADV iajs-3013 194 9 by by ADP iajs-3013 194 10 theorem theorem NOUN iajs-3013 194 11 4.10 4.10 NUM iajs-3013 194 12 𝐻 𝐻 PROPN iajs-3013 194 13 βŠ† βŠ† NUM iajs-3013 194 14 𝐹𝑗 𝐹𝑗 PROPN iajs-3013 194 15 for for ADP iajs-3013 194 16 some some DET iajs-3013 194 17 𝑗. 𝑗. NOUN iajs-3013 194 18 but but CCONJ iajs-3013 194 19 𝐻 𝐻 PROPN iajs-3013 194 20 is be AUX iajs-3013 194 21 maximal maximal ADJ iajs-3013 194 22 submodule submodule NOUN iajs-3013 194 23 then then ADV iajs-3013 195 1 𝐻 𝐻 PROPN iajs-3013 195 2 = = SYM iajs-3013 195 3 𝐹𝑗 𝐹𝑗 PROPN iajs-3013 195 4 and and CCONJ iajs-3013 195 5 this this PRON iajs-3013 195 6 implies imply VERB iajs-3013 195 7 ⋃ ⋃ PROPN iajs-3013 195 8 𝐹𝑖 𝐹𝑖 PROPN iajs-3013 195 9 βŠ† βŠ† NUM iajs-3013 195 10 𝐹𝑗𝑖 𝐹𝑗𝑖 PROPN iajs-3013 195 11 that that PRON iajs-3013 195 12 is be AUX iajs-3013 195 13 𝑄 𝑄 PROPN iajs-3013 195 14 βŠ† βŠ† NUM iajs-3013 195 15 𝐹𝑗 𝐹𝑗 PROPN iajs-3013 195 16 which which PRON iajs-3013 195 17 is be AUX iajs-3013 195 18 a a DET iajs-3013 195 19 contradiction contradiction NOUN iajs-3013 195 20 . . PUNCT iajs-3013 196 1 so so ADV iajs-3013 196 2 𝐿 𝐿 PROPN iajs-3013 196 3 βŠ‚ βŠ‚ PROPN iajs-3013 196 4 𝑄 𝑄 PROPN iajs-3013 196 5 and and CCONJ iajs-3013 196 6 by by ADP iajs-3013 196 7 theorem theorem NOUN iajs-3013 196 8 4.10 4.10 NUM iajs-3013 196 9 there there ADV iajs-3013 196 10 exists exist VERB iajs-3013 196 11 𝑗 𝑗 PRON iajs-3013 196 12 such such ADJ iajs-3013 196 13 that that SCONJ iajs-3013 196 14 𝐿 𝐿 PROPN iajs-3013 196 15 βŠ† βŠ† NUM iajs-3013 196 16 𝐹𝑗 𝐹𝑗 PROPN iajs-3013 196 17 , , PUNCT iajs-3013 196 18 so so ADV iajs-3013 196 19 𝐹1 𝐹1 PROPN iajs-3013 196 20 βŠ† βŠ† NUM iajs-3013 196 21 𝐹2 𝐹2 PROPN iajs-3013 196 22 βŠ† βŠ† NUM iajs-3013 196 23 β‹― β‹― ADP iajs-3013 196 24 βŠ† βŠ† NUM iajs-3013 196 25 𝐹𝑗 𝐹𝑗 PROPN iajs-3013 196 26 that that PRON iajs-3013 196 27 is is ADV iajs-3013 196 28 𝑄 𝑄 PRON iajs-3013 196 29 satisfies satisfy VERB iajs-3013 196 30 the the DET iajs-3013 196 31 ascending ascend VERB iajs-3013 196 32 chain chain NOUN iajs-3013 196 33 condition condition NOUN iajs-3013 196 34 on on ADP iajs-3013 196 35 alappnq alappnq ADJ iajs-3013 196 36 - - PUNCT iajs-3013 196 37 prime prime ADJ iajs-3013 196 38 radical radical ADJ iajs-3013 196 39 submodules submodule NOUN iajs-3013 196 40 . . PUNCT iajs-3013 197 1 the the DET iajs-3013 197 2 following follow VERB iajs-3013 197 3 corollaries corollary NOUN iajs-3013 197 4 are be AUX iajs-3013 197 5 direct direct ADJ iajs-3013 197 6 consequence consequence NOUN iajs-3013 197 7 of of ADP iajs-3013 197 8 proposition proposition NOUN iajs-3013 197 9 4.13 4.13 NUM iajs-3013 197 10 . . PUNCT iajs-3013 198 1 corollary corollary ADJ iajs-3013 198 2 4.14 4.14 NUM iajs-3013 198 3 let let VERB iajs-3013 198 4 𝑄 𝑄 PRON iajs-3013 198 5 be be AUX iajs-3013 198 6 strongly strongly ADV iajs-3013 198 7 alappnq alappnq ADJ iajs-3013 198 8 compactly compactly ADV iajs-3013 198 9 packed pack VERB iajs-3013 199 1 𝑅-module 𝑅-module PROPN iajs-3013 199 2 such such ADJ iajs-3013 199 3 that that DET iajs-3013 199 4 𝐽(𝑄 𝐽(𝑄 NOUN iajs-3013 199 5 ) ) PUNCT iajs-3013 199 6 β‰  β‰  PROPN iajs-3013 199 7 𝑄. 𝑄. NOUN iajs-3013 199 8 then then ADV iajs-3013 199 9 𝑄 𝑄 PRON iajs-3013 199 10 satisfies satisfy VERB iajs-3013 199 11 the the DET iajs-3013 199 12 ascending ascend VERB iajs-3013 199 13 chain chain NOUN iajs-3013 199 14 condition condition NOUN iajs-3013 199 15 on on ADP iajs-3013 199 16 alappnq alappnq ADJ iajs-3013 199 17 - - PUNCT iajs-3013 199 18 prime prime ADJ iajs-3013 199 19 radical radical ADJ iajs-3013 199 20 submodules submodule NOUN iajs-3013 199 21 . . PUNCT iajs-3013 200 1 corollary corollary ADJ iajs-3013 200 2 4.15 4.15 NUM iajs-3013 200 3 if if SCONJ iajs-3013 200 4 𝑄 𝑄 PRON iajs-3013 200 5 is be AUX iajs-3013 200 6 finitely finitely ADV iajs-3013 200 7 generated generate VERB iajs-3013 200 8 strongly strongly ADV iajs-3013 200 9 alappnq alappnq ADJ iajs-3013 200 10 compactly compactly ADV iajs-3013 200 11 packed pack VERB iajs-3013 200 12 𝑅-module 𝑅-module PROPN iajs-3013 200 13 , , PUNCT iajs-3013 200 14 then then ADV iajs-3013 200 15 𝑄 𝑄 PRON iajs-3013 200 16 satisfies satisfy VERB iajs-3013 200 17 the the DET iajs-3013 200 18 ascending ascend VERB iajs-3013 200 19 chain chain NOUN iajs-3013 200 20 condition condition NOUN iajs-3013 200 21 on on ADP iajs-3013 200 22 alappnq alappnq ADJ iajs-3013 200 23 - - PUNCT iajs-3013 200 24 prime prime ADJ iajs-3013 200 25 radical radical ADJ iajs-3013 200 26 submodules submodule NOUN iajs-3013 200 27 . . PUNCT iajs-3013 201 1 corollary corollary ADJ iajs-3013 201 2 4.16 4.16 NUM iajs-3013 201 3 if if SCONJ iajs-3013 201 4 𝑄 𝑄 PRON iajs-3013 201 5 is be AUX iajs-3013 201 6 multiplication multiplication NOUN iajs-3013 201 7 strongly strongly ADV iajs-3013 201 8 alappnq alappnq ADJ iajs-3013 201 9 compactly compactly ADV iajs-3013 201 10 packed pack VERB iajs-3013 201 11 𝑅-module 𝑅-module PROPN iajs-3013 201 12 , , PUNCT iajs-3013 201 13 then then ADV iajs-3013 201 14 𝑄 𝑄 PRON iajs-3013 201 15 satisfies satisfy VERB iajs-3013 201 16 the the DET iajs-3013 201 17 ascending ascend VERB iajs-3013 201 18 chain chain NOUN iajs-3013 201 19 condition condition NOUN iajs-3013 201 20 on on ADP iajs-3013 201 21 alappnq alappnq ADJ iajs-3013 201 22 - - PUNCT iajs-3013 201 23 prime prime ADJ iajs-3013 201 24 radical radical ADJ iajs-3013 201 25 submodules submodule NOUN iajs-3013 201 26 . . PUNCT iajs-3013 202 1 in in ADP iajs-3013 202 2 the the DET iajs-3013 202 3 following follow VERB iajs-3013 202 4 proposition proposition NOUN iajs-3013 202 5 we we PRON iajs-3013 202 6 give give VERB iajs-3013 202 7 a a DET iajs-3013 202 8 condition condition NOUN iajs-3013 202 9 under under ADP iajs-3013 202 10 which which PRON iajs-3013 202 11 the the DET iajs-3013 202 12 convers conver NOUN iajs-3013 202 13 of of ADP iajs-3013 202 14 proposition proposition NOUN iajs-3013 202 15 4.13 4.13 NUM iajs-3013 202 16 is be AUX iajs-3013 202 17 hold hold NOUN iajs-3013 202 18 . . PUNCT iajs-3013 203 1 ihjpas ihjpas PROPN iajs-3013 203 2 . . PUNCT iajs-3013 204 1 36(1)2023 36(1)2023 NUM iajs-3013 204 2 308 308 NUM iajs-3013 204 3 proposition proposition NOUN iajs-3013 204 4 4.17 4.17 NUM iajs-3013 204 5 let let VERB iajs-3013 204 6 𝑄 𝑄 PRON iajs-3013 204 7 be be AUX iajs-3013 204 8 a a DET iajs-3013 204 9 bezout bezout NOUN iajs-3013 204 10 𝑅-module 𝑅-module NOUN iajs-3013 204 11 . . PUNCT iajs-3013 205 1 if if SCONJ iajs-3013 205 2 𝑄 𝑄 PRON iajs-3013 205 3 satisfies satisfy VERB iajs-3013 205 4 the the DET iajs-3013 205 5 ascending ascend VERB iajs-3013 205 6 chain chain NOUN iajs-3013 205 7 condition condition NOUN iajs-3013 205 8 for for ADP iajs-3013 205 9 alappnq alappnq ADJ iajs-3013 205 10 - - PUNCT iajs-3013 205 11 prime prime ADJ iajs-3013 205 12 radical radical ADJ iajs-3013 205 13 submodules submodule NOUN iajs-3013 205 14 , , PUNCT iajs-3013 205 15 then then ADV iajs-3013 205 16 𝑄 𝑄 PRON iajs-3013 205 17 is be AUX iajs-3013 205 18 strongly strongly ADV iajs-3013 205 19 alappnq alappnq ADJ iajs-3013 205 20 compactly compactly ADV iajs-3013 205 21 packed pack VERB iajs-3013 205 22 module module NOUN iajs-3013 205 23 . . PUNCT iajs-3013 206 1 proof proof NOUN iajs-3013 206 2 let let VERB iajs-3013 206 3 𝐹 𝐹 PRON iajs-3013 206 4 be be AUX iajs-3013 206 5 a a DET iajs-3013 206 6 proper proper ADJ iajs-3013 206 7 submodule submodule NOUN iajs-3013 206 8 of of ADP iajs-3013 206 9 𝑄 𝑄 PROPN iajs-3013 206 10 , , PUNCT iajs-3013 206 11 then then ADV iajs-3013 206 12 by by ADP iajs-3013 206 13 proposition proposition NOUN iajs-3013 206 14 4.5 4.5 NUM iajs-3013 206 15 there there ADV iajs-3013 206 16 exists exist VERB iajs-3013 206 17 a a DET iajs-3013 206 18 finitely finitely ADV iajs-3013 206 19 generated generate VERB iajs-3013 206 20 submodule submodule PROPN iajs-3013 206 21 𝐿 𝐿 PROPN iajs-3013 206 22 of of ADP iajs-3013 206 23 𝐹 𝐹 PROPN iajs-3013 206 24 such such ADJ iajs-3013 206 25 that that SCONJ iajs-3013 206 26 𝐹 𝐹 PROPN iajs-3013 206 27 = = SYM iajs-3013 206 28 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐿 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐿 NOUN iajs-3013 206 29 ) ) PUNCT iajs-3013 206 30 and and CCONJ iajs-3013 206 31 hence hence ADV iajs-3013 206 32 by by ADP iajs-3013 206 33 proposition proposition NOUN iajs-3013 206 34 4.3 4.3 NUM iajs-3013 206 35 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 206 36 ) ) PUNCT iajs-3013 207 1 = = SYM iajs-3013 207 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„ PROPN iajs-3013 207 3 ( ( PUNCT iajs-3013 207 4 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐿 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐿 NOUN iajs-3013 207 5 ) ) PUNCT iajs-3013 207 6 ) ) PUNCT iajs-3013 207 7 = = SYM iajs-3013 207 8 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐿 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐿 NOUN iajs-3013 207 9 ) ) PUNCT iajs-3013 207 10 . . PUNCT iajs-3013 208 1 but but CCONJ iajs-3013 208 2 𝑄 𝑄 PRON iajs-3013 208 3 is be AUX iajs-3013 208 4 bezout bezout NOUN iajs-3013 208 5 module module NOUN iajs-3013 208 6 then then ADV iajs-3013 208 7 𝐿 𝐿 PROPN iajs-3013 208 8 is be AUX iajs-3013 208 9 cyclic cyclic ADJ iajs-3013 208 10 submodule submodule NOUN iajs-3013 208 11 , , PUNCT iajs-3013 208 12 then then ADV iajs-3013 208 13 there there PRON iajs-3013 208 14 exists exist VERB iajs-3013 208 15 π‘ž π‘ž PROPN iajs-3013 208 16 ∈ ∈ PROPN iajs-3013 208 17 𝐿 𝐿 PROPN iajs-3013 208 18 such such ADJ iajs-3013 208 19 that that DET iajs-3013 208 20 𝐿 𝐿 PROPN iajs-3013 208 21 = = SYM iajs-3013 208 22 π‘…π‘ž π‘…π‘ž NOUN iajs-3013 208 23 , , PUNCT iajs-3013 208 24 thus thus ADV iajs-3013 208 25 implies imply VERB iajs-3013 208 26 that that SCONJ iajs-3013 208 27 π‘ž π‘ž PROPN iajs-3013 208 28 ∈ ∈ PROPN iajs-3013 208 29 𝐹 𝐹 PROPN iajs-3013 208 30 and and CCONJ iajs-3013 208 31 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 208 32 ) ) PUNCT iajs-3013 209 1 = = PUNCT iajs-3013 209 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž NOUN iajs-3013 209 3 ) ) PUNCT iajs-3013 209 4 . . PUNCT iajs-3013 210 1 therefore therefore ADV iajs-3013 210 2 , , PUNCT iajs-3013 210 3 by by ADP iajs-3013 210 4 theorem theorem NOUN iajs-3013 210 5 4.10 4.10 NUM iajs-3013 210 6 𝑄 𝑄 PROPN iajs-3013 210 7 is be AUX iajs-3013 210 8 a a DET iajs-3013 210 9 strongly strongly ADV iajs-3013 210 10 alappnq alappnq NOUN iajs-3013 210 11 compactly compactly ADV iajs-3013 210 12 packed pack VERB iajs-3013 210 13 module module NOUN iajs-3013 210 14 . . PUNCT iajs-3013 211 1 proposition proposition NOUN iajs-3013 211 2 4.18 4.18 NUM iajs-3013 211 3 let let VERB iajs-3013 211 4 𝑓 𝑓 PRON iajs-3013 211 5 : : PUNCT iajs-3013 211 6 𝑄 𝑄 PROPN iajs-3013 211 7 β†’ β†’ SYM iajs-3013 211 8 𝑄′ 𝑄′ ADJ iajs-3013 211 9 be be VERB iajs-3013 211 10 an an DET iajs-3013 211 11 𝑅-epimorphism 𝑅-epimorphism PROPN iajs-3013 211 12 , , PUNCT iajs-3013 211 13 and and CCONJ iajs-3013 211 14 ker ker X iajs-3013 211 15 𝑓 𝑓 PRON iajs-3013 211 16 is be AUX iajs-3013 211 17 a a DET iajs-3013 211 18 small small ADJ iajs-3013 211 19 submodule submodule NOUN iajs-3013 211 20 of of ADP iajs-3013 211 21 𝑄 𝑄 PRON iajs-3013 211 22 such such DET iajs-3013 211 23 that that DET iajs-3013 211 24 ker ker NOUN iajs-3013 212 1 𝑓 𝑓 PRON iajs-3013 212 2 βŠ† βŠ† NUM iajs-3013 212 3 𝑃 𝑃 NOUN iajs-3013 212 4 for for ADP iajs-3013 212 5 each each DET iajs-3013 212 6 alappnq alappnq ADJ iajs-3013 212 7 - - PUNCT iajs-3013 212 8 prime prime NOUN iajs-3013 212 9 submodule submodule NOUN iajs-3013 212 10 𝑃 𝑃 PROPN iajs-3013 212 11 of of ADP iajs-3013 212 12 𝑄. 𝑄. NOUN iajs-3013 212 13 then then ADV iajs-3013 212 14 𝑄 𝑄 PRON iajs-3013 212 15 is be AUX iajs-3013 212 16 strongly strongly ADV iajs-3013 212 17 alappnq alappnq ADJ iajs-3013 212 18 compactly compactly ADV iajs-3013 212 19 packed pack VERB iajs-3013 212 20 if if SCONJ iajs-3013 212 21 and and CCONJ iajs-3013 212 22 only only ADV iajs-3013 212 23 if if SCONJ iajs-3013 212 24 𝑄′ 𝑄′ ADJ iajs-3013 212 25 is be AUX iajs-3013 212 26 strongly strongly ADV iajs-3013 212 27 alappnq alappnq ADJ iajs-3013 212 28 compactly compactly ADV iajs-3013 212 29 packed pack VERB iajs-3013 212 30 . . PUNCT iajs-3013 213 1 proof proof NOUN iajs-3013 213 2 ( ( PUNCT iajs-3013 213 3 ⟾ ⟾ ADJ iajs-3013 213 4 ) ) PUNCT iajs-3013 213 5 suppose suppose VERB iajs-3013 213 6 that that SCONJ iajs-3013 213 7 𝑄 𝑄 PRON iajs-3013 213 8 is be AUX iajs-3013 213 9 strongly strongly ADV iajs-3013 213 10 alappnq alappnq ADJ iajs-3013 213 11 compactly compactly ADV iajs-3013 213 12 packed pack VERB iajs-3013 213 13 𝑅-module 𝑅-module PROPN iajs-3013 213 14 , , PUNCT iajs-3013 213 15 and and CCONJ iajs-3013 213 16 𝐹′ 𝐹′ NUM iajs-3013 213 17 βŠ† βŠ† NUM iajs-3013 213 18 ⋃ ⋃ NOUN iajs-3013 213 19 π‘ƒβ€²π›Όπ›ΌβˆˆΚŒ π‘ƒβ€²π›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 213 20 , , PUNCT iajs-3013 213 21 where where SCONJ iajs-3013 213 22 𝐹′ 𝐹′ PROPN iajs-3013 213 23 is be AUX iajs-3013 213 24 proper proper ADJ iajs-3013 213 25 submodule submodule NOUN iajs-3013 213 26 of of ADP iajs-3013 213 27 𝑄′ 𝑄′ PROPN iajs-3013 213 28 and and CCONJ iajs-3013 213 29 𝑃′ 𝑃′ NOUN iajs-3013 213 30 is be AUX iajs-3013 213 31 an an DET iajs-3013 213 32 alappnq alappnq ADJ iajs-3013 213 33 - - PUNCT iajs-3013 213 34 prime prime NOUN iajs-3013 213 35 submodule submodule NOUN iajs-3013 213 36 of of ADP iajs-3013 213 37 𝑄′ 𝑄′ PROPN iajs-3013 213 38 for for ADP iajs-3013 213 39 all all DET iajs-3013 213 40 𝛼 𝛼 PROPN iajs-3013 213 41 ∈ ∈ PROPN iajs-3013 213 42 ʌ ʌ X iajs-3013 213 43 . . PUNCT iajs-3013 214 1 since since SCONJ iajs-3013 214 2 𝑓 𝑓 PRON iajs-3013 214 3 is be AUX iajs-3013 214 4 an an DET iajs-3013 214 5 epimorphism epimorphism NOUN iajs-3013 214 6 , , PUNCT iajs-3013 214 7 then then ADV iajs-3013 214 8 π‘“βˆ’1(𝐹′ π‘“βˆ’1(𝐹′ NOUN iajs-3013 214 9 ) ) PUNCT iajs-3013 214 10 βŠ† βŠ† NUM iajs-3013 214 11 π‘“βˆ’1(⋃ π‘“βˆ’1(⋃ NOUN iajs-3013 214 12 π‘ƒβ€²π›Όπ›ΌβˆˆΚŒ π‘ƒβ€²π›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 214 13 ) ) PUNCT iajs-3013 214 14 . . PUNCT iajs-3013 215 1 thus thus ADV iajs-3013 215 2 π‘“βˆ’1(𝐹′ π‘“βˆ’1(𝐹′ NOUN iajs-3013 215 3 ) ) PUNCT iajs-3013 215 4 βŠ† βŠ† NUM iajs-3013 215 5 ⋃ ⋃ NOUN iajs-3013 215 6 π‘“βˆ’1(𝑃′𝛼)π›ΌβˆˆΚŒ π‘“βˆ’1(𝑃′𝛼)π›ΌβˆˆΚŒ NOUN iajs-3013 215 7 . . PUNCT iajs-3013 216 1 but but CCONJ iajs-3013 216 2 𝑃′𝛼 𝑃′𝛼 NOUN iajs-3013 216 3 is be AUX iajs-3013 216 4 an an DET iajs-3013 216 5 alappnq alappnq ADJ iajs-3013 216 6 - - PUNCT iajs-3013 216 7 prime prime NOUN iajs-3013 216 8 submodule submodule NOUN iajs-3013 216 9 of of ADP iajs-3013 216 10 𝑄′ 𝑄′ PROPN iajs-3013 216 11 , , PUNCT iajs-3013 216 12 then then ADV iajs-3013 216 13 by by ADP iajs-3013 216 14 proposition proposition NOUN iajs-3013 216 15 2.5 2.5 NUM iajs-3013 216 16 we we PRON iajs-3013 216 17 have have VERB iajs-3013 216 18 π‘“βˆ’1(𝑃′𝛼 π‘“βˆ’1(𝑃′𝛼 NOUN iajs-3013 216 19 ) ) PUNCT iajs-3013 216 20 is be AUX iajs-3013 216 21 an an DET iajs-3013 216 22 alappnq alappnq ADJ iajs-3013 216 23 - - PUNCT iajs-3013 216 24 prime prime NOUN iajs-3013 216 25 submodule submodule NOUN iajs-3013 216 26 of of ADP iajs-3013 216 27 𝑄 𝑄 PROPN iajs-3013 216 28 for for ADP iajs-3013 216 29 all all DET iajs-3013 216 30 𝛼 𝛼 PRON iajs-3013 216 31 ∈ ∈ PROPN iajs-3013 216 32 ʌ ʌ X iajs-3013 216 33 . . PUNCT iajs-3013 217 1 since since SCONJ iajs-3013 217 2 𝑄 𝑄 PRON iajs-3013 217 3 is be AUX iajs-3013 217 4 strongly strongly ADV iajs-3013 217 5 alappnq alappnq ADJ iajs-3013 217 6 compactly compactly ADV iajs-3013 217 7 packed pack VERB iajs-3013 217 8 then then ADV iajs-3013 217 9 π‘“βˆ’1(𝐹′ π‘“βˆ’1(𝐹′ NOUN iajs-3013 217 10 ) ) PUNCT iajs-3013 218 1 βŠ† βŠ† NUM iajs-3013 218 2 π‘“βˆ’1(𝑃′𝛽 π‘“βˆ’1(𝑃′𝛽 NUM iajs-3013 218 3 ) ) PUNCT iajs-3013 218 4 for for ADP iajs-3013 218 5 some some DET iajs-3013 218 6 𝛽 𝛽 PROPN iajs-3013 218 7 ∈ ∈ PROPN iajs-3013 218 8 ʌ ʌ X iajs-3013 218 9 . . PUNCT iajs-3013 219 1 therefore therefore ADV iajs-3013 219 2 𝐹′ 𝐹′ NUM iajs-3013 219 3 βŠ† βŠ† NUM iajs-3013 219 4 𝑃′𝛽 𝑃′𝛽 ADJ iajs-3013 219 5 for for ADP iajs-3013 219 6 some some DET iajs-3013 219 7 𝛽 𝛽 NOUN iajs-3013 219 8 ∈ ∈ PROPN iajs-3013 219 9 ʌ ʌ PROPN iajs-3013 219 10 . . PUNCT iajs-3013 220 1 hence hence ADV iajs-3013 220 2 𝐹′ 𝐹′ PROPN iajs-3013 220 3 is be AUX iajs-3013 220 4 strongly strongly ADV iajs-3013 220 5 alappnq alappnq ADJ iajs-3013 220 6 compactly compactly ADV iajs-3013 220 7 packed pack VERB iajs-3013 220 8 submodule submodule NOUN iajs-3013 220 9 of of ADP iajs-3013 220 10 𝑄′. 𝑄′. NOUN iajs-3013 220 11 thus thus ADV iajs-3013 220 12 𝑄′ 𝑄′ ADJ iajs-3013 220 13 is be AUX iajs-3013 220 14 strongly strongly ADV iajs-3013 220 15 alappnq alappnq ADJ iajs-3013 220 16 compactly compactly ADV iajs-3013 220 17 packed pack VERB iajs-3013 220 18 . . PUNCT iajs-3013 221 1 ( ( PUNCT iajs-3013 221 2 ⟽ ⟽ X iajs-3013 221 3 ) ) PUNCT iajs-3013 221 4 suppose suppose VERB iajs-3013 221 5 that that SCONJ iajs-3013 221 6 𝑄′ 𝑄′ PROPN iajs-3013 221 7 is be AUX iajs-3013 221 8 strongly strongly ADV iajs-3013 221 9 alappnq alappnq ADJ iajs-3013 221 10 compactly compactly ADV iajs-3013 221 11 packed pack VERB iajs-3013 221 12 𝑅-module 𝑅-module PROPN iajs-3013 221 13 and and CCONJ iajs-3013 221 14 ker ker VERB iajs-3013 221 15 𝑓 𝑓 PRON iajs-3013 221 16 βŠ† βŠ† NUM iajs-3013 221 17 𝑃 𝑃 NOUN iajs-3013 221 18 for for ADP iajs-3013 221 19 each each DET iajs-3013 221 20 alappnq alappnq ADJ iajs-3013 221 21 - - PUNCT iajs-3013 221 22 prime prime NOUN iajs-3013 221 23 submodule submodule NOUN iajs-3013 221 24 𝑃 𝑃 PROPN iajs-3013 221 25 of of ADP iajs-3013 221 26 𝑄. 𝑄. NOUN iajs-3013 221 27 let let VERB iajs-3013 221 28 𝐹 𝐹 PRON iajs-3013 221 29 be be AUX iajs-3013 221 30 a a DET iajs-3013 221 31 proper proper ADJ iajs-3013 221 32 submodule submodule NOUN iajs-3013 221 33 of of ADP iajs-3013 221 34 𝑄 𝑄 PRON iajs-3013 221 35 such such ADJ iajs-3013 221 36 that that SCONJ iajs-3013 221 37 𝐹 𝐹 PROPN iajs-3013 221 38 βŠ† βŠ† NUM iajs-3013 221 39 ⋃ ⋃ ADP iajs-3013 221 40 π‘ƒπ›Όπ›ΌβˆˆΚŒ π‘ƒπ›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 221 41 , , PUNCT iajs-3013 221 42 where where SCONJ iajs-3013 221 43 𝑃𝛼 𝑃𝛼 PROPN iajs-3013 221 44 is be AUX iajs-3013 221 45 an an DET iajs-3013 221 46 alappnq alappnq ADJ iajs-3013 221 47 - - PUNCT iajs-3013 221 48 prime prime NOUN iajs-3013 221 49 submodule submodule NOUN iajs-3013 221 50 of of ADP iajs-3013 221 51 𝑄 𝑄 PROPN iajs-3013 221 52 for for ADP iajs-3013 221 53 all all DET iajs-3013 221 54 𝛼 𝛼 PRON iajs-3013 221 55 ∈ ∈ PROPN iajs-3013 221 56 ʌ ʌ X iajs-3013 221 57 . . PUNCT iajs-3013 222 1 then then ADV iajs-3013 222 2 𝑓(𝐹 𝑓(𝐹 NOUN iajs-3013 222 3 ) ) PUNCT iajs-3013 223 1 βŠ† βŠ† NUM iajs-3013 223 2 𝑓(⋃ 𝑓(⋃ PROPN iajs-3013 223 3 𝑃𝛼)π›ΌβˆˆΚŒ 𝑃𝛼)π›ΌβˆˆΚŒ NOUN iajs-3013 223 4 implies imply VERB iajs-3013 223 5 that that SCONJ iajs-3013 223 6 𝑓(𝐹 𝑓(𝐹 NOUN iajs-3013 223 7 ) ) PUNCT iajs-3013 223 8 βŠ† βŠ† NUM iajs-3013 223 9 ⋃ ⋃ NOUN iajs-3013 223 10 𝑓(𝑃𝛼)π›ΌβˆˆΚŒ 𝑓(𝑃𝛼)π›ΌβˆˆΚŒ NOUN iajs-3013 223 11 . . PUNCT iajs-3013 224 1 but but CCONJ iajs-3013 224 2 ker ker VERB iajs-3013 224 3 𝑓 𝑓 PRON iajs-3013 224 4 βŠ† βŠ† NUM iajs-3013 224 5 𝑃𝛼 𝑃𝛼 PROPN iajs-3013 224 6 for for ADP iajs-3013 224 7 each each DET iajs-3013 224 8 𝛼. 𝛼. NOUN iajs-3013 224 9 then then ADV iajs-3013 224 10 by by ADP iajs-3013 224 11 proposition proposition NOUN iajs-3013 224 12 2.6 2.6 NUM iajs-3013 224 13 𝑓(𝑃𝛼 𝑓(𝑃𝛼 NOUN iajs-3013 224 14 ) ) PUNCT iajs-3013 224 15 is be AUX iajs-3013 224 16 an an DET iajs-3013 224 17 alappnqprime alappnqprime NOUN iajs-3013 224 18 submodule submodule NOUN iajs-3013 224 19 of of ADP iajs-3013 224 20 𝑄′ 𝑄′ PROPN iajs-3013 224 21 for for ADP iajs-3013 224 22 all all DET iajs-3013 224 23 𝛼 𝛼 PROPN iajs-3013 224 24 ∈ ∈ PROPN iajs-3013 224 25 ʌ ʌ X iajs-3013 224 26 . . PUNCT iajs-3013 225 1 since since SCONJ iajs-3013 225 2 𝑄′ 𝑄′ PROPN iajs-3013 225 3 is be AUX iajs-3013 225 4 strongly strongly ADV iajs-3013 225 5 alappnq alappnq ADJ iajs-3013 225 6 compactly compactly ADV iajs-3013 225 7 packed pack VERB iajs-3013 225 8 𝑅-module 𝑅-module PROPN iajs-3013 225 9 , , PUNCT iajs-3013 225 10 then then ADV iajs-3013 225 11 𝑓(𝐹 𝑓(𝐹 NOUN iajs-3013 225 12 ) ) PUNCT iajs-3013 225 13 βŠ† βŠ† NUM iajs-3013 225 14 𝑓(𝑃𝛽 𝑓(𝑃𝛽 NOUN iajs-3013 225 15 ) ) PUNCT iajs-3013 225 16 for for ADP iajs-3013 225 17 some some DET iajs-3013 225 18 𝛽 𝛽 PROPN iajs-3013 225 19 ∈ ∈ PROPN iajs-3013 225 20 ʌ ʌ PROPN iajs-3013 225 21 . . PUNCT iajs-3013 226 1 thus thus ADV iajs-3013 226 2 , , PUNCT iajs-3013 226 3 for for ADP iajs-3013 226 4 every every DET iajs-3013 226 5 π‘₯ π‘₯ PROPN iajs-3013 226 6 ∈ ∈ PROPN iajs-3013 226 7 𝐹 𝐹 PROPN iajs-3013 226 8 , , PUNCT iajs-3013 226 9 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3013 226 10 ) ) PUNCT iajs-3013 226 11 ∈ ∈ PROPN iajs-3013 226 12 𝑓(𝐹 𝑓(𝐹 NOUN iajs-3013 226 13 ) ) PUNCT iajs-3013 226 14 βŠ† βŠ† NUM iajs-3013 226 15 𝑓(𝑃𝛽 𝑓(𝑃𝛽 NOUN iajs-3013 226 16 ) ) PUNCT iajs-3013 226 17 , , PUNCT iajs-3013 226 18 then then ADV iajs-3013 226 19 𝑓(π‘₯ 𝑓(π‘₯ VERB iajs-3013 226 20 ) ) PUNCT iajs-3013 226 21 ∈ ∈ PROPN iajs-3013 226 22 𝑓(𝑃𝛽 𝑓(𝑃𝛽 NOUN iajs-3013 226 23 ) ) PUNCT iajs-3013 226 24 . . PUNCT iajs-3013 227 1 therefore therefore ADV iajs-3013 227 2 , , PUNCT iajs-3013 227 3 there there PRON iajs-3013 227 4 exists exist VERB iajs-3013 227 5 𝑏 𝑏 PRON iajs-3013 227 6 ∈ ∈ PROPN iajs-3013 227 7 𝑃𝛽 𝑃𝛽 INTJ iajs-3013 227 8 such such ADJ iajs-3013 227 9 that that SCONJ iajs-3013 227 10 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3013 227 11 ) ) PUNCT iajs-3013 227 12 = = SYM iajs-3013 227 13 𝑓(𝑏 𝑓(𝑏 NOUN iajs-3013 227 14 ) ) PUNCT iajs-3013 227 15 , , PUNCT iajs-3013 227 16 then then ADV iajs-3013 227 17 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3013 227 18 ) ) PUNCT iajs-3013 227 19 βˆ’ βˆ’ PROPN iajs-3013 227 20 𝑓(𝑏 𝑓(𝑏 NOUN iajs-3013 227 21 ) ) PUNCT iajs-3013 227 22 = = SYM iajs-3013 227 23 0 0 NUM iajs-3013 227 24 , , PUNCT iajs-3013 227 25 and and CCONJ iajs-3013 227 26 𝑓(π‘₯ 𝑓(π‘₯ VERB iajs-3013 227 27 βˆ’ βˆ’ PROPN iajs-3013 227 28 𝑏 𝑏 NOUN iajs-3013 227 29 ) ) PUNCT iajs-3013 227 30 = = SYM iajs-3013 227 31 0 0 NUM iajs-3013 228 1 so so ADV iajs-3013 228 2 π‘₯ π‘₯ ADP iajs-3013 228 3 βˆ’ βˆ’ PROPN iajs-3013 228 4 𝑏 𝑏 PROPN iajs-3013 228 5 ∈ ∈ PROPN iajs-3013 228 6 ker ker NOUN iajs-3013 229 1 𝑓 𝑓 PROPN iajs-3013 229 2 βŠ† βŠ† NUM iajs-3013 229 3 𝑃𝛽. 𝑃𝛽. PROPN iajs-3013 229 4 that that PRON iajs-3013 229 5 is be AUX iajs-3013 229 6 π‘₯ π‘₯ DET iajs-3013 229 7 ∈ ∈ PROPN iajs-3013 229 8 𝑃𝛽. 𝑃𝛽. PROPN iajs-3013 229 9 therefore therefore ADV iajs-3013 229 10 𝐹 𝐹 PROPN iajs-3013 229 11 βŠ† βŠ† NUM iajs-3013 229 12 𝑃𝛽 𝑃𝛽 PROPN iajs-3013 229 13 for for ADP iajs-3013 229 14 some some DET iajs-3013 229 15 𝛽 𝛽 NOUN iajs-3013 229 16 ∈ ∈ NOUN iajs-3013 229 17 ʌ ʌ NOUN iajs-3013 229 18 and and CCONJ iajs-3013 229 19 hence hence ADV iajs-3013 229 20 𝐹 𝐹 PROPN iajs-3013 229 21 is be AUX iajs-3013 229 22 strongly strongly ADV iajs-3013 229 23 alappnq alappnq ADJ iajs-3013 229 24 compactly compactly ADV iajs-3013 229 25 packed pack VERB iajs-3013 229 26 submodule submodule NOUN iajs-3013 229 27 . . PUNCT iajs-3013 230 1 hence hence ADV iajs-3013 230 2 𝑄 𝑄 PROPN iajs-3013 230 3 is be AUX iajs-3013 230 4 strongly strongly ADV iajs-3013 230 5 alappnq alappnq ADJ iajs-3013 230 6 compactly compactly ADV iajs-3013 230 7 packed pack VERB iajs-3013 230 8 𝑅-module 𝑅-module PROPN iajs-3013 230 9 . . PUNCT iajs-3013 231 1 proposition proposition NOUN iajs-3013 231 2 4.19 4.19 NUM iajs-3013 231 3 let let VERB iajs-3013 231 4 𝑄 𝑄 PRON iajs-3013 231 5 be be AUX iajs-3013 231 6 an an DET iajs-3013 231 7 𝑅-module 𝑅-module NOUN iajs-3013 231 8 , , PUNCT iajs-3013 231 9 and and CCONJ iajs-3013 231 10 𝑆 𝑆 PROPN iajs-3013 231 11 a a DET iajs-3013 231 12 multiplicatively multiplicatively ADV iajs-3013 231 13 closed close VERB iajs-3013 231 14 set set VERB iajs-3013 231 15 in in ADP iajs-3013 231 16 𝑅. 𝑅. NOUN iajs-3013 231 17 if if SCONJ iajs-3013 231 18 𝑄 𝑄 PRON iajs-3013 231 19 is be AUX iajs-3013 231 20 strongly strongly ADV iajs-3013 231 21 alappnq alappnq ADJ iajs-3013 231 22 compactly compactly ADV iajs-3013 231 23 packed pack VERB iajs-3013 231 24 module module NOUN iajs-3013 231 25 , , PUNCT iajs-3013 231 26 then then ADV iajs-3013 231 27 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 231 28 is be AUX iajs-3013 231 29 strongly strongly ADV iajs-3013 231 30 alappnq alappnq ADJ iajs-3013 231 31 compactly compactly ADV iajs-3013 231 32 packed pack VERB iajs-3013 231 33 module module NOUN iajs-3013 231 34 . . PUNCT iajs-3013 232 1 proof proof NOUN iajs-3013 232 2 assume assume VERB iajs-3013 232 3 𝑄 𝑄 PRON iajs-3013 232 4 is be AUX iajs-3013 232 5 strongly strongly ADV iajs-3013 232 6 alappnq alappnq ADJ iajs-3013 232 7 compactly compactly ADV iajs-3013 232 8 packed pack VERB iajs-3013 232 9 . . PUNCT iajs-3013 233 1 let let VERB iajs-3013 233 2 𝐹 𝐹 PRON iajs-3013 233 3 be be AUX iajs-3013 233 4 proper proper ADJ iajs-3013 233 5 submodule submodule NOUN iajs-3013 233 6 of of ADP iajs-3013 233 7 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 233 8 with with ADP iajs-3013 233 9 𝐹 𝐹 PROPN iajs-3013 233 10 βŠ† βŠ† NUM iajs-3013 233 11 ⋃ ⋃ ADP iajs-3013 233 12 π‘ƒπ›Όπ›ΌβˆˆΚŒ π‘ƒπ›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 233 13 , , PUNCT iajs-3013 233 14 where where SCONJ iajs-3013 233 15 𝑃𝛼 𝑃𝛼 PROPN iajs-3013 233 16 is be AUX iajs-3013 233 17 an an DET iajs-3013 233 18 alappnq alappnq ADJ iajs-3013 233 19 - - PUNCT iajs-3013 233 20 prime prime NOUN iajs-3013 233 21 submodule submodule NOUN iajs-3013 233 22 of of ADP iajs-3013 233 23 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 233 24 for for ADP iajs-3013 233 25 all all DET iajs-3013 233 26 𝛼 𝛼 PROPN iajs-3013 233 27 ∈ ∈ PROPN iajs-3013 233 28 ʌ ʌ X iajs-3013 233 29 . . PUNCT iajs-3013 233 30 define define VERB iajs-3013 233 31 𝑓 𝑓 DET iajs-3013 233 32 : : PUNCT iajs-3013 233 33 𝑄 𝑄 PROPN iajs-3013 233 34 β†’ β†’ SYM iajs-3013 233 35 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 233 36 by by ADP iajs-3013 233 37 𝑓(π‘ž 𝑓(π‘ž NOUN iajs-3013 233 38 ) ) PUNCT iajs-3013 234 1 = = PUNCT iajs-3013 234 2 π‘ž π‘ž X iajs-3013 234 3 1 1 NUM iajs-3013 234 4 for for ADP iajs-3013 234 5 every every DET iajs-3013 234 6 π‘ž π‘ž PROPN iajs-3013 234 7 ∈ ∈ PROPN iajs-3013 234 8 𝑄. 𝑄. PROPN iajs-3013 234 9 thus thus ADV iajs-3013 234 10 𝑓 𝑓 PRON iajs-3013 234 11 is be AUX iajs-3013 234 12 an an DET iajs-3013 234 13 epimorphism epimorphism NOUN iajs-3013 234 14 . . PUNCT iajs-3013 235 1 therefore therefore ADV iajs-3013 235 2 π‘“βˆ’1(𝐹 π‘“βˆ’1(𝐹 NUM iajs-3013 235 3 ) ) PUNCT iajs-3013 235 4 βŠ† βŠ† NUM iajs-3013 235 5 π‘“βˆ’1(⋃ π‘“βˆ’1(⋃ NOUN iajs-3013 235 6 π‘ƒπ›Όπ›ΌβˆˆΚŒ π‘ƒπ›Όπ›ΌβˆˆΚŒ NOUN iajs-3013 235 7 ) ) PUNCT iajs-3013 235 8 , , PUNCT iajs-3013 235 9 implies imply VERB iajs-3013 235 10 that that SCONJ iajs-3013 235 11 π‘“βˆ’1(𝐹 π‘“βˆ’1(𝐹 NOUN iajs-3013 235 12 ) ) PUNCT iajs-3013 235 13 βŠ† βŠ† NUM iajs-3013 235 14 ⋃ ⋃ NOUN iajs-3013 235 15 π‘“βˆ’1(𝑃𝛼)π›ΌβˆˆΚŒ π‘“βˆ’1(𝑃𝛼)π›ΌβˆˆΚŒ NOUN iajs-3013 235 16 . . PUNCT iajs-3013 236 1 since since SCONJ iajs-3013 236 2 𝑃𝛼 𝑃𝛼 PROPN iajs-3013 236 3 is be AUX iajs-3013 236 4 an an DET iajs-3013 236 5 alappnq alappnq ADJ iajs-3013 236 6 - - PUNCT iajs-3013 236 7 prime prime NOUN iajs-3013 236 8 submodule submodule NOUN iajs-3013 236 9 of of ADP iajs-3013 236 10 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 236 11 for for ADP iajs-3013 236 12 all all DET iajs-3013 236 13 𝛼 𝛼 PRON iajs-3013 236 14 ∈ ∈ PROPN iajs-3013 236 15 ʌ ʌ NOUN iajs-3013 236 16 and and CCONJ iajs-3013 236 17 𝑓 𝑓 PRON iajs-3013 236 18 is be AUX iajs-3013 236 19 an an DET iajs-3013 236 20 epimorphism epimorphism NOUN iajs-3013 236 21 , , PUNCT iajs-3013 236 22 then then ADV iajs-3013 236 23 by by ADP iajs-3013 236 24 proposition proposition NOUN iajs-3013 236 25 2.5 2.5 NUM iajs-3013 236 26 we we PRON iajs-3013 236 27 have have VERB iajs-3013 236 28 π‘“βˆ’1(𝑃𝛼 π‘“βˆ’1(𝑃𝛼 ADV iajs-3013 236 29 ) ) PUNCT iajs-3013 236 30 is be AUX iajs-3013 236 31 an an DET iajs-3013 236 32 alappnq alappnq ADJ iajs-3013 236 33 - - PUNCT iajs-3013 236 34 prime prime NOUN iajs-3013 236 35 submodule submodule NOUN iajs-3013 236 36 of of ADP iajs-3013 236 37 𝑄 𝑄 PROPN iajs-3013 236 38 for for ADP iajs-3013 236 39 all all DET iajs-3013 236 40 𝛼 𝛼 PRON iajs-3013 236 41 ∈ ∈ PROPN iajs-3013 236 42 ʌ ʌ X iajs-3013 236 43 . . PUNCT iajs-3013 237 1 but but CCONJ iajs-3013 237 2 𝑄 𝑄 PRON iajs-3013 237 3 is be AUX iajs-3013 237 4 strongly strongly ADV iajs-3013 237 5 alappnq alappnq ADJ iajs-3013 237 6 compactly compactly ADV iajs-3013 237 7 packed pack VERB iajs-3013 237 8 , , PUNCT iajs-3013 237 9 then then ADV iajs-3013 237 10 π‘“βˆ’1(𝐹 π‘“βˆ’1(𝐹 NUM iajs-3013 237 11 ) ) PUNCT iajs-3013 237 12 βŠ† βŠ† NUM iajs-3013 237 13 π‘“βˆ’1(𝑃𝛽 π‘“βˆ’1(𝑃𝛽 NOUN iajs-3013 237 14 ) ) PUNCT iajs-3013 237 15 for for ADP iajs-3013 237 16 some some DET iajs-3013 237 17 𝛽 𝛽 NOUN iajs-3013 237 18 ∈ ∈ PROPN iajs-3013 237 19 ʌ ʌ X iajs-3013 237 20 . . PUNCT iajs-3013 238 1 therefore therefore ADV iajs-3013 238 2 ( ( PUNCT iajs-3013 238 3 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 NOUN iajs-3013 238 4 βŠ† βŠ† NUM iajs-3013 238 5 ( ( PUNCT iajs-3013 238 6 𝑓 𝑓 DET iajs-3013 238 7 βˆ’1(𝑃𝛽))𝑆 βˆ’1(𝑃𝛽))𝑆 PROPN iajs-3013 238 8 . . PUNCT iajs-3013 239 1 we we PRON iajs-3013 239 2 need need VERB iajs-3013 239 3 to to PART iajs-3013 239 4 show show VERB iajs-3013 239 5 that that SCONJ iajs-3013 239 6 ( ( PUNCT iajs-3013 239 7 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 PROPN iajs-3013 239 8 = = SYM iajs-3013 239 9 𝐹 𝐹 PROPN iajs-3013 239 10 for for ADP iajs-3013 239 11 any any DET iajs-3013 239 12 submodule submodule NOUN iajs-3013 239 13 𝐹 𝐹 PROPN iajs-3013 239 14 of of ADP iajs-3013 239 15 𝑄𝑆. 𝑄𝑆. NOUN iajs-3013 239 16 let let VERB iajs-3013 239 17 π‘₯ π‘₯ PRON iajs-3013 239 18 𝑠 𝑠 X iajs-3013 239 19 ∈ ∈ PROPN iajs-3013 239 20 ( ( PUNCT iajs-3013 239 21 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 NOUN iajs-3013 239 22 , , PUNCT iajs-3013 239 23 where where SCONJ iajs-3013 239 24 π‘₯ π‘₯ DET iajs-3013 239 25 ∈ ∈ PROPN iajs-3013 239 26 π‘“βˆ’1(𝐹 π‘“βˆ’1(𝐹 NUM iajs-3013 239 27 ) ) PUNCT iajs-3013 239 28 and and CCONJ iajs-3013 239 29 𝑠 𝑠 PROPN iajs-3013 239 30 ∈ ∈ PROPN iajs-3013 239 31 𝑆. 𝑆. PROPN iajs-3013 239 32 then then ADV iajs-3013 239 33 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3013 239 34 ) ) PUNCT iajs-3013 239 35 ∈ ∈ PROPN iajs-3013 239 36 𝐹 𝐹 PROPN iajs-3013 239 37 , , PUNCT iajs-3013 239 38 therefore therefore ADV iajs-3013 239 39 ihjpas ihjpa VERB iajs-3013 239 40 . . PUNCT iajs-3013 240 1 36(1)2023 36(1)2023 NUM iajs-3013 240 2 309 309 NUM iajs-3013 240 3 π‘₯ π‘₯ SYM iajs-3013 240 4 1 1 NUM iajs-3013 240 5 ∈ ∈ PROPN iajs-3013 240 6 𝐹 𝐹 PROPN iajs-3013 240 7 , , PUNCT iajs-3013 240 8 hence hence ADV iajs-3013 240 9 π‘₯ π‘₯ X iajs-3013 240 10 𝑠 𝑠 NOUN iajs-3013 240 11 = = SYM iajs-3013 240 12 1 1 NUM iajs-3013 240 13 𝑠 𝑠 PART iajs-3013 240 14 π‘₯ π‘₯ PROPN iajs-3013 240 15 1 1 NUM iajs-3013 240 16 ∈ ∈ PROPN iajs-3013 240 17 𝐹. 𝐹. NOUN iajs-3013 240 18 thus thus ADV iajs-3013 240 19 ( ( PUNCT iajs-3013 240 20 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 NOUN iajs-3013 240 21 βŠ† βŠ† NUM iajs-3013 240 22 𝐹. 𝐹. PROPN iajs-3013 240 23 now now ADV iajs-3013 240 24 let let VERB iajs-3013 240 25 π‘₯ π‘₯ PRON iajs-3013 240 26 𝑠 𝑠 PRON iajs-3013 240 27 ∈ ∈ PROPN iajs-3013 240 28 𝐹 𝐹 PROPN iajs-3013 240 29 , , PUNCT iajs-3013 240 30 then then ADV iajs-3013 240 31 1 1 NUM iajs-3013 240 32 𝑠 𝑠 SYM iajs-3013 240 33 π‘₯ π‘₯ NOUN iajs-3013 240 34 1 1 NUM iajs-3013 240 35 ∈ ∈ NOUN iajs-3013 240 36 𝐹 𝐹 PROPN iajs-3013 240 37 and and CCONJ iajs-3013 240 38 hence hence ADV iajs-3013 240 39 π‘₯ π‘₯ PRON iajs-3013 240 40 1 1 NUM iajs-3013 240 41 ∈ ∈ PROPN iajs-3013 240 42 𝐹 𝐹 PROPN iajs-3013 240 43 , , PUNCT iajs-3013 240 44 implies imply VERB iajs-3013 240 45 that that SCONJ iajs-3013 240 46 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3013 240 47 ) ) PUNCT iajs-3013 240 48 ∈ ∈ PROPN iajs-3013 240 49 𝐹 𝐹 PROPN iajs-3013 240 50 , , PUNCT iajs-3013 240 51 therefore therefore ADV iajs-3013 240 52 π‘₯ π‘₯ X iajs-3013 240 53 ∈ ∈ NOUN iajs-3013 240 54 π‘“βˆ’1(𝐹 π‘“βˆ’1(𝐹 NUM iajs-3013 240 55 ) ) PUNCT iajs-3013 240 56 and and CCONJ iajs-3013 240 57 π‘₯ π‘₯ X iajs-3013 240 58 𝑠 𝑠 X iajs-3013 240 59 ∈ ∈ PROPN iajs-3013 240 60 ( ( PUNCT iajs-3013 240 61 π‘“βˆ’1(𝐹))𝑆. π‘“βˆ’1(𝐹))𝑆. NOUN iajs-3013 240 62 thus thus ADV iajs-3013 240 63 𝐹 𝐹 PROPN iajs-3013 240 64 βŠ† βŠ† NUM iajs-3013 240 65 ( ( PUNCT iajs-3013 240 66 π‘“βˆ’1(𝐹))𝑆. π‘“βˆ’1(𝐹))𝑆. NOUN iajs-3013 240 67 therefore therefore ADV iajs-3013 240 68 𝐹 𝐹 PROPN iajs-3013 240 69 = = PRON iajs-3013 240 70 ( ( PUNCT iajs-3013 240 71 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 NOUN iajs-3013 240 72 for for ADP iajs-3013 240 73 any any DET iajs-3013 240 74 submodule submodule NOUN iajs-3013 240 75 𝐹 𝐹 PROPN iajs-3013 240 76 of of ADP iajs-3013 240 77 𝑄𝑆. 𝑄𝑆. NOUN iajs-3013 240 78 now now ADV iajs-3013 240 79 since since SCONJ iajs-3013 240 80 ( ( PUNCT iajs-3013 240 81 π‘“βˆ’1(𝐹))𝑆 π‘“βˆ’1(𝐹))𝑆 NOUN iajs-3013 240 82 βŠ† βŠ† NUM iajs-3013 240 83 ( ( PUNCT iajs-3013 240 84 𝑓 𝑓 DET iajs-3013 240 85 βˆ’1(𝑃𝛽))𝑆 βˆ’1(𝑃𝛽))𝑆 PROPN iajs-3013 240 86 for for ADP iajs-3013 240 87 some some DET iajs-3013 240 88 𝛽 𝛽 NOUN iajs-3013 240 89 ∈ ∈ NOUN iajs-3013 240 90 ʌ ʌ NOUN iajs-3013 240 91 we we PRON iajs-3013 240 92 have have VERB iajs-3013 240 93 𝐹 𝐹 PROPN iajs-3013 240 94 βŠ† βŠ† NUM iajs-3013 240 95 𝑃𝛽 𝑃𝛽 PROPN iajs-3013 240 96 for for ADP iajs-3013 240 97 some some DET iajs-3013 240 98 𝛽 𝛽 NOUN iajs-3013 240 99 ∈ ∈ PROPN iajs-3013 240 100 ʌ ʌ X iajs-3013 240 101 . . PUNCT iajs-3013 241 1 thus thus ADV iajs-3013 241 2 𝐹 𝐹 PROPN iajs-3013 241 3 is be AUX iajs-3013 241 4 strongly strongly ADV iajs-3013 241 5 alappnq alappnq ADJ iajs-3013 241 6 compactly compactly ADV iajs-3013 241 7 packed pack VERB iajs-3013 241 8 submodule submodule NOUN iajs-3013 241 9 . . PUNCT iajs-3013 242 1 therefore therefore ADV iajs-3013 242 2 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 242 3 is be AUX iajs-3013 242 4 strongly strongly ADV iajs-3013 242 5 alappnq alappnq ADJ iajs-3013 242 6 compactly compactly ADV iajs-3013 242 7 packed pack VERB iajs-3013 242 8 module module NOUN iajs-3013 242 9 . . PUNCT iajs-3013 243 1 5 5 X iajs-3013 243 2 . . X iajs-3013 243 3 conclusion conclusion VERB iajs-3013 243 4 the the DET iajs-3013 243 5 main main ADJ iajs-3013 243 6 results result NOUN iajs-3013 243 7 of of ADP iajs-3013 243 8 this this DET iajs-3013 243 9 paper paper NOUN iajs-3013 243 10 are be AUX iajs-3013 243 11 : : PUNCT iajs-3013 243 12 β€’ β€’ ADV iajs-3013 243 13 let let VERB iajs-3013 243 14 𝑄 𝑄 PRON iajs-3013 243 15 be be AUX iajs-3013 243 16 alappnq alappnq NOUN iajs-3013 243 17 compactly compactly ADV iajs-3013 243 18 packed pack VERB iajs-3013 243 19 module module NOUN iajs-3013 243 20 with with ADP iajs-3013 243 21 𝐽(𝑄 𝐽(𝑄 NOUN iajs-3013 243 22 ) ) PUNCT iajs-3013 243 23 β‰  β‰  PROPN iajs-3013 243 24 𝑄 𝑄 PROPN iajs-3013 243 25 , , PUNCT iajs-3013 243 26 then then ADV iajs-3013 243 27 , , PUNCT iajs-3013 243 28 𝑄 𝑄 PRON iajs-3013 243 29 satisfies satisfy VERB iajs-3013 243 30 the the DET iajs-3013 243 31 ascending ascend VERB iajs-3013 243 32 chain chain NOUN iajs-3013 243 33 condition condition NOUN iajs-3013 243 34 for for ADP iajs-3013 243 35 alappnq alappnq ADJ iajs-3013 243 36 - - PUNCT iajs-3013 243 37 prime prime NOUN iajs-3013 243 38 submodules submodule NOUN iajs-3013 243 39 . . PUNCT iajs-3013 244 1 β€’ β€’ NOUN iajs-3013 244 2 if if SCONJ iajs-3013 244 3 𝑄 𝑄 PRON iajs-3013 244 4 is be AUX iajs-3013 244 5 an an DET iajs-3013 244 6 alappnq alappnq NOUN iajs-3013 244 7 compactly compactly ADV iajs-3013 244 8 packed pack VERB iajs-3013 244 9 module module NOUN iajs-3013 244 10 , , PUNCT iajs-3013 244 11 then then ADV iajs-3013 244 12 , , PUNCT iajs-3013 244 13 𝑄𝑆 𝑄𝑆 PROPN iajs-3013 244 14 is be AUX iajs-3013 244 15 an an DET iajs-3013 244 16 alappnq alappnq NOUN iajs-3013 244 17 compactly compactly ADV iajs-3013 244 18 packed pack VERB iajs-3013 244 19 module module NOUN iajs-3013 244 20 , , PUNCT iajs-3013 244 21 for for ADP iajs-3013 244 22 each each DET iajs-3013 244 23 multiplicatively multiplicatively ADV iajs-3013 244 24 closed close VERB iajs-3013 244 25 set set VERB iajs-3013 244 26 𝑆 𝑆 PROPN iajs-3013 244 27 of of ADP iajs-3013 244 28 𝑅. 𝑅. NOUN iajs-3013 244 29 β€’ β€’ NOUN iajs-3013 244 30 an an DET iajs-3013 244 31 𝑅-module 𝑅-module PROPN iajs-3013 244 32 𝑄 𝑄 PROPN iajs-3013 244 33 is be AUX iajs-3013 244 34 strongly strongly ADV iajs-3013 244 35 alappnq alappnq ADJ iajs-3013 244 36 compactly compactly ADV iajs-3013 244 37 packed pack VERB iajs-3013 244 38 if if SCONJ iajs-3013 245 1 and and CCONJ iajs-3013 245 2 only only ADV iajs-3013 245 3 if if SCONJ iajs-3013 245 4 every every DET iajs-3013 245 5 proper proper ADJ iajs-3013 245 6 submodule submodule NOUN iajs-3013 245 7 of of ADP iajs-3013 245 8 𝑄 𝑄 PROPN iajs-3013 245 9 is be AUX iajs-3013 245 10 alappnq alappnq ADJ iajs-3013 245 11 - - PUNCT iajs-3013 245 12 prime prime NOUN iajs-3013 245 13 radical radical NOUN iajs-3013 245 14 of of ADP iajs-3013 245 15 a a DET iajs-3013 245 16 cyclic cyclic ADJ iajs-3013 245 17 submodule submodule NOUN iajs-3013 245 18 of of ADP iajs-3013 245 19 it it PRON iajs-3013 245 20 . . PUNCT iajs-3013 246 1 β€’ β€’ INTJ iajs-3013 246 2 let let VERB iajs-3013 246 3 𝑄 𝑄 PRON iajs-3013 246 4 be be AUX iajs-3013 246 5 an an DET iajs-3013 246 6 𝑅-module 𝑅-module PROPN iajs-3013 246 7 . . PUNCT iajs-3013 247 1 then then ADV iajs-3013 247 2 the the DET iajs-3013 247 3 following follow VERB iajs-3013 247 4 statements statement NOUN iajs-3013 247 5 are be AUX iajs-3013 247 6 equivalent equivalent ADJ iajs-3013 247 7 : : PUNCT iajs-3013 247 8 1 1 X iajs-3013 247 9 . . X iajs-3013 248 1 𝑄 𝑄 PRON iajs-3013 248 2 is be AUX iajs-3013 248 3 strongly strongly ADV iajs-3013 248 4 alappnq alappnq ADJ iajs-3013 248 5 compactly compactly ADV iajs-3013 248 6 packed pack VERB iajs-3013 248 7 module module NOUN iajs-3013 248 8 . . PUNCT iajs-3013 249 1 2 2 X iajs-3013 249 2 . . X iajs-3013 249 3 for for ADP iajs-3013 249 4 each each DET iajs-3013 249 5 𝐹 𝐹 PROPN iajs-3013 249 6 βŠ‚ βŠ‚ PROPN iajs-3013 249 7 𝑄 𝑄 PROPN iajs-3013 249 8 , , PUNCT iajs-3013 249 9 there there PRON iajs-3013 249 10 exists exist VERB iajs-3013 249 11 π‘ž π‘ž PROPN iajs-3013 249 12 ∈ ∈ PROPN iajs-3013 249 13 𝐹 𝐹 PROPN iajs-3013 249 14 such such ADJ iajs-3013 249 15 that that PRON iajs-3013 249 16 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹 PROPN iajs-3013 249 17 ) ) PUNCT iajs-3013 250 1 = = PUNCT iajs-3013 250 2 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(π‘…π‘ž NOUN iajs-3013 250 3 ) ) PUNCT iajs-3013 250 4 . . PUNCT iajs-3013 251 1 3 3 X iajs-3013 251 2 . . X iajs-3013 251 3 for for ADP iajs-3013 251 4 each each DET iajs-3013 251 5 𝐹 𝐹 PROPN iajs-3013 251 6 βŠ‚ βŠ‚ PROPN iajs-3013 251 7 𝑄 𝑄 PROPN iajs-3013 251 8 , , PUNCT iajs-3013 251 9 if if SCONJ iajs-3013 251 10 { { PUNCT iajs-3013 251 11 𝐹𝛼}π›ΌβˆˆΚŒ 𝐹𝛼}π›ΌβˆˆΚŒ PUNCT iajs-3013 251 12 is be AUX iajs-3013 251 13 a a DET iajs-3013 251 14 family family NOUN iajs-3013 251 15 of of ADP iajs-3013 251 16 submodules submodule NOUN iajs-3013 251 17 of of ADP iajs-3013 251 18 𝑄 𝑄 PROPN iajs-3013 251 19 , , PUNCT iajs-3013 251 20 such such ADJ iajs-3013 251 21 that that SCONJ iajs-3013 251 22 𝐹 𝐹 PROPN iajs-3013 251 23 βŠ† βŠ† NUM iajs-3013 251 24 ⋃ ⋃ ADP iajs-3013 251 25 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 251 26 , , PUNCT iajs-3013 251 27 then then ADV iajs-3013 251 28 there there PRON iajs-3013 251 29 exists exist VERB iajs-3013 251 30 𝛽 𝛽 PROPN iajs-3013 251 31 ∈ ∈ PROPN iajs-3013 251 32 ʌ ʌ NOUN iajs-3013 251 33 such such ADJ iajs-3013 251 34 that that SCONJ iajs-3013 251 35 𝐹 𝐹 PROPN iajs-3013 251 36 βŠ† βŠ† NUM iajs-3013 251 37 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛽 π΄π‘™π‘Žπ‘π‘π‘›π‘žπ‘Ÿπ‘Žπ‘‘π‘„(𝐹𝛽 NOUN iajs-3013 251 38 ) ) PUNCT iajs-3013 251 39 . . PUNCT iajs-3013 252 1 4 4 X iajs-3013 252 2 . . X iajs-3013 252 3 for for ADP iajs-3013 252 4 each each DET iajs-3013 252 5 𝐹 𝐹 PROPN iajs-3013 252 6 βŠ‚ βŠ‚ PROPN iajs-3013 252 7 𝑄 𝑄 PROPN iajs-3013 252 8 , , PUNCT iajs-3013 252 9 if if SCONJ iajs-3013 252 10 { { PUNCT iajs-3013 252 11 𝐹𝛼}π›ΌβˆˆΚŒ 𝐹𝛼}π›ΌβˆˆΚŒ PUNCT iajs-3013 252 12 is be AUX iajs-3013 252 13 a a DET iajs-3013 252 14 family family NOUN iajs-3013 252 15 of of ADP iajs-3013 252 16 alappnq alappnq ADJ iajs-3013 252 17 - - PUNCT iajs-3013 252 18 prime prime ADJ iajs-3013 252 19 radical radical ADJ iajs-3013 252 20 submodule submodule NOUN iajs-3013 252 21 of of ADP iajs-3013 252 22 𝑄 𝑄 PROPN iajs-3013 252 23 , , PUNCT iajs-3013 252 24 with with ADP iajs-3013 252 25 𝐹 𝐹 PROPN iajs-3013 252 26 βŠ† βŠ† NUM iajs-3013 252 27 ⋃ ⋃ ADP iajs-3013 252 28 πΉπ›Όπ›ΌβˆˆΚŒ πΉπ›Όπ›ΌβˆˆΚŒ PROPN iajs-3013 252 29 , , PUNCT iajs-3013 252 30 then then ADV iajs-3013 252 31 there there PRON iajs-3013 252 32 exists exist VERB iajs-3013 252 33 𝛽 𝛽 PROPN iajs-3013 252 34 ∈ ∈ PROPN iajs-3013 252 35 ʌ ʌ NOUN iajs-3013 252 36 such such ADJ iajs-3013 252 37 that that SCONJ iajs-3013 252 38 𝐹 𝐹 PROPN iajs-3013 252 39 βŠ† βŠ† NUM iajs-3013 252 40 𝐹𝛽. 𝐹𝛽. NUM iajs-3013 252 41 references reference NOUN iajs-3013 252 42 1 1 NUM iajs-3013 252 43 . . PUNCT iajs-3013 252 44 ali ali PROPN iajs-3013 252 45 sh sh PROPN iajs-3013 252 46 . . PROPN iajs-3013 252 47 a. a. PROPN iajs-3013 252 48 ; ; PUNCT iajs-3013 252 49 haibat haibat PROPN iajs-3013 252 50 k. k. PROPN iajs-3013 252 51 m. m. PROPN iajs-3013 252 52 almost almost ADV iajs-3013 252 53 approximaitly approximaitly ADV iajs-3013 253 1 nearly nearly ADV iajs-3013 253 2 quasiprime quasiprime ADJ iajs-3013 253 3 submodules submodule NOUN iajs-3013 253 4 , , PUNCT iajs-3013 253 5 j. j. PROPN iajs-3013 253 6 of of ADP iajs-3013 253 7 alqadisiyah alqadisiyah NOUN iajs-3013 253 8 for for ADP iajs-3013 253 9 computer computer NOUN iajs-3013 253 10 science science NOUN iajs-3013 253 11 and and CCONJ iajs-3013 253 12 mathematics mathematic NOUN iajs-3013 253 13 , , PUNCT iajs-3013 253 14 14(3 14(3 NUM iajs-3013 253 15 ) ) PUNCT iajs-3013 253 16 , , PUNCT iajs-3013 253 17 will will AUX iajs-3013 253 18 be be AUX iajs-3013 253 19 published publish VERB iajs-3013 253 20 in in ADP iajs-3013 253 21 2022 2022 NUM iajs-3013 253 22 . . PUNCT iajs-3013 254 1 2 2 X iajs-3013 254 2 . . X iajs-3013 254 3 abdul abdul PROPN iajs-3013 254 4 - - PUNCT iajs-3013 254 5 razak razak PROPN iajs-3013 254 6 h. h. PROPN iajs-3013 254 7 m. m. PROPN iajs-3013 254 8 quasi quasi ADJ iajs-3013 254 9 - - ADJ iajs-3013 254 10 prime prime ADJ iajs-3013 254 11 modules module NOUN iajs-3013 254 12 and and CCONJ iajs-3013 254 13 quasi quasi ADJ iajs-3013 254 14 - - ADJ iajs-3013 254 15 prime prime ADJ iajs-3013 254 16 submodules submodule NOUN iajs-3013 254 17 , , PUNCT iajs-3013 254 18 m.sc m.sc PROPN iajs-3013 254 19 . . PUNCT iajs-3013 255 1 thesis thesis PROPN iajs-3013 255 2 , , PUNCT iajs-3013 255 3 univ univ PROPN iajs-3013 255 4 . . PROPN iajs-3013 255 5 of of ADP iajs-3013 255 6 baghdad baghdad PROPN iajs-3013 255 7 . . PUNCT iajs-3013 256 1 1999 1999 NUM iajs-3013 256 2 . . PUNCT iajs-3013 257 1 3 3 X iajs-3013 257 2 . . X iajs-3013 257 3 nuhad nuhad PROPN iajs-3013 257 4 , , PUNCT iajs-3013 257 5 s. s. PROPN iajs-3013 257 6 a. a. PROPN iajs-3013 257 7 ; ; PUNCT iajs-3013 257 8 adwia adwia VERB iajs-3013 257 9 , , PUNCT iajs-3013 257 10 j. j. PROPN iajs-3013 257 11 a. a. PROPN iajs-3013 257 12 nearly nearly ADV iajs-3013 257 13 quasi quasi VERB iajs-3013 257 14 prime prime ADJ iajs-3013 257 15 submodules submodule NOUN iajs-3013 257 16 , , PUNCT iajs-3013 257 17 international international ADJ iajs-3013 257 18 journal journal NOUN iajs-3013 257 19 of of ADP iajs-3013 257 20 advanced advanced ADJ iajs-3013 257 21 research research NOUN iajs-3013 257 22 , , PUNCT iajs-3013 257 23 2017 2017 NUM iajs-3013 257 24 , , PUNCT iajs-3013 257 25 5(1 5(1 NUM iajs-3013 257 26 ) ) PUNCT iajs-3013 257 27 , , PUNCT iajs-3013 257 28 170 170 NUM iajs-3013 257 29 - - SYM iajs-3013 257 30 180 180 NUM iajs-3013 257 31 . . NOUN iajs-3013 258 1 4 4 NUM iajs-3013 258 2 . . X iajs-3013 258 3 ali ali PROPN iajs-3013 258 4 sh sh PROPN iajs-3013 258 5 . . PROPN iajs-3013 258 6 a. a. PROPN iajs-3013 258 7 ; ; PUNCT iajs-3013 258 8 omar omar PROPN iajs-3013 258 9 a. a. PROPN iajs-3013 258 10 a. a. PROPN iajs-3013 258 11 ; ; PUNCT iajs-3013 259 1 haibat haibat PROPN iajs-3013 259 2 k. k. PROPN iajs-3013 259 3 m. m. PROPN iajs-3013 259 4 approximaitly approximaitly ADV iajs-3013 259 5 quasi quasi VERB iajs-3013 259 6 primary primary ADJ iajs-3013 259 7 submodules submodule NOUN iajs-3013 259 8 , , PUNCT iajs-3013 259 9 ibn ibn PROPN iajs-3013 259 10 alhaitham alhaitham NOUN iajs-3013 259 11 journal journal NOUN iajs-3013 259 12 for for ADP iajs-3013 259 13 pure pure ADJ iajs-3013 259 14 and and CCONJ iajs-3013 259 15 applied apply VERB iajs-3013 259 16 sci sci PROPN iajs-3013 259 17 . . PROPN iajs-3013 259 18 2020 2020 NUM iajs-3013 259 19 , , PUNCT iajs-3013 259 20 33(4 33(4 NUM iajs-3013 259 21 ) ) PUNCT iajs-3013 259 22 , , PUNCT iajs-3013 259 23 92 92 NUM iajs-3013 259 24 - - SYM iajs-3013 259 25 101 101 NUM iajs-3013 259 26 . . NOUN iajs-3013 260 1 5 5 NUM iajs-3013 260 2 . . NOUN iajs-3013 260 3 athab athab PROPN iajs-3013 260 4 , , PUNCT iajs-3013 260 5 e. e. 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