id sid tid token lemma pos iajs-779 1 1 2011 2011 NUM iajs-779 1 2 ) ) PUNCT iajs-779 1 3 2 2 NUM iajs-779 1 4 ( ( PUNCT iajs-779 1 5 24مجلة 24مجلة NUM iajs-779 1 6 ابن ابن VERB iajs-779 1 7 الهیثم الهیثم ADJ iajs-779 1 8 للعلوم للعلوم PROPN iajs-779 1 9 الصرفة الصرفة PROPN iajs-779 1 10 والتطبیقیة والتطبیقیة PROPN iajs-779 1 11 المجلد المجلد PROPN iajs-779 1 12 المقاسات المقاسات PROPN iajs-779 1 13 الجزئیة الجزئیة PROPN iajs-779 1 14 شبه شبه VERB iajs-779 1 15 االولیة االولیة PROPN iajs-779 1 16 التامة التامة PROPN iajs-779 1 17 والمقاسات والمقاسات PROPN iajs-779 1 18 شبه شبه VERB iajs-779 1 19 االولیة االولیة PROPN iajs-779 1 20 التامة التامة PROPN iajs-779 1 21 بثینة بثینة PROPN iajs-779 1 22 نجاد نجاد PROPN iajs-779 1 23 شهاب شهاب PROPN iajs-779 1 24 هادي هادي NOUN iajs-779 1 25 و و PRON iajs-779 1 26 انعام انعام ADJ iajs-779 1 27 محمد محمد ADJ iajs-779 1 28 علي علي NOUN iajs-779 1 29 جامعة جامعة NOUN iajs-779 1 30 بغداد،ابن بغداد،ابن PROPN iajs-779 1 31 الهیثم الهیثم VERB iajs-779 1 32 -كلیة -كلیة PROPN iajs-779 1 33 التربیة،قسم التربیة،قسم PROPN iajs-779 1 34 الریاضیات الریاضیات NOUN iajs-779 1 35 2010 2010 NUM iajs-779 1 36 ، ، NOUN iajs-779 1 37 اب اب X iajs-779 1 38 ، ، NOUN iajs-779 1 39 23 23 NUM iajs-779 1 40 : : PUNCT iajs-779 1 41 استلم استلم PROPN iajs-779 1 42 البحث البحث VERB iajs-779 1 43 في في ADP iajs-779 1 44 2010،تشرین 2010،تشرین NOUN iajs-779 1 45 الثاني الثاني X iajs-779 1 46 ، ، X iajs-779 2 1 9 9 NUM iajs-779 2 2 : : PUNCT iajs-779 2 3 قبل قبل NOUN iajs-779 2 4 البحث البحث NOUN iajs-779 2 5 في في ADP iajs-779 2 6 الخالصة الخالصة NOUN iajs-779 2 7 في في ADP iajs-779 2 8 هذا هذا NOUN iajs-779 2 9 البحـث البحـث NOUN iajs-779 2 10 درسـنا درسـنا ADJ iajs-779 2 11 مفهـومي مفهـومي ADJ iajs-779 2 12 المقاسـات المقاسـات ADJ iajs-779 2 13 الجزئیـة الجزئیـة PROPN iajs-779 2 14 . . PUNCT iajs-779 3 1 مقاساً مقاساً PROPN iajs-779 3 2 احادیا احادیا VERB iajs-779 3 3 ً ً PROPN iajs-779 3 4 mحلقة mحلقة NOUN iajs-779 3 5 ابدالیة ابدالیة NOUN iajs-779 4 1 ذا ذا PROPN iajs-779 4 2 عنصر عنصر PROPN iajs-779 4 3 محاید محاید PROPN iajs-779 4 4 ولیكن ولیكن ADJ iajs-779 4 5 rلتكن rلتكن NOUN iajs-779 4 6 ه ه PROPN iajs-779 4 7 مقـاس مقـاس PROPN iajs-779 4 8 جزئـي جزئـي ADJ iajs-779 4 9 شـبه شـبه PROPN iajs-779 4 10 wالتـام wالتـام PROPN iajs-779 4 11 شبه شبه VERB iajs-779 4 12 االولیة االولیة PROPN iajs-779 4 13 التامة التامة PROPN iajs-779 4 14 والمقاسات والمقاسات PROPN iajs-779 4 15 شبه شبه VERB iajs-779 4 16 االولیة االولیة PROPN iajs-779 4 17 التامة التامة PROPN iajs-779 4 18 إذ إذ PROPN iajs-779 4 19 یقال یقال PROPN iajs-779 4 20 عن عن PROPN iajs-779 4 21 المقاس المقاس PROPN iajs-779 4 22 الجزئـي الجزئـي NOUN iajs-779 4 23 الفعلـي الفعلـي PROPN iajs-779 4 24 المتغیـر المتغیـر NOUN iajs-779 4 25 انـ انـ X iajs-779 4 26 مقاسـاً مقاسـاً PROPN iajs-779 4 27 شـبه شـبه PROPN iajs-779 4 28 اولـي اولـي VERB iajs-779 4 29 تـام تـام ADJ iajs-779 4 30 اذا اذا PROPN iajs-779 4 31 mویسـمى mویسـمى PROPN iajs-779 4 32 xwمقاس xwمقاس PROPN iajs-779 4 33 جزئي جزئي NOUN iajs-779 4 34 متغیر متغیر VERB iajs-779 4 35 تام تام NOUN iajs-779 4 36 یـؤدي یـؤدي NOUN iajs-779 4 37 الـى الـى VERB iajs-779 4 38 ان ان ADP iajs-779 4 39 xلكل xلكل PROPN iajs-779 4 40 xxwاولي xxwاولي PROPN iajs-779 4 41 تام تام SCONJ iajs-779 4 42 اذا اذا PROPN iajs-779 4 43 كان كان PROPN iajs-779 4 44 ـاس ـاس PROPN iajs-779 4 45 شـبه شـبه PROPN iajs-779 4 46 اولــي اولــي PROPN iajs-779 4 47 تـام تـام PROPN iajs-779 4 48 ( ( PUNCT iajs-779 4 49 0)كـان 0)كـان NUM iajs-779 4 50 المقــاس المقــاس PROPN iajs-779 4 51 الجزئـي الجزئـي VERB iajs-779 4 52 اعطینــا اعطینــا ADJ iajs-779 4 53 الخــواص الخــواص ADV iajs-779 4 54 االساسـیة االساسـیة PROPN iajs-779 4 55 لهــذین لهــذین ADJ iajs-779 4 56 المفهـومین المفهـومین NOUN iajs-779 4 57 وكــذلك وكــذلك NOUN iajs-779 4 58 درسـنا درسـنا VERB iajs-779 4 59 العالقــات العالقــات PROPN iajs-779 4 60 بــین بــین PROPN iajs-779 4 61 . . PUNCT iajs-779 5 1 مقـ مقـ PROPN iajs-779 5 2 ذات ذات NOUN iajs-779 5 3 ) ) PUNCT iajs-779 5 4 المقاسـات(مع المقاسـات(مع NOUN iajs-779 5 5 انواع انواع ADJ iajs-779 5 6 اخـرى اخـرى PROPN iajs-779 5 7 مـن مـن PROPN iajs-779 5 8 المقاسـات المقاسـات PROPN iajs-779 5 9 الجزئیـة الجزئیـة PROPN iajs-779 5 10 ) ) PUNCT iajs-779 5 11 المقاسات المقاسات PROPN iajs-779 5 12 شبه شبه VERB iajs-779 5 13 االولیة االولیة ADJ iajs-779 5 14 التامة(المقاسات التامة(المقاسات PROPN iajs-779 5 15 الجزئیة الجزئیة NOUN iajs-779 5 16 شبه شبه VERB iajs-779 5 17 االولیة االولیة ADJ iajs-779 5 18 التامة التامة NOUN iajs-779 6 1 .العالقة .العالقة PROPN iajs-779 6 2 معهما معهما PROPN iajs-779 6 3 ـات ـات PROPN iajs-779 6 4 الجزئیــة الجزئیــة PROPN iajs-779 6 5 شــبه شــبه PROPN iajs-779 6 6 اال اال NOUN iajs-779 6 7 : : PUNCT iajs-779 6 8 الكلمــات الكلمــات PROPN iajs-779 6 9 المفتاحـیـة المفتاحـیـة PROPN iajs-779 6 10 ـةالمقاسـ ـةالمقاسـ ADV iajs-779 6 11 ـة ـة PROPN iajs-779 6 12 مــن مــن PROPN iajs-779 6 13 الـــنمط الـــنمط NOUN iajs-779 6 14 -ولیــة -ولیــة PROPN iajs-779 6 15 التامـ التامـ VERB iajs-779 6 16 ـات ـات PROPN iajs-779 6 17 الجزئیـ الجزئیـ PROPN iajs-779 6 18 المقاســات المقاســات PROPN iajs-779 6 19 شــبه شــبه PROPN iajs-779 6 20 االولیــة االولیــة VERB iajs-779 6 21 التامــة التامــة ADJ iajs-779 6 22 المقاسـ المقاسـ ADJ iajs-779 6 23 invarian invarian ADJ iajs-779 6 24 المقاسات المقاسات PROPN iajs-779 6 25 االولیة االولیة PROPN iajs-779 6 26 التامة التامة PROPN iajs-779 6 27 -التامة -التامة PROPN iajs-779 6 28 ibn ibn PROPN iajs-779 6 29 alhaitham alhaitham PROPN iajs-779 6 30 j. j. PROPN iajs-779 6 31 for for ADP iajs-779 6 32 pure pure ADJ iajs-779 6 33 & & CCONJ iajs-779 6 34 appl appl PROPN iajs-779 6 35 . . PUNCT iajs-779 7 1 sci sci PROPN iajs-779 7 2 . . PUNCT iajs-779 7 3 vol.24 vol.24 NOUN iajs-779 7 4 ( ( PUNCT iajs-779 7 5 2 2 NUM iajs-779 7 6 ) ) PUNCT iajs-779 7 7 2011 2011 NUM iajs-779 7 8 on on ADP iajs-779 7 9 fully fully ADV iajs-779 7 10 semiprime semiprime NOUN iajs-779 7 11 submodules submodule NOUN iajs-779 7 12 and and CCONJ iajs-779 7 13 fully fully ADV iajs-779 7 14 semiprime semiprime NOUN iajs-779 7 15 modules module NOUN iajs-779 7 16 i.m.a.hadi i.m.a.hadi PROPN iajs-779 7 17 and and CCONJ iajs-779 7 18 b.n b.n PROPN iajs-779 7 19 . . PROPN iajs-779 7 20 shihab shihab PROPN iajs-779 7 21 department department PROPN iajs-779 7 22 of of ADP iajs-779 7 23 mathematics mathematics PROPN iajs-779 7 24 , , PUNCT iajs-779 7 25 ibn ibn PROPN iajs-779 7 26 - - PUNCT iajs-779 7 27 al al PROPN iajs-779 7 28 - - PUNCT iajs-779 7 29 haitham haitham PROPN iajs-779 7 30 , , PUNCT iajs-779 7 31 college college NOUN iajs-779 7 32 of of ADP iajs-779 7 33 education education NOUN iajs-779 7 34 , , PUNCT iajs-779 7 35 university university PROPN iajs-779 7 36 of of ADP iajs-779 7 37 baghdad baghdad PROPN iajs-779 7 38 received receive VERB iajs-779 7 39 in in ADP iajs-779 7 40 : : PUNCT iajs-779 7 41 23 23 NUM iajs-779 7 42 , , PUNCT iajs-779 7 43 august august PROPN iajs-779 7 44 , , PUNCT iajs-779 7 45 2010 2010 NUM iajs-779 7 46 accepted accept VERB iajs-779 7 47 in in ADP iajs-779 7 48 : : PUNCT iajs-779 7 49 9 9 NUM iajs-779 7 50 , , PUNCT iajs-779 7 51 november november PROPN iajs-779 7 52 , , PUNCT iajs-779 7 53 2010 2010 NUM iajs-779 7 54 abstract abstract ADV iajs-779 7 55 let let VERB iajs-779 7 56 r r PRON iajs-779 7 57 be be AUX iajs-779 7 58 a a DET iajs-779 7 59 commutative commutative ADJ iajs-779 7 60 ring ring NOUN iajs-779 7 61 with with ADP iajs-779 7 62 unity unity NOUN iajs-779 7 63 and and CCONJ iajs-779 7 64 let let VERB iajs-779 7 65 m m PRON iajs-779 7 66 be be AUX iajs-779 7 67 a a DET iajs-779 7 68 unitary unitary ADJ iajs-779 7 69 r r NOUN iajs-779 7 70 - - PUNCT iajs-779 7 71 module module NOUN iajs-779 7 72 . . PUNCT iajs-779 8 1 in in ADP iajs-779 8 2 this this DET iajs-779 8 3 paper paper NOUN iajs-779 8 4 we we PRON iajs-779 8 5 study study VERB iajs-779 8 6 fully fully ADV iajs-779 8 7 semiprime semiprime NOUN iajs-779 8 8 submodules submodule NOUN iajs-779 8 9 and and CCONJ iajs-779 8 10 fully fully ADV iajs-779 8 11 semiprime semiprime NOUN iajs-779 8 12 modules module NOUN iajs-779 8 13 , , PUNCT iajs-779 8 14 where where SCONJ iajs-779 8 15 a a DET iajs-779 8 16 proper proper ADJ iajs-779 8 17 fully fully ADV iajs-779 8 18 invariant invariant ADJ iajs-779 8 19 r r NOUN iajs-779 8 20 - - PUNCT iajs-779 8 21 submodule submodule NOUN iajs-779 8 22 w w NOUN iajs-779 8 23 of of ADP iajs-779 8 24 m m PROPN iajs-779 8 25 is be AUX iajs-779 8 26 called call VERB iajs-779 8 27 fully fully ADV iajs-779 8 28 semiprime semiprime NOUN iajs-779 8 29 in in ADP iajs-779 8 30 m m NOUN iajs-779 8 31 if if SCONJ iajs-779 8 32 whenever whenever SCONJ iajs-779 8 33 xxw xxw PROPN iajs-779 8 34 for for ADP iajs-779 8 35 all all DET iajs-779 8 36 fully fully ADV iajs-779 8 37 invariant invariant ADJ iajs-779 8 38 r r NOUN iajs-779 8 39 - - PUNCT iajs-779 8 40 submodule submodule NOUN iajs-779 8 41 x x PUNCT iajs-779 8 42 of of ADP iajs-779 8 43 m m PROPN iajs-779 8 44 , , PUNCT iajs-779 8 45 implies imply VERB iajs-779 8 46 xw xw PROPN iajs-779 8 47 . . PUNCT iajs-779 9 1 m m PROPN iajs-779 9 2 is be AUX iajs-779 9 3 called call VERB iajs-779 9 4 fully fully ADV iajs-779 9 5 semiprime semiprime NOUN iajs-779 9 6 if if SCONJ iajs-779 9 7 ( ( PUNCT iajs-779 9 8 0 0 X iajs-779 9 9 ) ) PUNCT iajs-779 9 10 is be AUX iajs-779 9 11 a a DET iajs-779 9 12 fully fully ADV iajs-779 9 13 semiprime semiprime NOUN iajs-779 9 14 submodule submodule NOUN iajs-779 9 15 of of ADP iajs-779 9 16 m. m. NOUN iajs-779 9 17 we we PRON iajs-779 9 18 give give VERB iajs-779 9 19 basic basic ADJ iajs-779 9 20 properties property NOUN iajs-779 9 21 of of ADP iajs-779 9 22 these these DET iajs-779 9 23 concepts concept NOUN iajs-779 9 24 . . PUNCT iajs-779 10 1 also also ADV iajs-779 10 2 we we PRON iajs-779 10 3 study study VERB iajs-779 10 4 the the DET iajs-779 10 5 relationships relationship NOUN iajs-779 10 6 between between ADP iajs-779 10 7 fully fully ADV iajs-779 10 8 semiprime semiprime NOUN iajs-779 10 9 submodules submodule NOUN iajs-779 10 10 ( ( PUNCT iajs-779 10 11 modules module NOUN iajs-779 10 12 ) ) PUNCT iajs-779 10 13 and and CCONJ iajs-779 10 14 other other ADJ iajs-779 10 15 related related ADJ iajs-779 10 16 submodules submodule NOUN iajs-779 10 17 ( ( PUNCT iajs-779 10 18 modules module NOUN iajs-779 10 19 ) ) PUNCT iajs-779 10 20 respectively respectively ADV iajs-779 10 21 . . PUNCT iajs-779 11 1 key key ADJ iajs-779 11 2 words word NOUN iajs-779 11 3 : : PUNCT iajs-779 11 4 fully fully ADV iajs-779 11 5 semiprime semiprime NOUN iajs-779 11 6 submodule submodule NOUN iajs-779 11 7 , , PUNCT iajs-779 11 8 fully fully ADV iajs-779 11 9 semiprime semiprime NOUN iajs-779 11 10 modules module NOUN iajs-779 11 11 , , PUNCT iajs-779 11 12 fully fully ADV iajs-779 11 13 invariant invariant ADJ iajs-779 11 14 submodule submodule NOUN iajs-779 11 15 , , PUNCT iajs-779 11 16 fully fully ADV iajs-779 11 17 prime prime ADJ iajs-779 11 18 modules module NOUN iajs-779 11 19 . . PUNCT iajs-779 12 1 introduction introduction NOUN iajs-779 12 2 j.abuihlail j.abuihlail NOUN iajs-779 12 3 in in ADP iajs-779 12 4 [ [ X iajs-779 12 5 1 1 NUM iajs-779 12 6 ] ] PUNCT iajs-779 12 7 , , PUNCT iajs-779 12 8 suggested suggest VERB iajs-779 12 9 the the DET iajs-779 12 10 definition definition NOUN iajs-779 12 11 of of ADP iajs-779 12 12 fully fully ADV iajs-779 12 13 semiprime semiprime NOUN iajs-779 12 14 submodule submodule NOUN iajs-779 12 15 and and CCONJ iajs-779 12 16 fully fully ADV iajs-779 12 17 semiprime semiprime NOUN iajs-779 12 18 module module NOUN iajs-779 12 19 as as ADP iajs-779 12 20 projects project NOUN iajs-779 12 21 , , PUNCT iajs-779 12 22 where where SCONJ iajs-779 12 23 a a DET iajs-779 12 24 proper proper ADJ iajs-779 12 25 fully fully ADV iajs-779 12 26 invarianr invarianr NOUN iajs-779 12 27 r r NOUN iajs-779 12 28 - - PUNCT iajs-779 12 29 module module NOUN iajs-779 12 30 w w PROPN iajs-779 12 31   NOUN iajs-779 12 32 m m ADP iajs-779 12 33 is be AUX iajs-779 12 34 fully fully ADV iajs-779 12 35 semiprime semiprime NOUN iajs-779 12 36 in in ADP iajs-779 12 37 m m PROPN iajs-779 12 38 , , PUNCT iajs-779 12 39 if if SCONJ iajs-779 12 40 whenever whenever SCONJ iajs-779 12 41 xxw xxw PROPN iajs-779 12 42 for for ADP iajs-779 12 43 all all DET iajs-779 12 44 fully fully ADV iajs-779 12 45 invariant invariant ADJ iajs-779 12 46 r r NOUN iajs-779 12 47 - - PUNCT iajs-779 12 48 submodules submodules NOUN iajs-779 12 49 xm xm NOUN iajs-779 12 50 , , PUNCT iajs-779 12 51 it it PRON iajs-779 12 52 follows follow VERB iajs-779 12 53 that that SCONJ iajs-779 12 54 xw xw PROPN iajs-779 12 55 . . PUNCT iajs-779 13 1 an an DET iajs-779 13 2 r r NOUN iajs-779 13 3 - - PUNCT iajs-779 13 4 module module NOUN iajs-779 13 5 m m NOUN iajs-779 13 6 is be AUX iajs-779 13 7 called call VERB iajs-779 13 8 fully fully ADV iajs-779 13 9 semiprime semiprime NOUN iajs-779 13 10 if if SCONJ iajs-779 13 11 whenever whenever SCONJ iajs-779 13 12 xx=0 xx=0 PROPN iajs-779 13 13 for for ADP iajs-779 13 14 all all DET iajs-779 13 15 fully fully ADV iajs-779 13 16 invariant invariant ADJ iajs-779 13 17 rsubmodule rsubmodule NOUN iajs-779 13 18 x x PUNCT iajs-779 13 19 of of ADP iajs-779 13 20 m m PRON iajs-779 13 21 , , PUNCT iajs-779 13 22 it it PRON iajs-779 13 23 follows follow VERB iajs-779 13 24 that that SCONJ iajs-779 13 25 x=0 x=0 PROPN iajs-779 13 26 ; ; PUNCT iajs-779 13 27 that that PRON iajs-779 13 28 is be AUX iajs-779 13 29 m m VERB iajs-779 13 30 is be AUX iajs-779 13 31 a a DET iajs-779 13 32 fully fully ADV iajs-779 13 33 semiprime semiprime NOUN iajs-779 13 34 module module NOUN iajs-779 13 35 if if SCONJ iajs-779 13 36 0 0 NUM iajs-779 13 37   NOUN iajs-779 13 38 m m ADP iajs-779 13 39 is be AUX iajs-779 13 40 fully fully ADV iajs-779 13 41 semiprime semiprime NOUN iajs-779 13 42 . . PUNCT iajs-779 14 1 also also ADV iajs-779 14 2 for for ADP iajs-779 14 3 r r NOUN iajs-779 14 4 - - PUNCT iajs-779 14 5 submodules submodules NOUN iajs-779 14 6 x x NOUN iajs-779 14 7 , , PUNCT iajs-779 14 8 y y PROPN iajs-779 14 9   PROPN iajs-779 14 10 m m PROPN iajs-779 14 11 , , PUNCT iajs-779 14 12 the the DET iajs-779 14 13 internal internal ADJ iajs-779 14 14 product product NOUN iajs-779 14 15 xy xy PROPN iajs-779 14 16 is be AUX iajs-779 14 17 defined define VERB iajs-779 14 18 by by ADP iajs-779 14 19 {f(x):fhom(m {f(x):fhom(m PROPN iajs-779 14 20 , , PUNCT iajs-779 14 21 y y NOUN iajs-779 14 22 ) ) PUNCT iajs-779 14 23 } } PUNCT iajs-779 14 24 . . PUNCT iajs-779 15 1 notice notice VERB iajs-779 15 2 that that SCONJ iajs-779 15 3 , , PUNCT iajs-779 15 4 if if SCONJ iajs-779 15 5 ym ym PROPN iajs-779 15 6 is be AUX iajs-779 15 7 fully fully ADV iajs-779 15 8 invariant invariant ADJ iajs-779 15 9 , , PUNCT iajs-779 15 10 then then ADV iajs-779 15 11 xym xym ADV iajs-779 15 12 is be AUX iajs-779 15 13 also also ADV iajs-779 15 14 fully fully ADV iajs-779 15 15 invariant invariant ADJ iajs-779 15 16 , , PUNCT iajs-779 15 17 and and CCONJ iajs-779 15 18 if if SCONJ iajs-779 15 19 xm xm PROPN iajs-779 15 20 is be AUX iajs-779 15 21 fully fully ADV iajs-779 15 22 invariant invariant ADJ iajs-779 15 23 , , PUNCT iajs-779 15 24 then then ADV iajs-779 15 25 xy xy PROPN iajs-779 15 26 xy xy PROPN iajs-779 15 27 . . PUNCT iajs-779 16 1 the the DET iajs-779 16 2 internal internal ADJ iajs-779 16 3 product product NOUN iajs-779 16 4 of of ADP iajs-779 16 5 submodules submodule NOUN iajs-779 16 6 of of ADP iajs-779 16 7 a a DET iajs-779 16 8 given give VERB iajs-779 16 9 module module NOUN iajs-779 16 10 over over ADP iajs-779 16 11 an an DET iajs-779 16 12 associative associative NOUN iajs-779 16 13 not not PART iajs-779 16 14 necessarily necessarily ADV iajs-779 16 15 commutative commutative ADJ iajs-779 16 16 ring ring NOUN iajs-779 16 17 was be AUX iajs-779 16 18 first first ADV iajs-779 16 19 introduced introduce VERB iajs-779 16 20 by by ADP iajs-779 16 21 bican bican PROPN iajs-779 16 22 et.al et.al PROPN iajs-779 16 23 , , PUNCT iajs-779 16 24 [ [ X iajs-779 16 25 2 2 NUM iajs-779 16 26 ] ] PUNCT iajs-779 16 27 to to PART iajs-779 16 28 present present VERB iajs-779 16 29 the the DET iajs-779 16 30 notion notion NOUN iajs-779 16 31 of of ADP iajs-779 16 32 prime prime ADJ iajs-779 16 33 modules module NOUN iajs-779 16 34 . . PUNCT iajs-779 17 1 the the DET iajs-779 17 2 definition definition NOUN iajs-779 17 3 is be AUX iajs-779 17 4 modified modify VERB iajs-779 17 5 in in ADP iajs-779 17 6 [ [ X iajs-779 17 7 3 3 NUM iajs-779 17 8 ] ] PUNCT iajs-779 17 9 , , PUNCT iajs-779 17 10 where where SCONJ iajs-779 17 11 arbitrary arbitrary ADJ iajs-779 17 12 submodules submodule NOUN iajs-779 17 13 are be AUX iajs-779 17 14 replaced replace VERB iajs-779 17 15 by by ADP iajs-779 17 16 fully fully ADV iajs-779 17 17 invariant invariant ADJ iajs-779 17 18 ones one NOUN iajs-779 17 19 . . PUNCT iajs-779 18 1 to to PART iajs-779 18 2 avoid avoid VERB iajs-779 18 3 any any DET iajs-779 18 4 possible possible ADJ iajs-779 18 5 confusion confusion NOUN iajs-779 18 6 , , PUNCT iajs-779 18 7 such such ADJ iajs-779 18 8 modules module NOUN iajs-779 18 9 are be AUX iajs-779 18 10 referred refer VERB iajs-779 18 11 to to ADP iajs-779 18 12 as as ADP iajs-779 18 13 fully fully ADV iajs-779 18 14 prime prime ADJ iajs-779 18 15 modules module NOUN iajs-779 18 16 , , PUNCT iajs-779 18 17 where where SCONJ iajs-779 18 18 a a DET iajs-779 18 19 proper proper ADJ iajs-779 18 20 fully fully ADV iajs-779 18 21 invariant invariant ADJ iajs-779 18 22 submodule submodule NOUN iajs-779 18 23 w w PROPN iajs-779 18 24   PROPN iajs-779 18 25 m m SCONJ iajs-779 18 26 is be AUX iajs-779 18 27 fully fully ADV iajs-779 18 28 prime prime ADJ iajs-779 18 29 , , PUNCT iajs-779 18 30 if if SCONJ iajs-779 18 31 whenever whenever SCONJ iajs-779 18 32 xyw xyw PROPN iajs-779 18 33 , , PUNCT iajs-779 18 34 for for ADP iajs-779 18 35 all all DET iajs-779 18 36 fully fully ADV iajs-779 18 37 invariant invariant ADJ iajs-779 18 38 r r NOUN iajs-779 18 39 - - PUNCT iajs-779 18 40 submodule submodule NOUN iajs-779 18 41 xm xm PROPN iajs-779 18 42 , , PUNCT iajs-779 18 43 ym ym PROPN iajs-779 18 44 , , PUNCT iajs-779 18 45 it it PRON iajs-779 18 46 follows follow VERB iajs-779 18 47 that that SCONJ iajs-779 18 48 xw xw PROPN iajs-779 18 49 or or CCONJ iajs-779 18 50 yw yw PROPN iajs-779 18 51 . . PUNCT iajs-779 19 1 an an DET iajs-779 19 2 r r NOUN iajs-779 19 3 - - PUNCT iajs-779 19 4 module module NOUN iajs-779 19 5 is be AUX iajs-779 19 6 called call VERB iajs-779 19 7 fully fully ADV iajs-779 19 8 prime prime ADJ iajs-779 19 9 if if SCONJ iajs-779 19 10 ( ( PUNCT iajs-779 19 11 0 0 NUM iajs-779 19 12 ) ) PUNCT iajs-779 19 13   NOUN iajs-779 20 1   PROPN iajs-779 20 2 m m NOUN iajs-779 20 3 is be AUX iajs-779 20 4 a a DET iajs-779 20 5 fully fully ADV iajs-779 20 6 prime prime ADJ iajs-779 20 7 submodule submodule NOUN iajs-779 20 8 ; ; PUNCT iajs-779 20 9 that that PRON iajs-779 20 10 is be AUX iajs-779 20 11 whenever whenever SCONJ iajs-779 20 12 xy=(0 xy=(0 PROPN iajs-779 20 13 ) ) PUNCT iajs-779 20 14 for for ADP iajs-779 20 15 all all DET iajs-779 20 16 fully fully ADV iajs-779 20 17 invariant invariant ADJ iajs-779 20 18 r r NOUN iajs-779 20 19 - - PUNCT iajs-779 20 20 submodules submodules NOUN iajs-779 20 21 xm xm PROPN iajs-779 20 22 , , PUNCT iajs-779 20 23 ym ym PROPN iajs-779 20 24 , , PUNCT iajs-779 20 25 it it PRON iajs-779 20 26 follows follow VERB iajs-779 20 27 that that SCONJ iajs-779 20 28 x=(0 x=(0 NUM iajs-779 20 29 ) ) PUNCT iajs-779 20 30 or or CCONJ iajs-779 20 31 y=(0 y=(0 PROPN iajs-779 20 32 ) ) PUNCT iajs-779 20 33 . . PUNCT iajs-779 21 1 in in ADP iajs-779 21 2 this this DET iajs-779 21 3 paper paper NOUN iajs-779 21 4 we we PRON iajs-779 21 5 give give VERB iajs-779 21 6 a a DET iajs-779 21 7 comprehensive comprehensive ADJ iajs-779 21 8 study study NOUN iajs-779 21 9 of of ADP iajs-779 21 10 the the DET iajs-779 21 11 concepts concept NOUN iajs-779 21 12 fully fully ADV iajs-779 21 13 semiprime semiprime NOUN iajs-779 21 14 submodules submodule NOUN iajs-779 21 15 and and CCONJ iajs-779 21 16 fully fully ADV iajs-779 21 17 semiprime semiprime NOUN iajs-779 21 18 modules module NOUN iajs-779 21 19 , , PUNCT iajs-779 21 20 where where SCONJ iajs-779 21 21 this this DET iajs-779 21 22 paper paper NOUN iajs-779 21 23 consists consist VERB iajs-779 21 24 of of ADP iajs-779 21 25 two two NUM iajs-779 21 26 sections section NOUN iajs-779 21 27 . . PUNCT iajs-779 22 1 in in ADP iajs-779 22 2 section section NOUN iajs-779 22 3 one one NUM iajs-779 22 4 , , PUNCT iajs-779 22 5 we we PRON iajs-779 22 6 give give VERB iajs-779 22 7 the the DET iajs-779 22 8 basic basic ADJ iajs-779 22 9 properties property NOUN iajs-779 22 10 of of ADP iajs-779 22 11 fully fully ADV iajs-779 22 12 semiprime semiprime NOUN iajs-779 22 13 submodules submodule NOUN iajs-779 22 14 and and CCONJ iajs-779 22 15 fully fully ADV iajs-779 22 16 semiprime semiprime NOUN iajs-779 22 17 modules module NOUN iajs-779 22 18 . . PUNCT iajs-779 23 1 section section NOUN iajs-779 23 2 two two NUM iajs-779 23 3 is be AUX iajs-779 23 4 devoted devote VERB iajs-779 23 5 to to PART iajs-779 23 6 study study VERB iajs-779 23 7 the the DET iajs-779 23 8 relationships relationship NOUN iajs-779 23 9 between between ADP iajs-779 23 10 fully fully ADV iajs-779 23 11 semiprime semiprime NOUN iajs-779 23 12 modules module NOUN iajs-779 23 13 and and CCONJ iajs-779 23 14 other other ADJ iajs-779 23 15 modules module NOUN iajs-779 23 16 such such ADJ iajs-779 23 17 as as ADP iajs-779 23 18 uniform uniform ADJ iajs-779 23 19 module module NOUN iajs-779 23 20 , , PUNCT iajs-779 23 21 chained chain VERB iajs-779 23 22 module module NOUN iajs-779 23 23 , , PUNCT iajs-779 23 24 z z NOUN iajs-779 23 25 - - PUNCT iajs-779 23 26 regular regular ADJ iajs-779 23 27 module module NOUN iajs-779 23 28 , , PUNCT iajs-779 23 29 quasi quasi ADJ iajs-779 23 30 - - ADJ iajs-779 23 31 dedekind dedekind ADJ iajs-779 23 32 module module NOUN iajs-779 23 33 , , PUNCT iajs-779 23 34 multiplication multiplication NOUN iajs-779 23 35 module module NOUN iajs-779 23 36 and and CCONJ iajs-779 23 37 retractable retractable ADJ iajs-779 23 38 module module NOUN iajs-779 23 39 . . PUNCT iajs-779 24 1 ibn ibn PROPN iajs-779 24 2 alhaitham alhaitham PROPN iajs-779 24 3 j. j. PROPN iajs-779 24 4 for for ADP iajs-779 24 5 pure pure ADJ iajs-779 24 6 & & CCONJ iajs-779 24 7 appl appl PROPN iajs-779 24 8 . . PUNCT iajs-779 25 1 sci sci PROPN iajs-779 25 2 . . PUNCT iajs-779 25 3 vol.24 vol.24 NOUN iajs-779 25 4 ( ( PUNCT iajs-779 25 5 2 2 NUM iajs-779 25 6 ) ) PUNCT iajs-779 25 7 2011 2011 NUM iajs-779 25 8 next next ADV iajs-779 25 9 throughout throughout ADP iajs-779 25 10 this this DET iajs-779 25 11 paper paper NOUN iajs-779 25 12 , , PUNCT iajs-779 25 13 r r NOUN iajs-779 25 14 is be AUX iajs-779 25 15 commutative commutative ADJ iajs-779 25 16 ring ring NOUN iajs-779 25 17 with with ADP iajs-779 25 18 unity unity NOUN iajs-779 25 19 and and CCONJ iajs-779 25 20 m m AUX iajs-779 25 21 be be AUX iajs-779 25 22 a a DET iajs-779 25 23 unitary unitary ADJ iajs-779 25 24 r r NOUN iajs-779 25 25 - - PUNCT iajs-779 25 26 module module NOUN iajs-779 25 27 . . PUNCT iajs-779 26 1 1fully 1fully NUM iajs-779 26 2 semiprime semiprime NOUN iajs-779 26 3 submodules submodule NOUN iajs-779 26 4 and and CCONJ iajs-779 26 5 fully fully ADV iajs-779 26 6 semiprime semiprime NOUN iajs-779 26 7 modules module NOUN iajs-779 26 8 - - PUNCT iajs-779 26 9 basic basic ADJ iajs-779 26 10 results result NOUN iajs-779 26 11 in in ADP iajs-779 26 12 this this DET iajs-779 26 13 section section NOUN iajs-779 26 14 we we PRON iajs-779 26 15 study study VERB iajs-779 26 16 the the DET iajs-779 26 17 concepts concept NOUN iajs-779 26 18 of of ADP iajs-779 26 19 fully fully ADV iajs-779 26 20 semiprime semiprime NOUN iajs-779 26 21 submodules submodule NOUN iajs-779 26 22 and and CCONJ iajs-779 26 23 fully fully ADV iajs-779 26 24 semiprime semiprime NOUN iajs-779 26 25 modules module NOUN iajs-779 26 26 which which PRON iajs-779 26 27 are be AUX iajs-779 26 28 introduced introduce VERB iajs-779 26 29 in in ADP iajs-779 26 30 [ [ X iajs-779 26 31 1 1 NUM iajs-779 26 32 ] ] PUNCT iajs-779 26 33 as as ADP iajs-779 26 34 projects project NOUN iajs-779 26 35 . . PUNCT iajs-779 27 1 the the DET iajs-779 27 2 concepts concept NOUN iajs-779 27 3 are be AUX iajs-779 27 4 generalizations generalization NOUN iajs-779 27 5 of of ADP iajs-779 27 6 fully fully ADV iajs-779 27 7 prime prime ADJ iajs-779 27 8 submodules submodule NOUN iajs-779 27 9 and and CCONJ iajs-779 27 10 fully fully ADV iajs-779 27 11 prime prime ADJ iajs-779 27 12 modules module NOUN iajs-779 27 13 which which PRON iajs-779 27 14 are be AUX iajs-779 27 15 studied study VERB iajs-779 27 16 in in ADP iajs-779 27 17 [ [ X iajs-779 27 18 3 3 NUM iajs-779 27 19 ] ] PUNCT iajs-779 27 20 . . PUNCT iajs-779 28 1 we we PRON iajs-779 28 2 give give VERB iajs-779 28 3 characterizations characterization NOUN iajs-779 28 4 about about ADP iajs-779 28 5 theses thesis NOUN iajs-779 28 6 concepts concept NOUN iajs-779 28 7 and and CCONJ iajs-779 28 8 establishe establishe PRON iajs-779 28 9 some some DET iajs-779 28 10 basic basic ADJ iajs-779 28 11 properties property NOUN iajs-779 28 12 about about ADP iajs-779 28 13 them they PRON iajs-779 28 14 . . PUNCT iajs-779 29 1 we we PRON iajs-779 29 2 begin begin VERB iajs-779 29 3 with with ADP iajs-779 29 4 the the DET iajs-779 29 5 following follow VERB iajs-779 29 6 definition definition NOUN iajs-779 29 7 . . PUNCT iajs-779 30 1 1.1 1.1 NUM iajs-779 30 2 definition definition NOUN iajs-779 30 3 , , PUNCT iajs-779 30 4 [ [ X iajs-779 30 5 2 2 NUM iajs-779 30 6 ] ] PUNCT iajs-779 30 7 : : PUNCT iajs-779 30 8 let let VERB iajs-779 30 9 k k X iajs-779 30 10 , , PUNCT iajs-779 30 11 l l NOUN iajs-779 30 12 be be VERB iajs-779 30 13 two two NUM iajs-779 30 14 fully fully ADV iajs-779 30 15 invariant invariant ADJ iajs-779 30 16 submodules submodule NOUN iajs-779 30 17 of of ADP iajs-779 30 18 r r NOUN iajs-779 30 19 - - PUNCT iajs-779 30 20 module module NOUN iajs-779 30 21 m. m. NOUN iajs-779 30 22 then then ADV iajs-779 30 23 kl={f(k kl={f(k VERB iajs-779 30 24 ): ): PUNCT iajs-779 30 25 f f X iajs-779 30 26 : : PUNCT iajs-779 30 27 ml ml X iajs-779 30 28 } } PUNCT iajs-779 30 29 a a DET iajs-779 30 30 proper proper ADJ iajs-779 30 31 submodule submodule NOUN iajs-779 30 32 n n PROPN iajs-779 30 33 of of ADP iajs-779 30 34 an an DET iajs-779 30 35 r r NOUN iajs-779 30 36 - - PUNCT iajs-779 30 37 module module NOUN iajs-779 30 38 m m NOUN iajs-779 30 39 is be AUX iajs-779 30 40 called call VERB iajs-779 30 41 invariant invariant ADJ iajs-779 30 42 if if SCONJ iajs-779 30 43 for for ADP iajs-779 30 44 each each DET iajs-779 30 45 f f NOUN iajs-779 30 46 r r NOUN iajs-779 30 47 end(m end(m PROPN iajs-779 30 48 ) ) PUNCT iajs-779 30 49 , , PUNCT iajs-779 30 50 f(n)n f(n)n PROPN iajs-779 30 51 . . PUNCT iajs-779 31 1 m m PROPN iajs-779 31 2 is be AUX iajs-779 31 3 called call VERB iajs-779 31 4 fully fully ADV iajs-779 31 5 invariant invariant ADJ iajs-779 31 6 if if SCONJ iajs-779 31 7 every every DET iajs-779 31 8 submodule submodule NOUN iajs-779 31 9 of of ADP iajs-779 31 10 m m PROPN iajs-779 31 11 is be AUX iajs-779 31 12 invariant invariant ADJ iajs-779 31 13 , , PUNCT iajs-779 31 14 see see VERB iajs-779 31 15 [ [ X iajs-779 31 16 4 4 NUM iajs-779 31 17 ] ] PUNCT iajs-779 31 18 . . PUNCT iajs-779 32 1 invariant invariant PROPN iajs-779 32 2 submodule submodule PROPN iajs-779 32 3 is be AUX iajs-779 32 4 called call VERB iajs-779 32 5 fully fully ADV iajs-779 32 6 invariant invariant ADJ iajs-779 32 7 submodule submodule NOUN iajs-779 32 8 by by ADP iajs-779 32 9 some some DET iajs-779 32 10 authors author NOUN iajs-779 32 11 , , PUNCT iajs-779 32 12 see see VERB iajs-779 32 13 [ [ X iajs-779 32 14 3,p.14 3,p.14 NUM iajs-779 32 15 ] ] PUNCT iajs-779 32 16 . . PUNCT iajs-779 33 1 1.2 1.2 NUM iajs-779 33 2 definition definition NOUN iajs-779 33 3 , , PUNCT iajs-779 33 4 [ [ X iajs-779 33 5 3 3 NUM iajs-779 33 6 ] ] X iajs-779 33 7 : : PUNCT iajs-779 33 8 a a DET iajs-779 33 9 fully fully ADV iajs-779 33 10 invariant invariant ADJ iajs-779 33 11 submodule submodule NOUN iajs-779 33 12 n n PROPN iajs-779 33 13 of of ADP iajs-779 33 14 an an DET iajs-779 33 15 r r NOUN iajs-779 33 16 - - PUNCT iajs-779 33 17 module module NOUN iajs-779 33 18 m m NOUN iajs-779 33 19 is be AUX iajs-779 33 20 called call VERB iajs-779 33 21 fully fully ADV iajs-779 33 22 prime prime ADJ iajs-779 33 23 if if SCONJ iajs-779 33 24 for for ADP iajs-779 33 25 all all DET iajs-779 33 26 fully fully ADV iajs-779 33 27 invariant invariant ADJ iajs-779 33 28 submodules submodule NOUN iajs-779 33 29 k k PROPN iajs-779 33 30 and and CCONJ iajs-779 33 31 l l NOUN iajs-779 33 32 of of ADP iajs-779 33 33 m m PRON iajs-779 33 34 such such ADJ iajs-779 33 35 that that SCONJ iajs-779 33 36 kln kln PROPN iajs-779 33 37 , , PUNCT iajs-779 33 38 implies imply VERB iajs-779 33 39 kn kn NOUN iajs-779 33 40 or or CCONJ iajs-779 33 41 ln ln PROPN iajs-779 33 42 . . PUNCT iajs-779 34 1 now now ADV iajs-779 34 2 , , PUNCT iajs-779 34 3 we we PRON iajs-779 34 4 give give VERB iajs-779 34 5 the the DET iajs-779 34 6 following follow VERB iajs-779 34 7 concept concept NOUN iajs-779 34 8 . . PUNCT iajs-779 35 1 1.3 1.3 NUM iajs-779 35 2 definition definition NOUN iajs-779 35 3 , , PUNCT iajs-779 35 4 [ [ X iajs-779 35 5 1 1 NUM iajs-779 35 6 ] ] X iajs-779 35 7 : : PUNCT iajs-779 35 8 a a DET iajs-779 35 9 fully fully ADV iajs-779 35 10 invariant invariant ADJ iajs-779 35 11 submodule submodule NOUN iajs-779 35 12 n n PROPN iajs-779 35 13 of of ADP iajs-779 35 14 an an DET iajs-779 35 15 r r NOUN iajs-779 35 16 - - PUNCT iajs-779 35 17 module module NOUN iajs-779 35 18 m m NOUN iajs-779 35 19 is be AUX iajs-779 35 20 called call VERB iajs-779 35 21 fully fully ADV iajs-779 35 22 semiprime semiprime NOUN iajs-779 35 23 if if SCONJ iajs-779 35 24 for for ADP iajs-779 35 25 all all DET iajs-779 35 26 fully fully ADV iajs-779 35 27 invariant invariant ADJ iajs-779 35 28 submodules submodule NOUN iajs-779 35 29 k k PROPN iajs-779 35 30 of of ADP iajs-779 35 31 m m PROPN iajs-779 35 32 such such ADJ iajs-779 35 33 that that SCONJ iajs-779 35 34 kkn kkn PROPN iajs-779 35 35 , , PUNCT iajs-779 35 36 implies imply VERB iajs-779 35 37 kn kn NOUN iajs-779 35 38 . . PUNCT iajs-779 36 1 we we PRON iajs-779 36 2 call call VERB iajs-779 36 3 m m VERB iajs-779 36 4 fully fully ADV iajs-779 36 5 prime prime ADJ iajs-779 36 6 ( ( PUNCT iajs-779 36 7 fully fully ADV iajs-779 36 8 semiprime semiprime NOUN iajs-779 36 9 ) ) PUNCT iajs-779 36 10 module module NOUN iajs-779 36 11 if if SCONJ iajs-779 36 12 ( ( PUNCT iajs-779 36 13 0 0 NUM iajs-779 36 14 ) ) PUNCT iajs-779 36 15 is be AUX iajs-779 36 16 fully fully ADV iajs-779 36 17 prime prime ADJ iajs-779 36 18 ( ( PUNCT iajs-779 36 19 fully fully ADV iajs-779 36 20 semiprime semiprime ADJ iajs-779 36 21 ) ) PUNCT iajs-779 36 22 submodule submodule NOUN iajs-779 36 23 , , PUNCT iajs-779 36 24 see see VERB iajs-779 36 25 [ [ X iajs-779 36 26 1 1 NUM iajs-779 36 27 ] ] PUNCT iajs-779 36 28 . . PUNCT iajs-779 37 1 recall recall VERB iajs-779 37 2 that that PRON iajs-779 37 3 : : PUNCT iajs-779 37 4 an an DET iajs-779 37 5 r r NOUN iajs-779 37 6 - - PUNCT iajs-779 37 7 module module NOUN iajs-779 37 8 m m NOUN iajs-779 37 9 is be AUX iajs-779 37 10 said say VERB iajs-779 37 11 to to PART iajs-779 37 12 be be AUX iajs-779 37 13 a a DET iajs-779 37 14 prime prime ADJ iajs-779 37 15 module module NOUN iajs-779 37 16 if if SCONJ iajs-779 37 17 annrm annrm NOUN iajs-779 37 18 = = NOUN iajs-779 37 19 annrn annrn NOUN iajs-779 37 20 for for ADP iajs-779 37 21 every every DET iajs-779 37 22 nonzero nonzero PROPN iajs-779 37 23 submodule submodule PROPN iajs-779 37 24 n n PROPN iajs-779 37 25 of of ADP iajs-779 37 26 m m PROPN iajs-779 37 27 , , PUNCT iajs-779 37 28 where where SCONJ iajs-779 37 29 annrm={rr annrm={rr ADV iajs-779 37 30 : : PUNCT iajs-779 37 31 rx=0 rx=0 NOUN iajs-779 37 32 for for ADP iajs-779 37 33 each each PRON iajs-779 37 34 xm xm NOUN iajs-779 37 35 } } PUNCT iajs-779 37 36 , , PUNCT iajs-779 37 37 see see VERB iajs-779 37 38 [ [ X iajs-779 37 39 5 5 NUM iajs-779 37 40 ] ] PUNCT iajs-779 37 41 . . PUNCT iajs-779 38 1 an an DET iajs-779 38 2 r r NOUN iajs-779 38 3 - - PUNCT iajs-779 38 4 module module NOUN iajs-779 38 5 m m NOUN iajs-779 38 6 is be AUX iajs-779 38 7 called call VERB iajs-779 38 8 semiprime semiprime NOUN iajs-779 38 9 if if SCONJ iajs-779 39 1 and and CCONJ iajs-779 39 2 only only ADV iajs-779 39 3 if if SCONJ iajs-779 39 4 annrn annrn NOUN iajs-779 39 5 is be AUX iajs-779 39 6 a a DET iajs-779 39 7 semiprime semiprime NOUN iajs-779 39 8 ideal ideal NOUN iajs-779 39 9 of of ADP iajs-779 39 10 r r NOUN iajs-779 39 11 for for ADP iajs-779 39 12 each each DET iajs-779 39 13 non non ADJ iajs-779 39 14 - - ADJ iajs-779 39 15 zero zero ADJ iajs-779 39 16 r r NOUN iajs-779 39 17 - - PUNCT iajs-779 39 18 submodule submodule NOUN iajs-779 39 19 n n PROPN iajs-779 39 20 of of ADP iajs-779 39 21 m m PRON iajs-779 39 22 , , PUNCT iajs-779 39 23 see see VERB iajs-779 39 24 [ [ X iajs-779 39 25 6 6 NUM iajs-779 39 26 ] ] PUNCT iajs-779 39 27 . . PUNCT iajs-779 40 1 next next ADV iajs-779 40 2 , , PUNCT iajs-779 40 3 we we PRON iajs-779 40 4 give give VERB iajs-779 40 5 some some DET iajs-779 40 6 remarks remark NOUN iajs-779 40 7 and and CCONJ iajs-779 40 8 examples example NOUN iajs-779 40 9 . . PUNCT iajs-779 41 1 1.4 1.4 NUM iajs-779 41 2 note note NOUN iajs-779 41 3 : : PUNCT iajs-779 41 4 consider consider VERB iajs-779 41 5 r r NOUN iajs-779 41 6 as as ADP iajs-779 41 7 a a DET iajs-779 41 8 left left ADJ iajs-779 41 9 r r NOUN iajs-779 41 10 - - PUNCT iajs-779 41 11 module module NOUN iajs-779 41 12 , , PUNCT iajs-779 41 13 let let VERB iajs-779 41 14 i i PRON iajs-779 41 15 , , PUNCT iajs-779 41 16 j j PROPN iajs-779 41 17 be be VERB iajs-779 41 18 two two NUM iajs-779 41 19 ideals ideal NOUN iajs-779 41 20 of of ADP iajs-779 41 21 r. r. PROPN iajs-779 41 22 then then ADV iajs-779 41 23 ij ij PROPN iajs-779 41 24 = = SYM iajs-779 41 25 ij ij NOUN iajs-779 41 26 , , PUNCT iajs-779 41 27 since since SCONJ iajs-779 41 28 every every DET iajs-779 41 29 ideal ideal NOUN iajs-779 41 30 of of ADP iajs-779 41 31 r r NOUN iajs-779 41 32 is be AUX iajs-779 41 33 a a DET iajs-779 41 34 fully fully ADV iajs-779 41 35 invariant invariant ADJ iajs-779 41 36 r r NOUN iajs-779 41 37 - - PUNCT iajs-779 41 38 submodule submodule NOUN iajs-779 41 39 . . PUNCT iajs-779 42 1 thus thus ADV iajs-779 42 2 i i PRON iajs-779 42 3 is be AUX iajs-779 42 4 a a DET iajs-779 42 5 fully fully ADV iajs-779 42 6 semiprime semiprime NOUN iajs-779 42 7 ideal ideal NOUN iajs-779 42 8 if if SCONJ iajs-779 42 9 and and CCONJ iajs-779 42 10 only only ADV iajs-779 42 11 if if SCONJ iajs-779 42 12 i i PRON iajs-779 42 13 is be AUX iajs-779 42 14 a a DET iajs-779 42 15 semiprime semiprime NOUN iajs-779 42 16 ideal ideal NOUN iajs-779 42 17 . . PUNCT iajs-779 43 1 1.5 1.5 NUM iajs-779 43 2 remarks remark NOUN iajs-779 43 3 and and CCONJ iajs-779 43 4 examples example NOUN iajs-779 43 5 : : PUNCT iajs-779 43 6 1 1 X iajs-779 43 7 . . X iajs-779 43 8 let let VERB iajs-779 43 9 n n PRON iajs-779 43 10 be be AUX iajs-779 43 11 a a DET iajs-779 43 12 submodule submodule NOUN iajs-779 43 13 of of ADP iajs-779 43 14 an an DET iajs-779 43 15 r r NOUN iajs-779 43 16 - - PUNCT iajs-779 43 17 module module NOUN iajs-779 43 18 m. m. NOUN iajs-779 43 19 if if SCONJ iajs-779 43 20 n n PRON iajs-779 43 21 is be AUX iajs-779 43 22 a a DET iajs-779 43 23 fully fully ADV iajs-779 43 24 prime prime ADJ iajs-779 43 25 submodule submodule NOUN iajs-779 43 26 , , PUNCT iajs-779 43 27 then then ADV iajs-779 43 28 n n PRON iajs-779 43 29 is be AUX iajs-779 43 30 a a DET iajs-779 43 31 fully fully ADV iajs-779 43 32 semiprime semiprime NOUN iajs-779 43 33 submodule submodule NOUN iajs-779 43 34 . . PUNCT iajs-779 44 1 2 2 X iajs-779 44 2 . . X iajs-779 44 3 if if SCONJ iajs-779 44 4 an an DET iajs-779 44 5 r r NOUN iajs-779 44 6 - - PUNCT iajs-779 44 7 module module NOUN iajs-779 44 8 m m NOUN iajs-779 44 9 is be AUX iajs-779 44 10 fully fully ADV iajs-779 44 11 prime prime ADJ iajs-779 44 12 module module NOUN iajs-779 44 13 , , PUNCT iajs-779 44 14 then then ADV iajs-779 44 15 m m NOUN iajs-779 44 16 is be AUX iajs-779 44 17 prime prime ADJ iajs-779 44 18 module module NOUN iajs-779 44 19 . . PUNCT iajs-779 45 1 3 3 X iajs-779 45 2 . . X iajs-779 45 3 a a DET iajs-779 45 4 submodule submodule NOUN iajs-779 45 5 n n PROPN iajs-779 45 6 of of ADP iajs-779 45 7 an an DET iajs-779 45 8 r r NOUN iajs-779 45 9 - - PUNCT iajs-779 45 10 module module NOUN iajs-779 45 11 m m NOUN iajs-779 45 12 is be AUX iajs-779 45 13 semiprime semiprime NOUN iajs-779 45 14 , , PUNCT iajs-779 45 15 if if SCONJ iajs-779 45 16 n n PRON iajs-779 45 17 is be AUX iajs-779 45 18 fully fully ADV iajs-779 45 19 semiprime semiprime NOUN iajs-779 45 20 submodule submodule NOUN iajs-779 45 21 . . PUNCT iajs-779 46 1 proof proof NOUN iajs-779 46 2 : : PUNCT iajs-779 46 3 suppose suppose VERB iajs-779 46 4 that that SCONJ iajs-779 46 5 rr rr ADP iajs-779 46 6 , , PUNCT iajs-779 46 7 xm xm NOUN iajs-779 46 8 such such ADJ iajs-779 46 9 that that SCONJ iajs-779 46 10 r r NOUN iajs-779 46 11 2 2 NUM iajs-779 46 12 xn xn PROPN iajs-779 46 13 . . PUNCT iajs-779 47 1 let let VERB iajs-779 47 2 k= > PROPN iajs-779 47 4 , , PUNCT iajs-779 47 5 k k PROPN iajs-779 47 6 is be AUX iajs-779 47 7 a a DET iajs-779 47 8 fully fully ADV iajs-779 47 9 invariant invariant ADJ iajs-779 47 10 submodule submodule NOUN iajs-779 47 11 , , PUNCT iajs-779 47 12 then then ADV iajs-779 47 13 kk={f(k kk={f(k VERB iajs-779 47 14 ): ): PUNCT iajs-779 47 15 f f X iajs-779 47 16 : : PUNCT iajs-779 47 17 mk= mk= PROPN iajs-779 47 18 < < X iajs-779 47 19 rx rx VERB iajs-779 47 20 > > NOUN iajs-779 47 21 } } PUNCT iajs-779 47 22 . . PUNCT iajs-779 48 1 now now ADV iajs-779 48 2 , , PUNCT iajs-779 48 3 f(k)=f f(k)=f NOUN iajs-779 48 4 < < X iajs-779 48 5 rx>=r rx>=r NUM iajs-779 48 6 < < X iajs-779 48 7 f(x)>n 2x>n NUM iajs-779 48 9 . . PUNCT iajs-779 49 1 thus thus ADV iajs-779 49 2 kkn kkn PROPN iajs-779 49 3 , , PUNCT iajs-779 49 4 implies imply VERB iajs-779 49 5 kn kn NOUN iajs-779 49 6 , , PUNCT iajs-779 49 7 so so ADV iajs-779 49 8 rxn rxn NOUN iajs-779 49 9 . . PUNCT iajs-779 50 1 4 4 X iajs-779 50 2 . . X iajs-779 50 3 if if SCONJ iajs-779 50 4 an an DET iajs-779 50 5 r r NOUN iajs-779 50 6 - - PUNCT iajs-779 50 7 module module NOUN iajs-779 50 8 m m NOUN iajs-779 50 9 is be AUX iajs-779 50 10 a a DET iajs-779 50 11 fully fully ADV iajs-779 50 12 semiprime semiprime NOUN iajs-779 50 13 module module NOUN iajs-779 50 14 , , PUNCT iajs-779 50 15 then then ADV iajs-779 50 16 m m NOUN iajs-779 50 17 is be AUX iajs-779 50 18 a a DET iajs-779 50 19 semiprime semiprime NOUN iajs-779 50 20 module module NOUN iajs-779 50 21 . . PUNCT iajs-779 51 1 5 5 NUM iajs-779 51 2 . . X iajs-779 51 3 z6 z6 PROPN iajs-779 51 4 as as ADP iajs-779 51 5 a a DET iajs-779 51 6 z z NOUN iajs-779 51 7 - - PUNCT iajs-779 51 8 module module NOUN iajs-779 51 9 is be AUX iajs-779 51 10 fully fully ADV iajs-779 51 11 semiprime semiprime NOUN iajs-779 51 12 , , PUNCT iajs-779 51 13 since since SCONJ iajs-779 51 14 for for ADP iajs-779 51 15 all all DET iajs-779 51 16 submodule submodule NOUN iajs-779 51 17 n n CCONJ iajs-779 51 18 , , PUNCT iajs-779 51 19 n(0 n(0 NUM iajs-779 51 20 ) ) PUNCT iajs-779 51 21 , , PUNCT iajs-779 51 22 then then ADV iajs-779 51 23 nn(0 nn(0 VERB iajs-779 51 24 ) ) PUNCT iajs-779 51 25 . . PUNCT iajs-779 52 1 thus thus ADV iajs-779 52 2 z6 z6 PROPN iajs-779 52 3 is be AUX iajs-779 52 4 a a DET iajs-779 52 5 semiprime semiprime NOUN iajs-779 52 6 z z NOUN iajs-779 52 7 - - PUNCT iajs-779 52 8 module module NOUN iajs-779 52 9 . . PUNCT iajs-779 53 1 but but CCONJ iajs-779 53 2 it it PRON iajs-779 53 3 is be AUX iajs-779 53 4 not not PART iajs-779 53 5 a a DET iajs-779 53 6 fully fully ADV iajs-779 53 7 prime prime ADJ iajs-779 53 8 because because SCONJ iajs-779 53 9 it it PRON iajs-779 53 10 is be AUX iajs-779 53 11 not not PART iajs-779 53 12 prime prime ADJ iajs-779 53 13 . . PUNCT iajs-779 54 1 6 6 X iajs-779 54 2 . . X iajs-779 54 3 z4 z4 PROPN iajs-779 54 4 as as ADP iajs-779 54 5 a a DET iajs-779 54 6 z z NOUN iajs-779 54 7 - - PUNCT iajs-779 54 8 module module NOUN iajs-779 54 9 is be AUX iajs-779 54 10 not not PART iajs-779 54 11 semiprime semiprime NOUN iajs-779 54 12 module module NOUN iajs-779 54 13 , , PUNCT iajs-779 54 14 since since SCONJ iajs-779 54 15 annzz4=4z annzz4=4z PUNCT iajs-779 54 16 is be AUX iajs-779 54 17 not not PART iajs-779 54 18 a a DET iajs-779 54 19 semiprime semiprime NOUN iajs-779 54 20 ideal ideal NOUN iajs-779 54 21 of of ADP iajs-779 54 22 z. z. PROPN iajs-779 54 23 hence hence PROPN iajs-779 54 24 z4 z4 PROPN iajs-779 54 25 is be AUX iajs-779 54 26 not not PART iajs-779 54 27 fully fully ADV iajs-779 54 28 semiprime semiprime NOUN iajs-779 54 29 . . PUNCT iajs-779 55 1 7 7 X iajs-779 55 2 . . X iajs-779 55 3 6z 6z NOUN iajs-779 55 4 as as ADP iajs-779 55 5 a a DET iajs-779 55 6 z z NOUN iajs-779 55 7 - - PUNCT iajs-779 55 8 submodule submodule NOUN iajs-779 55 9 of of ADP iajs-779 55 10 z z PROPN iajs-779 55 11 is be AUX iajs-779 55 12 semiprime semiprime NOUN iajs-779 55 13 , , PUNCT iajs-779 55 14 so so CCONJ iajs-779 55 15 it it PRON iajs-779 55 16 is be AUX iajs-779 55 17 fully fully ADV iajs-779 55 18 semiprime semiprime ADJ iajs-779 55 19 . . PUNCT iajs-779 56 1 8 8 X iajs-779 56 2 . . X iajs-779 57 1 let let VERB iajs-779 57 2 r r PRON iajs-779 57 3 be be AUX iajs-779 57 4 an an DET iajs-779 57 5 integral integral ADJ iajs-779 57 6 domain domain NOUN iajs-779 57 7 and and CCONJ iajs-779 57 8 k k PROPN iajs-779 57 9 be be AUX iajs-779 57 10 the the DET iajs-779 57 11 quotient quotient NOUN iajs-779 57 12 field field NOUN iajs-779 57 13 of of ADP iajs-779 57 14 r. r. PROPN iajs-779 57 15 then then ADV iajs-779 57 16 k k PROPN iajs-779 57 17 is be AUX iajs-779 57 18 an an DET iajs-779 57 19 r r NOUN iajs-779 57 20 - - PUNCT iajs-779 57 21 module module NOUN iajs-779 57 22 and and CCONJ iajs-779 57 23 the the DET iajs-779 57 24 zero zero NUM iajs-779 57 25 r r NOUN iajs-779 57 26 - - PUNCT iajs-779 57 27 submodule submodule NOUN iajs-779 57 28 of of ADP iajs-779 57 29 k k PROPN iajs-779 57 30 is be AUX iajs-779 57 31 the the DET iajs-779 57 32 only only ADJ iajs-779 57 33 semiprime semiprime NOUN iajs-779 57 34 in in ADP iajs-779 57 35 k. k. PROPN iajs-779 57 36 that that PRON iajs-779 57 37 is be AUX iajs-779 57 38 ( ( PUNCT iajs-779 57 39 0 0 NUM iajs-779 57 40 ) ) PUNCT iajs-779 57 41 is be AUX iajs-779 57 42 the the DET iajs-779 57 43 only only ADJ iajs-779 57 44 fully fully ADV iajs-779 57 45 semiprime semiprime NOUN iajs-779 57 46 submodule submodule NOUN iajs-779 57 47 in in ADP iajs-779 57 48 k k PROPN iajs-779 57 49 , , PUNCT iajs-779 57 50 because because SCONJ iajs-779 57 51 if if SCONJ iajs-779 57 52   ADP iajs-779 57 53 n n CCONJ iajs-779 57 54 < < X iajs-779 57 55 k k X iajs-779 57 56 , , PUNCT iajs-779 57 57 n(0 n(0 NUM iajs-779 57 58 ) ) PUNCT iajs-779 57 59 , , PUNCT iajs-779 57 60 n n PRON iajs-779 57 61 is be AUX iajs-779 57 62 fully fully ADV iajs-779 57 63 semiprime semiprime NOUN iajs-779 57 64 submodule submodule NOUN iajs-779 57 65 , , PUNCT iajs-779 57 66 then then ADV iajs-779 57 67 n n PROPN iajs-779 57 68 is be AUX iajs-779 57 69 semiprime semiprime NOUN iajs-779 57 70 , , PUNCT iajs-779 57 71 which which PRON iajs-779 57 72 is be AUX iajs-779 57 73 a a DET iajs-779 57 74 contradiction contradiction NOUN iajs-779 57 75 . . PUNCT iajs-779 58 1 ibn ibn PROPN iajs-779 58 2 alhaitham alhaitham PROPN iajs-779 58 3 j. j. PROPN iajs-779 58 4 for for ADP iajs-779 58 5 pure pure ADJ iajs-779 58 6 & & CCONJ iajs-779 58 7 appl appl PROPN iajs-779 58 8 . . PUNCT iajs-779 59 1 sci sci PROPN iajs-779 59 2 . . PUNCT iajs-779 59 3 vol.24 vol.24 NOUN iajs-779 59 4 ( ( PUNCT iajs-779 59 5 2 2 NUM iajs-779 59 6 ) ) PUNCT iajs-779 59 7 2011 2011 NUM iajs-779 59 8 9 9 NUM iajs-779 59 9 . . PUNCT iajs-779 60 1 p p X iajs-779 60 2 z z NOUN iajs-779 60 3   NOUN iajs-779 60 4 as as ADP iajs-779 60 5 a a DET iajs-779 60 6 z z NOUN iajs-779 60 7 - - PUNCT iajs-779 60 8 module module NOUN iajs-779 60 9 has have VERB iajs-779 60 10 no no DET iajs-779 60 11 fully fully ADV iajs-779 60 12 semiprime semiprime NOUN iajs-779 60 13 submodule submodule NOUN iajs-779 60 14 . . PUNCT iajs-779 61 1 10 10 NUM iajs-779 61 2 . . PUNCT iajs-779 62 1 the the DET iajs-779 62 2 homomorphic homomorphic ADJ iajs-779 62 3 image image NOUN iajs-779 62 4 of of ADP iajs-779 62 5 a a DET iajs-779 62 6 fully fully ADV iajs-779 62 7 semiprime semiprime NOUN iajs-779 62 8 module module NOUN iajs-779 62 9 need need AUX iajs-779 62 10 not not PART iajs-779 62 11 fully fully ADV iajs-779 62 12 semiprime semiprime NOUN iajs-779 62 13 module module NOUN iajs-779 62 14 , , PUNCT iajs-779 62 15 for for ADP iajs-779 62 16 example example NOUN iajs-779 62 17 : : PUNCT iajs-779 62 18 z z NOUN iajs-779 62 19 as as SCONJ iajs-779 62 20 a a DET iajs-779 62 21 z z NOUN iajs-779 62 22 - - PUNCT iajs-779 62 23 module module NOUN iajs-779 62 24 is be AUX iajs-779 62 25 a a DET iajs-779 62 26 fully fully ADV iajs-779 62 27 prime prime ADJ iajs-779 62 28 . . PUNCT iajs-779 63 1 then then ADV iajs-779 63 2 z z PROPN iajs-779 63 3 is be AUX iajs-779 63 4 fully fully ADV iajs-779 63 5 semiprime semiprime NOUN iajs-779 63 6 . . PUNCT iajs-779 64 1 now now ADV iajs-779 64 2 , , PUNCT iajs-779 64 3 let let VERB iajs-779 64 4 :zz/(4 :zz/(4 PROPN iajs-779 64 5 ) ) PUNCT iajs-779 64 6 z4 z4 PROPN iajs-779 64 7 . . PUNCT iajs-779 65 1 z4 z4 PROPN iajs-779 65 2 as as ADP iajs-779 65 3 a a DET iajs-779 65 4 z z NOUN iajs-779 65 5 - - PUNCT iajs-779 65 6 module module NOUN iajs-779 65 7 is be AUX iajs-779 65 8 not not PART iajs-779 65 9 a a DET iajs-779 65 10 fully fully ADV iajs-779 65 11 semiprime semiprime NOUN iajs-779 65 12 . . PUNCT iajs-779 66 1 now now ADV iajs-779 66 2 , , PUNCT iajs-779 66 3 we we PRON iajs-779 66 4 have have VERB iajs-779 66 5 the the DET iajs-779 66 6 following follow VERB iajs-779 66 7 proposition proposition NOUN iajs-779 66 8 . . PUNCT iajs-779 67 1 1.6 1.6 NUM iajs-779 67 2 proposition proposition NOUN iajs-779 67 3 : : PUNCT iajs-779 67 4 if if SCONJ iajs-779 67 5 n n PRON iajs-779 67 6 is be AUX iajs-779 67 7 fully fully ADV iajs-779 67 8 semiprime semiprime ADJ iajs-779 67 9 r r NOUN iajs-779 67 10 - - PUNCT iajs-779 67 11 submodule submodule NOUN iajs-779 67 12 of of ADP iajs-779 67 13 m m PROPN iajs-779 67 14 , , PUNCT iajs-779 67 15 then then ADV iajs-779 67 16 [ [ X iajs-779 67 17 n n X iajs-779 67 18 r r NOUN iajs-779 67 19 : : PUNCT iajs-779 67 20 k k X iajs-779 67 21 ] ] X iajs-779 67 22 is be AUX iajs-779 67 23 a a DET iajs-779 67 24 semiprime semiprime NOUN iajs-779 67 25 ideal ideal NOUN iajs-779 67 26 of of ADP iajs-779 67 27 r r NOUN iajs-779 67 28 for for ADP iajs-779 67 29 all all DET iajs-779 67 30 n n PRON iajs-779 67 31   NOUN iajs-779 68 1   PROPN iajs-779 68 2 k. k. PROPN iajs-779 68 3 proof proof NOUN iajs-779 68 4 : : PUNCT iajs-779 68 5 we we PRON iajs-779 68 6 have have AUX iajs-779 68 7 n n ADV iajs-779 68 8 is be AUX iajs-779 68 9 fully fully ADV iajs-779 68 10 semiprime semiprime NOUN iajs-779 68 11 submodule submodule NOUN iajs-779 68 12 , , PUNCT iajs-779 68 13 then then ADV iajs-779 68 14 n n PROPN iajs-779 68 15 is be AUX iajs-779 68 16 semiprime semiprime NOUN iajs-779 68 17 submodule submodule NOUN iajs-779 68 18 ( ( PUNCT iajs-779 68 19 by by ADP iajs-779 68 20 remarks remark NOUN iajs-779 68 21 and and CCONJ iajs-779 68 22 examples example NOUN iajs-779 68 23 ( ( PUNCT iajs-779 68 24 1.4),3 1.4),3 NUM iajs-779 68 25 ) ) PUNCT iajs-779 68 26 . . PUNCT iajs-779 69 1 then then ADV iajs-779 69 2 it it PRON iajs-779 69 3 is be AUX iajs-779 69 4 easy easy ADJ iajs-779 69 5 to to PART iajs-779 69 6 show show VERB iajs-779 69 7 that that SCONJ iajs-779 69 8 [ [ X iajs-779 69 9 n n X iajs-779 69 10 r r NOUN iajs-779 69 11 : : PUNCT iajs-779 69 12 k k X iajs-779 69 13 ] ] X iajs-779 69 14 is be AUX iajs-779 69 15 semiprime semiprime NOUN iajs-779 69 16 ideal ideal NOUN iajs-779 69 17 for for ADP iajs-779 69 18 all all DET iajs-779 69 19 kn kn NOUN iajs-779 69 20 . . PUNCT iajs-779 70 1 the the DET iajs-779 70 2 following following ADJ iajs-779 70 3 result result NOUN iajs-779 70 4 is be AUX iajs-779 70 5 a a DET iajs-779 70 6 consequence consequence NOUN iajs-779 70 7 of of ADP iajs-779 70 8 proposition proposition NOUN iajs-779 70 9 ( ( PUNCT iajs-779 70 10 1.6 1.6 NUM iajs-779 70 11 ) ) PUNCT iajs-779 70 12 . . PUNCT iajs-779 71 1 1.7 1.7 NUM iajs-779 71 2 corollary corollary NOUN iajs-779 71 3 : : PUNCT iajs-779 71 4 if if SCONJ iajs-779 71 5 n n PRON iajs-779 71 6 is be AUX iajs-779 71 7 a a DET iajs-779 71 8 fully fully ADV iajs-779 71 9 semiprime semiprime NOUN iajs-779 71 10 submodule submodule NOUN iajs-779 71 11 of of ADP iajs-779 71 12 an an DET iajs-779 71 13 r r NOUN iajs-779 71 14 - - PUNCT iajs-779 71 15 module module NOUN iajs-779 71 16 m m NOUN iajs-779 71 17 , , PUNCT iajs-779 71 18 then then ADV iajs-779 71 19 [ [ X iajs-779 71 20 n n X iajs-779 71 21 r r NOUN iajs-779 71 22 : : PUNCT iajs-779 71 23 m m VERB iajs-779 71 24 ] ] X iajs-779 71 25 is be AUX iajs-779 71 26 a a DET iajs-779 71 27 semiprime semiprime NOUN iajs-779 71 28 ideal ideal NOUN iajs-779 71 29 . . PUNCT iajs-779 72 1 the the DET iajs-779 72 2 following following ADJ iajs-779 72 3 result result NOUN iajs-779 72 4 is be AUX iajs-779 72 5 a a DET iajs-779 72 6 characterization characterization NOUN iajs-779 72 7 of of ADP iajs-779 72 8 fully fully ADV iajs-779 72 9 semiprime semiprime NOUN iajs-779 72 10 submodule submodule NOUN iajs-779 72 11 , , PUNCT iajs-779 72 12 but but CCONJ iajs-779 72 13 first first ADV iajs-779 72 14 the the DET iajs-779 72 15 following follow VERB iajs-779 72 16 lemma lemma PROPN iajs-779 72 17 is be AUX iajs-779 72 18 needed need VERB iajs-779 72 19 . . PUNCT iajs-779 73 1 1.8 1.8 NUM iajs-779 73 2 lemma lemma PROPN iajs-779 73 3 : : PUNCT iajs-779 73 4 let let VERB iajs-779 73 5 k k PRON iajs-779 73 6 be be AUX iajs-779 73 7 a a DET iajs-779 73 8 fully fully ADV iajs-779 73 9 invariant invariant ADJ iajs-779 73 10 submodule submodule NOUN iajs-779 73 11 of of ADP iajs-779 73 12 an an DET iajs-779 73 13 r r NOUN iajs-779 73 14 - - PUNCT iajs-779 73 15 module module NOUN iajs-779 73 16 m. m. NOUN iajs-779 73 17 then then ADV iajs-779 73 18 i(kk)ikik i(kk)ikik NOUN iajs-779 73 19 for for ADP iajs-779 73 20 every every DET iajs-779 73 21 ideal ideal NOUN iajs-779 73 22 i i PRON iajs-779 73 23 of of ADP iajs-779 73 24 r. r. PROPN iajs-779 73 25 proof proof PROPN iajs-779 73 26 : : PUNCT iajs-779 73 27 ikik={f(ik):f ikik={f(ik):f NOUN iajs-779 73 28 : : PUNCT iajs-779 73 29 mik mik NUM iajs-779 73 30 } } PUNCT iajs-779 73 31 = = SYM iajs-779 73 32 i{f(k):f i{f(k):f NOUN iajs-779 73 33 : : PUNCT iajs-779 73 34 mikk mikk X iajs-779 73 35 } } PUNCT iajs-779 73 36 since since SCONJ iajs-779 73 37 kk=={g(k):g kk=={g(k):g PROPN iajs-779 73 38 : : PUNCT iajs-779 73 39 mk mk NOUN iajs-779 73 40 } } PUNCT iajs-779 73 41 . . PUNCT iajs-779 74 1 it it PRON iajs-779 74 2 follows follow VERB iajs-779 74 3 that that SCONJ iajs-779 74 4 ikik ikik ADJ iajs-779 74 5   NOUN iajs-779 74 6 i(kk i(kk PROPN iajs-779 74 7 ) ) PUNCT iajs-779 74 8 1.9 1.9 NUM iajs-779 74 9 proposition proposition NOUN iajs-779 74 10 : : PUNCT iajs-779 74 11 let let VERB iajs-779 74 12 n n PRON iajs-779 74 13 be be AUX iajs-779 74 14 a a DET iajs-779 74 15 submodule submodule NOUN iajs-779 74 16 of of ADP iajs-779 74 17 an an DET iajs-779 74 18 r r NOUN iajs-779 74 19 - - PUNCT iajs-779 74 20 module module NOUN iajs-779 74 21 m. m. NOUN iajs-779 74 22 then then ADV iajs-779 74 23 n n PRON iajs-779 74 24 is be AUX iajs-779 74 25 a a DET iajs-779 74 26 fully fully ADV iajs-779 74 27 semiprime semiprime NOUN iajs-779 74 28 submodule submodule NOUN iajs-779 74 29 if if SCONJ iajs-779 74 30 and and CCONJ iajs-779 74 31 only only ADV iajs-779 74 32 if if SCONJ iajs-779 74 33 [ [ X iajs-779 74 34 n n X iajs-779 74 35 m m VERB iajs-779 74 36 : : PUNCT iajs-779 75 1 i i PRON iajs-779 75 2 ] ] PUNCT iajs-779 75 3 is be AUX iajs-779 75 4 a a DET iajs-779 75 5 fully fully ADV iajs-779 75 6 semiprime semiprime NOUN iajs-779 75 7 submodule submodule NOUN iajs-779 75 8 of of ADP iajs-779 75 9 m m PROPN iajs-779 75 10 for for ADP iajs-779 75 11 every every DET iajs-779 75 12 ideal ideal NOUN iajs-779 75 13 i i PRON iajs-779 75 14 of of ADP iajs-779 75 15 r. r. PROPN iajs-779 75 16 proof proof NOUN iajs-779 75 17 : : PUNCT iajs-779 75 18 ( ( PUNCT iajs-779 75 19   NOUN iajs-779 75 20 ) ) PUNCT iajs-779 75 21 suppose suppose VERB iajs-779 75 22 that that SCONJ iajs-779 75 23 n n PRON iajs-779 75 24 is be AUX iajs-779 75 25 a a DET iajs-779 75 26 fully fully ADV iajs-779 75 27 semiprime semiprime NOUN iajs-779 75 28 submodule submodule NOUN iajs-779 75 29 of of ADP iajs-779 75 30 m m PROPN iajs-779 75 31 , , PUNCT iajs-779 75 32 let let VERB iajs-779 75 33 k k PRON iajs-779 75 34 be be AUX iajs-779 75 35 a a DET iajs-779 75 36 fully fully ADV iajs-779 75 37 invariant invariant ADJ iajs-779 75 38 submodule submodule NOUN iajs-779 75 39 of of ADP iajs-779 75 40 m m PRON iajs-779 75 41 such such ADJ iajs-779 75 42 that that DET iajs-779 75 43 kk[n kk[n NOUN iajs-779 76 1 m m PROPN iajs-779 76 2 : : PUNCT iajs-779 77 1 i i PRON iajs-779 77 2 ] ] X iajs-779 77 3 , , PUNCT iajs-779 77 4 implies imply VERB iajs-779 77 5 i(kk)n i(kk)n ADV iajs-779 77 6 , , PUNCT iajs-779 77 7 then then ADV iajs-779 77 8 by by ADP iajs-779 77 9 lemma lemma PROPN iajs-779 77 10 ( ( PUNCT iajs-779 77 11 1.8 1.8 NUM iajs-779 77 12 ) ) PUNCT iajs-779 77 13 , , PUNCT iajs-779 77 14 ikik ikik X iajs-779 77 15 n n NOUN iajs-779 77 16 , , PUNCT iajs-779 77 17 but but CCONJ iajs-779 77 18 n n PRON iajs-779 77 19 is be AUX iajs-779 77 20 a a DET iajs-779 77 21 fully fully ADV iajs-779 77 22 semiprime semiprime NOUN iajs-779 77 23 submodule submodule NOUN iajs-779 77 24 . . PUNCT iajs-779 78 1 thus thus ADV iajs-779 78 2 ikn ikn ADV iajs-779 78 3 . . PUNCT iajs-779 79 1 therefore therefore ADV iajs-779 79 2 k[n k[n VERB iajs-779 79 3 m m PROPN iajs-779 79 4 : : PUNCT iajs-779 79 5 i i PRON iajs-779 79 6 ] ] X iajs-779 79 7 . . PUNCT iajs-779 80 1 hence hence ADV iajs-779 80 2 [ [ X iajs-779 80 3 n n NOUN iajs-779 80 4 m m VERB iajs-779 80 5 : : PUNCT iajs-779 81 1 i i PRON iajs-779 81 2 ] ] PUNCT iajs-779 81 3 is be AUX iajs-779 81 4 a a DET iajs-779 81 5 fully fully ADV iajs-779 81 6 semiprime semiprime NOUN iajs-779 81 7 submodule submodule NOUN iajs-779 81 8 . . PUNCT iajs-779 82 1 the the DET iajs-779 82 2 converse converse NOUN iajs-779 82 3 follows follow VERB iajs-779 82 4 by by ADP iajs-779 82 5 taking take VERB iajs-779 82 6 i i PRON iajs-779 82 7 = = NOUN iajs-779 82 8 r r NOUN iajs-779 82 9 , , PUNCT iajs-779 82 10 because because SCONJ iajs-779 82 11 [ [ X iajs-779 82 12 n n X iajs-779 82 13 m m VERB iajs-779 82 14 : : PUNCT iajs-779 82 15 r]=n r]=n X iajs-779 82 16 . . PUNCT iajs-779 83 1 next next ADV iajs-779 83 2 , , PUNCT iajs-779 83 3 we we PRON iajs-779 83 4 have have VERB iajs-779 83 5 the the DET iajs-779 83 6 following follow VERB iajs-779 83 7 proposition proposition NOUN iajs-779 83 8 . . PUNCT iajs-779 84 1 1.10 1.10 NUM iajs-779 84 2 proposition proposition NOUN iajs-779 84 3 : : PUNCT iajs-779 84 4 let let VERB iajs-779 84 5 n n PRON iajs-779 84 6 be be AUX iajs-779 84 7 a a DET iajs-779 84 8 fully fully ADV iajs-779 84 9 invariant invariant ADJ iajs-779 84 10 submodule submodule NOUN iajs-779 84 11 of of ADP iajs-779 84 12 m. m. NOUN iajs-779 84 13 if if SCONJ iajs-779 84 14 n n PRON iajs-779 84 15 is be AUX iajs-779 84 16 a a DET iajs-779 84 17 fully fully ADV iajs-779 84 18 semiprime semiprime NOUN iajs-779 84 19 submodule submodule NOUN iajs-779 84 20 , , PUNCT iajs-779 84 21 then then ADV iajs-779 84 22 m m PROPN iajs-779 84 23 / / SYM iajs-779 84 24 n n PRON iajs-779 84 25 is be AUX iajs-779 84 26 fully fully ADV iajs-779 84 27 semiprime semiprime NOUN iajs-779 84 28 module module NOUN iajs-779 84 29 . . PUNCT iajs-779 85 1 proof proof NOUN iajs-779 85 2 : : PUNCT iajs-779 85 3 let let VERB iajs-779 85 4 k k X iajs-779 85 5 / / SYM iajs-779 85 6 n n PRON iajs-779 85 7 be be VERB iajs-779 85 8 a a DET iajs-779 85 9 fully fully ADV iajs-779 85 10 invariant invariant ADJ iajs-779 85 11 submodule submodule NOUN iajs-779 85 12 of of ADP iajs-779 85 13 m m PROPN iajs-779 85 14 / / SYM iajs-779 85 15 n n PRON iajs-779 85 16 such such ADJ iajs-779 85 17 that that SCONJ iajs-779 85 18 k k PROPN iajs-779 85 19 / / SYM iajs-779 85 20 nk nk ADJ iajs-779 85 21 / / SYM iajs-779 85 22 n n CCONJ iajs-779 85 23 = = NOUN iajs-779 85 24 n n CCONJ iajs-779 85 25 = = NOUN iajs-779 85 26 om om PROPN iajs-779 85 27 / / SYM iajs-779 85 28 n. n. NOUN iajs-779 86 1 then then ADV iajs-779 86 2 k k PROPN iajs-779 86 3 is be AUX iajs-779 86 4 a a DET iajs-779 86 5 fully fully ADV iajs-779 86 6 invariant invariant ADJ iajs-779 86 7 submodules submodule NOUN iajs-779 86 8 of of ADP iajs-779 86 9 m m PROPN iajs-779 86 10 , , PUNCT iajs-779 86 11 with with ADP iajs-779 86 12 k k PROPN iajs-779 86 13 m m VERB iajs-779 86 14 kn kn NOUN iajs-779 87 1 [ [ X iajs-779 87 2 3 3 NUM iajs-779 87 3 , , PUNCT iajs-779 87 4 corollary corollary ADJ iajs-779 87 5 ( ( PUNCT iajs-779 87 6 1.1.21 1.1.21 NUM iajs-779 87 7 ) ) PUNCT iajs-779 87 8 ] ] PUNCT iajs-779 87 9 . . PUNCT iajs-779 88 1 hence hence ADV iajs-779 88 2 kn kn NOUN iajs-779 88 3 ( ( PUNCT iajs-779 88 4 since since SCONJ iajs-779 88 5 n n PRON iajs-779 88 6 is be AUX iajs-779 88 7 a a DET iajs-779 88 8 fully fully ADV iajs-779 88 9 semiprime semiprime NOUN iajs-779 88 10 submodule submodule NOUN iajs-779 88 11 ) ) PUNCT iajs-779 88 12 . . PUNCT iajs-779 89 1 that that PRON iajs-779 89 2 is be AUX iajs-779 89 3 m/ m/ NOUN iajs-779 90 1 n n X iajs-779 90 2 k k NOUN iajs-779 90 3 o o X iajs-779 91 1 n n CCONJ iajs-779 91 2   NOUN iajs-779 91 3 . . PUNCT iajs-779 92 1 the the DET iajs-779 92 2 converse converse NOUN iajs-779 92 3 of of ADP iajs-779 92 4 proposition proposition NOUN iajs-779 92 5 ( ( PUNCT iajs-779 92 6 1.10 1.10 NUM iajs-779 92 7 ) ) PUNCT iajs-779 92 8 holds hold VERB iajs-779 92 9 under under ADP iajs-779 92 10 the the DET iajs-779 92 11 condition condition NOUN iajs-779 92 12 m m VERB iajs-779 92 13 is be AUX iajs-779 92 14 self self NOUN iajs-779 92 15 projective projective ADJ iajs-779 92 16 . . PUNCT iajs-779 93 1 1.11 1.11 NUM iajs-779 93 2 proposition proposition NOUN iajs-779 93 3 : : PUNCT iajs-779 93 4 let let VERB iajs-779 93 5 m m PRON iajs-779 93 6 be be AUX iajs-779 93 7 a a DET iajs-779 93 8 self self NOUN iajs-779 93 9 projective projective NOUN iajs-779 93 10 module module NOUN iajs-779 93 11 and and CCONJ iajs-779 93 12 let let VERB iajs-779 93 13 n n PRON iajs-779 93 14 be be AUX iajs-779 93 15 a a DET iajs-779 93 16 fully fully ADV iajs-779 93 17 invariant invariant ADJ iajs-779 93 18 submodule submodule NOUN iajs-779 93 19 of of ADP iajs-779 93 20 m. m. NOUN iajs-779 93 21 if if SCONJ iajs-779 93 22 m m NOUN iajs-779 93 23 / / SYM iajs-779 93 24 n n PRON iajs-779 93 25 is be AUX iajs-779 93 26 fully fully ADV iajs-779 93 27 semiprime semiprime NOUN iajs-779 93 28 , , PUNCT iajs-779 93 29 then then ADV iajs-779 93 30 n n PRON iajs-779 93 31 is be AUX iajs-779 93 32 a a DET iajs-779 93 33 fully fully ADV iajs-779 93 34 semiprime semiprime NOUN iajs-779 93 35 submodule submodule NOUN iajs-779 93 36 in in ADP iajs-779 93 37 m. m. PROPN iajs-779 93 38 ibn ibn PROPN iajs-779 93 39 alhaitham alhaitham PROPN iajs-779 93 40 j. j. PROPN iajs-779 93 41 for for ADP iajs-779 93 42 pure pure ADJ iajs-779 93 43 & & CCONJ iajs-779 93 44 appl appl PROPN iajs-779 93 45 . . PUNCT iajs-779 94 1 sci sci PROPN iajs-779 94 2 . . PUNCT iajs-779 94 3 vol.24 vol.24 NOUN iajs-779 94 4 ( ( PUNCT iajs-779 94 5 2 2 NUM iajs-779 94 6 ) ) PUNCT iajs-779 94 7 2011 2011 NUM iajs-779 94 8 proof proof NOUN iajs-779 94 9 : : PUNCT iajs-779 94 10 let let VERB iajs-779 94 11 k k PRON iajs-779 94 12 be be AUX iajs-779 94 13 a a DET iajs-779 94 14 fully fully ADV iajs-779 94 15 invariant invariant ADJ iajs-779 94 16 submodule submodule NOUN iajs-779 94 17 of of ADP iajs-779 94 18 n n PRON iajs-779 94 19 such such ADJ iajs-779 94 20 that that DET iajs-779 94 21 kkn kkn PROPN iajs-779 94 22 . . PUNCT iajs-779 95 1 then then ADV iajs-779 95 2 k'= k'= VERB iajs-779 95 3 k k PROPN iajs-779 95 4 n n PRON iajs-779 95 5 n n PROPN iajs-779 95 6   ADV iajs-779 95 7 is be AUX iajs-779 95 8 a a DET iajs-779 95 9 fully fully ADV iajs-779 95 10 invariant invariant ADJ iajs-779 95 11 submodule submodule NOUN iajs-779 95 12 of of ADP iajs-779 95 13 m m PROPN iajs-779 95 14 n n X iajs-779 95 15 [ [ X iajs-779 95 16 3 3 NUM iajs-779 95 17 , , PUNCT iajs-779 95 18 lemma lemma X iajs-779 95 19 ( ( PUNCT iajs-779 95 20 1.1.20)(ii 1.1.20)(ii NUM iajs-779 95 21 ) ) PUNCT iajs-779 95 22 ] ] PUNCT iajs-779 95 23 and and CCONJ iajs-779 95 24 k k PROPN iajs-779 95 25 ' ' PART iajs-779 95 26 m/ m/ NOUN iajs-779 95 27 n n PROPN iajs-779 95 28   PROPN iajs-779 95 29 k'=0 k'=0 PROPN iajs-779 95 30 . . PROPN iajs-779 96 1 hence hence ADV iajs-779 96 2 k'=0 k'=0 PROPN iajs-779 96 3 , , PUNCT iajs-779 96 4 that that PRON iajs-779 96 5 is be AUX iajs-779 96 6 k+nn k+nn PROPN iajs-779 96 7 and and CCONJ iajs-779 96 8 so so ADV iajs-779 96 9 kn kn PROPN iajs-779 96 10 . . PUNCT iajs-779 97 1 thus thus ADV iajs-779 97 2 n n PRON iajs-779 97 3 is be AUX iajs-779 97 4 a a DET iajs-779 97 5 fully fully ADV iajs-779 97 6 invariant invariant ADJ iajs-779 97 7 semiprime semiprime NOUN iajs-779 97 8 submodule submodule NOUN iajs-779 97 9 . . PUNCT iajs-779 98 1 as as ADP iajs-779 98 2 an an DET iajs-779 98 3 application application NOUN iajs-779 98 4 of of ADP iajs-779 98 5 proposition proposition NOUN iajs-779 98 6 ( ( PUNCT iajs-779 98 7 1.10 1.10 NUM iajs-779 98 8 ) ) PUNCT iajs-779 98 9 we we PRON iajs-779 98 10 give give VERB iajs-779 98 11 the the DET iajs-779 98 12 following follow VERB iajs-779 98 13 corollary corollary NOUN iajs-779 98 14 . . PUNCT iajs-779 99 1 1.12 1.12 NUM iajs-779 99 2 corollary corollary NOUN iajs-779 99 3 : : PUNCT iajs-779 99 4 let let VERB iajs-779 99 5 :m :m PROPN iajs-779 99 6 m m AUX iajs-779 99 7 ' ' PUNCT iajs-779 99 8 be be AUX iajs-779 99 9 an an DET iajs-779 99 10 epimorphisim epimorphisim NOUN iajs-779 99 11 . . PUNCT iajs-779 100 1 if if SCONJ iajs-779 100 2 ker ker PROPN iajs-779 100 3   PROPN iajs-779 100 4 is be AUX iajs-779 100 5 a a DET iajs-779 100 6 fully fully ADV iajs-779 100 7 semiprime semiprime NOUN iajs-779 100 8 submodule submodule NOUN iajs-779 100 9 , , PUNCT iajs-779 100 10 then then ADV iajs-779 100 11 m m PROPN iajs-779 100 12 ' ' PUNCT iajs-779 100 13 is be AUX iajs-779 100 14 a a DET iajs-779 100 15 fully fully ADV iajs-779 100 16 semiprime semiprime NOUN iajs-779 100 17 module module NOUN iajs-779 100 18 . . PUNCT iajs-779 101 1 proof proof NOUN iajs-779 101 2 : : PUNCT iajs-779 101 3 since since SCONJ iajs-779 101 4   NOUN iajs-779 101 5 is be AUX iajs-779 101 6 an an DET iajs-779 101 7 epimorphisim epimorphisim NOUN iajs-779 101 8 , , PUNCT iajs-779 101 9 then then ADV iajs-779 101 10 m m PROPN iajs-779 101 11 / / SYM iajs-779 101 12 ker ker NOUN iajs-779 101 13 m m NOUN iajs-779 101 14 ' ' PUNCT iajs-779 101 15 . . PUNCT iajs-779 102 1 but but CCONJ iajs-779 102 2 ker ker NOUN iajs-779 102 3 is be AUX iajs-779 102 4 a a DET iajs-779 102 5 fully fully ADV iajs-779 102 6 semiprime semiprime NOUN iajs-779 102 7 submodule submodule NOUN iajs-779 102 8 , , PUNCT iajs-779 102 9 so so ADV iajs-779 102 10 by by ADP iajs-779 102 11 proposition proposition NOUN iajs-779 102 12 ( ( PUNCT iajs-779 102 13 1.10 1.10 NUM iajs-779 102 14 ) ) PUNCT iajs-779 102 15 , , PUNCT iajs-779 102 16 m m PROPN iajs-779 102 17 / / SYM iajs-779 102 18 ker ker NOUN iajs-779 102 19 is be AUX iajs-779 102 20 a a DET iajs-779 102 21 fully fully ADV iajs-779 102 22 semiprime semiprime NOUN iajs-779 102 23 module module NOUN iajs-779 102 24 . . PUNCT iajs-779 103 1 this this PRON iajs-779 103 2 completes complete VERB iajs-779 103 3 the the DET iajs-779 103 4 proof proof NOUN iajs-779 103 5 . . PUNCT iajs-779 104 1 before before SCONJ iajs-779 104 2 we we PRON iajs-779 104 3 give give VERB iajs-779 104 4 the the DET iajs-779 104 5 next next ADJ iajs-779 104 6 result result NOUN iajs-779 104 7 , , PUNCT iajs-779 104 8 we we PRON iajs-779 104 9 introduce introduce VERB iajs-779 104 10 the the DET iajs-779 104 11 following follow VERB iajs-779 104 12 lemma lemma PROPN iajs-779 104 13 . . PROPN iajs-779 105 1 1.13 1.13 NUM iajs-779 105 2 lemma lemma PROPN iajs-779 105 3 : : PUNCT iajs-779 105 4 let let VERB iajs-779 105 5 :mm :mm PRON iajs-779 105 6 ' ' PUNCT iajs-779 105 7 be be AUX iajs-779 105 8 an an DET iajs-779 105 9 isomorphism isomorphism NOUN iajs-779 105 10 , , PUNCT iajs-779 105 11 where where SCONJ iajs-779 105 12 m m VERB iajs-779 105 13 , , PUNCT iajs-779 105 14 m m VERB iajs-779 105 15 ' ' PUNCT iajs-779 105 16 be be VERB iajs-779 105 17 two two NUM iajs-779 105 18 r r NOUN iajs-779 105 19 - - PUNCT iajs-779 105 20 modules module NOUN iajs-779 105 21 , , PUNCT iajs-779 105 22 let let VERB iajs-779 105 23 k k PRON iajs-779 105 24 be be AUX iajs-779 105 25 a a DET iajs-779 105 26 fully fully ADV iajs-779 105 27 invariant invariant ADJ iajs-779 105 28 submodule submodule NOUN iajs-779 105 29 of of ADP iajs-779 105 30 m. m. NOUN iajs-779 105 31 then then ADV iajs-779 105 32 (kk)(k)(k (kk)(k)(k PROPN iajs-779 105 33 ) ) PUNCT iajs-779 105 34 . . PUNCT iajs-779 106 1 proof proof NOUN iajs-779 106 2 : : PUNCT iajs-779 106 3 (kk (kk X iajs-779 106 4 ) ) PUNCT iajs-779 106 5 = = PUNCT iajs-779 107 1 ({f(k):f ({f(k):f PROPN iajs-779 107 2 : : PUNCT iajs-779 107 3 mk mk NOUN iajs-779 107 4 } } PUNCT iajs-779 107 5 ) ) PUNCT iajs-779 108 1 = = SYM iajs-779 108 2 {f(k):f {f(k):f PROPN iajs-779 108 3 : : PUNCT iajs-779 108 4 mk mk NOUN iajs-779 108 5 } } PUNCT iajs-779 108 6 = = SYM iajs-779 108 7 {( {( NUM iajs-779 108 8 f)(k):f f)(k):f ADJ iajs-779 108 9 : : PUNCT iajs-779 108 10 mk mk NOUN iajs-779 108 11 } } PUNCT iajs-779 108 12 . . PUNCT iajs-779 109 1 but but CCONJ iajs-779 109 2   X iajs-779 109 3 1 1 NUM iajs-779 109 4 : : PUNCT iajs-779 109 5 m'm m'm PROPN iajs-779 109 6 is be AUX iajs-779 109 7 an an DET iajs-779 109 8 isomorphism isomorphism NOUN iajs-779 109 9 and and CCONJ iajs-779 109 10 km km NOUN iajs-779 109 11 , , PUNCT iajs-779 109 12 so so SCONJ iajs-779 109 13 there there PRON iajs-779 109 14 exists exist VERB iajs-779 109 15 k'm k'm NOUN iajs-779 109 16 ' ' PUNCT iajs-779 109 17 such such ADJ iajs-779 109 18 that that SCONJ iajs-779 109 19 1(k')=k 1(k')=k PROPN iajs-779 109 20 ; ; PUNCT iajs-779 109 21 that that PRON iajs-779 109 22 is be AUX iajs-779 109 23 (k)=k (k)=k PROPN iajs-779 109 24 ' ' PUNCT iajs-779 109 25 . . PUNCT iajs-779 110 1 hence hence ADV iajs-779 110 2 (kk)={ (kk)={ X iajs-779 110 3 f f NOUN iajs-779 110 4 1(k'):f 1(k'):f NUM iajs-779 110 5 : : PUNCT iajs-779 110 6 mk mk NOUN iajs-779 110 7 } } PUNCT iajs-779 110 8 . . PUNCT iajs-779 111 1 but but CCONJ iajs-779 111 2 1 1 NUM iajs-779 111 3 fm fm NOUN iajs-779 111 4 ' ' NOUN iajs-779 111 5 m m VERB iajs-779 111 6 k k X iajs-779 111 7   PROPN iajs-779 111 8   PROPN iajs-779 111 9   PROPN iajs-779 111 10   PROPN iajs-779 111 11   PROPN iajs-779 111 12   PROPN iajs-779 111 13   INTJ iajs-779 111 14 , , PUNCT iajs-779 111 15 hence hence ADV iajs-779 111 16 ( ( PUNCT iajs-779 111 17   NOUN iajs-779 111 18 f f X iajs-779 111 19   NOUN iajs-779 111 20 1 1 NUM iajs-779 111 21 ): ): PUNCT iajs-779 111 22 m'k'=(k m'k'=(k PROPN iajs-779 111 23 ) ) PUNCT iajs-779 111 24 . . PUNCT iajs-779 112 1 hence hence ADV iajs-779 112 2 (kk)={ (kk)={ X iajs-779 112 3 f f NOUN iajs-779 112 4 1(k 1(k PROPN iajs-779 112 5 ' ' PUNCT iajs-779 112 6 ): ): PUNCT iajs-779 112 7   NOUN iajs-779 112 8 f f X iajs-779 112 9   X iajs-779 112 10 1 1 NUM iajs-779 112 11 : : PUNCT iajs-779 112 12 m'(k)=k m'(k)=k NOUN iajs-779 112 13 ' ' PUNCT iajs-779 112 14 } } PUNCT iajs-779 112 15 . . PUNCT iajs-779 113 1 now now ADV iajs-779 113 2 (k)(k)=k'k'={h(k'):h (k)(k)=k'k'={h(k'):h ADV iajs-779 113 3 : : PUNCT iajs-779 113 4 m'k'=(k m'k'=(k PROPN iajs-779 113 5 ) ) PUNCT iajs-779 113 6 } } PUNCT iajs-779 113 7 . . PUNCT iajs-779 114 1 it it PRON iajs-779 114 2 follows follow VERB iajs-779 114 3 that that SCONJ iajs-779 114 4 (kk)(k)(k (kk)(k)(k PROPN iajs-779 114 5 ) ) PUNCT iajs-779 114 6 . . PUNCT iajs-779 115 1 however however ADV iajs-779 115 2 , , PUNCT iajs-779 115 3 we we PRON iajs-779 115 4 get get VERB iajs-779 115 5 the the DET iajs-779 115 6 following follow VERB iajs-779 115 7 proposition proposition NOUN iajs-779 115 8 . . PUNCT iajs-779 116 1 1.14 1.14 NUM iajs-779 116 2 proposition proposition NOUN iajs-779 116 3 : : PUNCT iajs-779 116 4 if if SCONJ iajs-779 116 5 m m VERB iajs-779 116 6 and and CCONJ iajs-779 116 7 m m NOUN iajs-779 116 8 ' ' PUNCT iajs-779 116 9 are be AUX iajs-779 116 10 two two NUM iajs-779 116 11 isomorphic isomorphic ADJ iajs-779 116 12 r r NOUN iajs-779 116 13 - - PUNCT iajs-779 116 14 modules module NOUN iajs-779 116 15 , , PUNCT iajs-779 116 16 then then ADV iajs-779 116 17 m m PROPN iajs-779 116 18 ' ' PUNCT iajs-779 116 19 is be AUX iajs-779 116 20 fully fully ADV iajs-779 116 21 semiprime semiprime NOUN iajs-779 116 22 module module NOUN iajs-779 116 23 if if SCONJ iajs-779 116 24 and and CCONJ iajs-779 116 25 only only ADV iajs-779 116 26 if if SCONJ iajs-779 116 27 m m NOUN iajs-779 116 28 is be AUX iajs-779 116 29 fully fully ADV iajs-779 116 30 semiprime semiprime NOUN iajs-779 116 31 module module NOUN iajs-779 116 32 . . PUNCT iajs-779 117 1 proof proof NOUN iajs-779 117 2 : : PUNCT iajs-779 117 3 let let VERB iajs-779 117 4 :mm :mm PROPN iajs-779 117 5 ' ' PART iajs-779 117 6 , , PUNCT iajs-779 117 7   PROPN iajs-779 117 8 is be AUX iajs-779 117 9 an an DET iajs-779 117 10 isomorphism isomorphism NOUN iajs-779 117 11 , , PUNCT iajs-779 117 12 and and CCONJ iajs-779 117 13 let let VERB iajs-779 117 14 l l NOUN iajs-779 117 15 be be AUX iajs-779 117 16 a a DET iajs-779 117 17 fully fully ADV iajs-779 117 18 invariant invariant ADJ iajs-779 117 19 submodule submodule NOUN iajs-779 117 20 of of ADP iajs-779 117 21 m m NOUN iajs-779 117 22 ' ' PART iajs-779 117 23 such such ADJ iajs-779 117 24 that that DET iajs-779 117 25 ll=0 ll=0 NOUN iajs-779 117 26 . . PUNCT iajs-779 118 1 to to PART iajs-779 118 2 prove prove VERB iajs-779 118 3 l=0 l=0 PROPN iajs-779 118 4 . . PUNCT iajs-779 119 1 let let VERB iajs-779 119 2 k= k= NOUN iajs-779 119 3 1(l 1(l NUM iajs-779 119 4 ) ) PUNCT iajs-779 119 5 , , PUNCT iajs-779 119 6 that that PRON iajs-779 119 7 is be AUX iajs-779 119 8 (k)=l (k)=l PROPN iajs-779 119 9 . . PUNCT iajs-779 120 1 then then ADV iajs-779 120 2 ll=(k)(k)(kk ll=(k)(k)(kk PROPN iajs-779 120 3 ) ) PUNCT iajs-779 120 4 ( ( PUNCT iajs-779 120 5 by by ADP iajs-779 120 6 lemma lemma PROPN iajs-779 120 7 ( ( PUNCT iajs-779 120 8 1.13 1.13 NUM iajs-779 120 9 ) ) PUNCT iajs-779 120 10 . . PUNCT iajs-779 121 1 but but CCONJ iajs-779 121 2 (k)(k)=0 (k)(k)=0 PROPN iajs-779 121 3 ( ( PUNCT iajs-779 121 4 since since SCONJ iajs-779 121 5 ll=0 ll=0 PROPN iajs-779 121 6 ) ) PUNCT iajs-779 121 7 , , PUNCT iajs-779 121 8 implies imply VERB iajs-779 121 9 (kk)=0 (kk)=0 PROPN iajs-779 121 10 . . PUNCT iajs-779 122 1 since since SCONJ iajs-779 122 2   PROPN iajs-779 122 3 is be AUX iajs-779 122 4 one one NUM iajs-779 122 5 to to ADP iajs-779 122 6 one one NUM iajs-779 122 7 , , PUNCT iajs-779 122 8 we we PRON iajs-779 122 9 have have VERB iajs-779 122 10 kk=0 kk=0 NOUN iajs-779 122 11 and and CCONJ iajs-779 122 12 since since SCONJ iajs-779 122 13 m m PROPN iajs-779 122 14 is be AUX iajs-779 122 15 fully fully ADV iajs-779 122 16 semiprime semiprime NOUN iajs-779 122 17 , , PUNCT iajs-779 122 18 we we PRON iajs-779 122 19 get get VERB iajs-779 122 20 k=0 k=0 PROPN iajs-779 122 21 . . PUNCT iajs-779 123 1 this this PRON iajs-779 123 2 implies imply VERB iajs-779 123 3 (k)=(0)=0 (k)=(0)=0 NOUN iajs-779 123 4 . . PUNCT iajs-779 123 5 therefore therefore ADV iajs-779 123 6 l=(0 l=(0 ADV iajs-779 123 7 ) ) PUNCT iajs-779 123 8 and and CCONJ iajs-779 123 9 hence hence ADV iajs-779 123 10 m m NOUN iajs-779 123 11 ' ' PUNCT iajs-779 123 12 is be AUX iajs-779 123 13 a a DET iajs-779 123 14 fully fully ADV iajs-779 123 15 semiprime semiprime NOUN iajs-779 123 16 module module NOUN iajs-779 123 17 . . PUNCT iajs-779 124 1 1.15 1.15 NUM iajs-779 124 2 proposition proposition NOUN iajs-779 124 3 : : PUNCT iajs-779 124 4 let let VERB iajs-779 124 5 n n PRON iajs-779 124 6 and and CCONJ iajs-779 124 7 k k PROPN iajs-779 124 8 be be AUX iajs-779 124 9 two two NUM iajs-779 124 10 fully fully ADV iajs-779 124 11 semiprime semiprime NOUN iajs-779 124 12 submodules submodule NOUN iajs-779 124 13 of of ADP iajs-779 124 14 an an DET iajs-779 124 15 r r NOUN iajs-779 124 16 - - PUNCT iajs-779 124 17 module module NOUN iajs-779 124 18 m. m. NOUN iajs-779 124 19 then then ADV iajs-779 124 20 nk nk NOUN iajs-779 124 21 is be AUX iajs-779 124 22 a a DET iajs-779 124 23 fully fully ADV iajs-779 124 24 semiprime semiprime NOUN iajs-779 124 25 submodule submodule NOUN iajs-779 124 26 of of ADP iajs-779 124 27 m. m. NOUN iajs-779 124 28 proof proof NOUN iajs-779 124 29 : : PUNCT iajs-779 124 30 let let VERB iajs-779 124 31 l l NOUN iajs-779 124 32 be be AUX iajs-779 124 33 a a DET iajs-779 124 34 fully fully ADV iajs-779 124 35 invariant invariant ADJ iajs-779 124 36 submodule submodule NOUN iajs-779 124 37 of of ADP iajs-779 124 38 m m PRON iajs-779 124 39 such such ADJ iajs-779 124 40 that that DET iajs-779 124 41 llnk llnk PROPN iajs-779 124 42 . . PUNCT iajs-779 125 1 but but CCONJ iajs-779 125 2 nkk nkk PROPN iajs-779 125 3 and and CCONJ iajs-779 125 4 nkn nkn PROPN iajs-779 125 5 . . PUNCT iajs-779 125 6 llk llk NOUN iajs-779 125 7 and and CCONJ iajs-779 125 8 lln lln PROPN iajs-779 125 9 . . PUNCT iajs-779 126 1 thus thus ADV iajs-779 126 2 lk lk PROPN iajs-779 126 3 and and CCONJ iajs-779 126 4 ln ln PROPN iajs-779 126 5 ( ( PUNCT iajs-779 126 6 since since SCONJ iajs-779 126 7 k k PROPN iajs-779 126 8 and and CCONJ iajs-779 126 9 l l NOUN iajs-779 126 10 are be AUX iajs-779 126 11 fully fully ADV iajs-779 126 12 semiprime semiprime NOUN iajs-779 126 13 ) ) PUNCT iajs-779 126 14 . . PUNCT iajs-779 127 1 therefore therefore ADV iajs-779 127 2 lnk lnk ADJ iajs-779 127 3 . . PROPN iajs-779 128 1 hence hence ADV iajs-779 128 2 nk nk NOUN iajs-779 128 3 is be AUX iajs-779 128 4 a a DET iajs-779 128 5 fully fully ADV iajs-779 128 6 semiprime semiprime NOUN iajs-779 128 7 submodule submodule NOUN iajs-779 128 8 of of ADP iajs-779 128 9 m. m. NOUN iajs-779 128 10 by by ADP iajs-779 128 11 using use VERB iajs-779 128 12 the the DET iajs-779 128 13 mathematical mathematical ADJ iajs-779 128 14 induction induction NOUN iajs-779 128 15 , , PUNCT iajs-779 128 16 we we PRON iajs-779 128 17 obtain obtain VERB iajs-779 128 18 the the DET iajs-779 128 19 following follow VERB iajs-779 128 20 result result NOUN iajs-779 128 21 . . PUNCT iajs-779 129 1 1.16 1.16 NUM iajs-779 129 2 corollary corollary NOUN iajs-779 129 3 : : PUNCT iajs-779 129 4 the the DET iajs-779 129 5 intersection intersection NOUN iajs-779 129 6 of of ADP iajs-779 129 7 a a DET iajs-779 129 8 finite finite ADJ iajs-779 129 9 collection collection NOUN iajs-779 129 10 of of ADP iajs-779 129 11 fully fully ADV iajs-779 129 12 semiprime semiprime NOUN iajs-779 129 13 submodules submodule NOUN iajs-779 129 14 of of ADP iajs-779 129 15 an an DET iajs-779 129 16 r r NOUN iajs-779 129 17 - - PUNCT iajs-779 129 18 module module NOUN iajs-779 129 19 is be AUX iajs-779 129 20 a a DET iajs-779 129 21 fully fully ADV iajs-779 129 22 semiprime semiprime NOUN iajs-779 129 23 submodule submodule NOUN iajs-779 129 24 . . PUNCT iajs-779 130 1 1.17 1.17 NUM iajs-779 130 2 proposition proposition NOUN iajs-779 130 3 : : PUNCT iajs-779 130 4 let let VERB iajs-779 130 5 m m PRON iajs-779 130 6 be be AUX iajs-779 130 7 an an DET iajs-779 130 8 r r NOUN iajs-779 130 9 - - PUNCT iajs-779 130 10 module module NOUN iajs-779 130 11 . . PUNCT iajs-779 131 1 then then ADV iajs-779 131 2 m m PROPN iajs-779 131 3 is be AUX iajs-779 131 4 fully fully ADV iajs-779 131 5 semiprime semiprime NOUN iajs-779 131 6 module module NOUN iajs-779 131 7 if if SCONJ iajs-779 131 8 and and CCONJ iajs-779 131 9 only only ADV iajs-779 131 10 if if SCONJ iajs-779 131 11 for for ADP iajs-779 131 12 all all DET iajs-779 131 13 mm mm NUM iajs-779 131 14 , , PUNCT iajs-779 131 15 ( ( PUNCT iajs-779 131 16 m)(m)=(0 m)(m)=(0 NOUN iajs-779 131 17 ) ) PUNCT iajs-779 131 18 , , PUNCT iajs-779 131 19 implies imply VERB iajs-779 131 20 m=0 m=0 PROPN iajs-779 131 21 . . PUNCT iajs-779 132 1 proof proof NOUN iajs-779 132 2 : : PUNCT iajs-779 132 3 ( ( PUNCT iajs-779 132 4   NOUN iajs-779 132 5 ) ) PUNCT iajs-779 132 6 it it PRON iajs-779 132 7 is be AUX iajs-779 132 8 clear clear ADJ iajs-779 132 9 . . PUNCT iajs-779 133 1 ibn ibn PROPN iajs-779 133 2 alhaitham alhaitham PROPN iajs-779 133 3 j. j. PROPN iajs-779 133 4 for for ADP iajs-779 133 5 pure pure ADJ iajs-779 133 6 & & CCONJ iajs-779 133 7 appl appl PROPN iajs-779 133 8 . . PUNCT iajs-779 134 1 sci sci PROPN iajs-779 134 2 . . PUNCT iajs-779 134 3 vol.24 vol.24 NOUN iajs-779 134 4 ( ( PUNCT iajs-779 134 5 2 2 NUM iajs-779 134 6 ) ) PUNCT iajs-779 134 7 2011 2011 NUM iajs-779 134 8 to to PART iajs-779 134 9 prove prove VERB iajs-779 134 10 the the DET iajs-779 134 11 other other ADJ iajs-779 134 12 side side NOUN iajs-779 134 13 , , PUNCT iajs-779 134 14 let let VERB iajs-779 134 15 nn=(0 nn=(0 NOUN iajs-779 134 16 ) ) PUNCT iajs-779 134 17 . . PUNCT iajs-779 135 1 suppose suppose VERB iajs-779 135 2 n(0 n(0 PUNCT iajs-779 135 3 ) ) PUNCT iajs-779 135 4 , , PUNCT iajs-779 135 5 so so CCONJ iajs-779 135 6 there there PRON iajs-779 135 7 exists exist VERB iajs-779 135 8 mn mn PROPN iajs-779 135 9 , , PUNCT iajs-779 135 10 m(0 m(0 PUNCT iajs-779 135 11 ) ) PUNCT iajs-779 135 12 such such ADJ iajs-779 135 13 that that SCONJ iajs-779 135 14 ( ( PUNCT iajs-779 135 15 m)n m)n NUM iajs-779 135 16 , , PUNCT iajs-779 135 17 implies imply VERB iajs-779 135 18 ( ( PUNCT iajs-779 135 19 m)(m)nn=(0 m)(m)nn=(0 PROPN iajs-779 135 20 ) ) PUNCT iajs-779 135 21 . . PUNCT iajs-779 136 1 then then ADV iajs-779 136 2 ( ( PUNCT iajs-779 136 3 m)(m)=(0 m)(m)=(0 NOUN iajs-779 136 4 ) ) PUNCT iajs-779 136 5 . . PUNCT iajs-779 137 1 hence hence ADV iajs-779 137 2 m=0 m=0 PROPN iajs-779 137 3 , , PUNCT iajs-779 137 4 which which PRON iajs-779 137 5 is be AUX iajs-779 137 6 a a DET iajs-779 137 7 contradiction contradiction NOUN iajs-779 137 8 . . PUNCT iajs-779 138 1 thus thus ADV iajs-779 138 2 n=0 n=0 NUM iajs-779 138 3 and and CCONJ iajs-779 138 4 this this PRON iajs-779 138 5 completes complete VERB iajs-779 138 6 the the DET iajs-779 138 7 proof proof NOUN iajs-779 138 8 . . PUNCT iajs-779 139 1 for for ADP iajs-779 139 2 our our PRON iajs-779 139 3 next next ADJ iajs-779 139 4 proposition proposition NOUN iajs-779 139 5 , , PUNCT iajs-779 139 6 the the DET iajs-779 139 7 following follow VERB iajs-779 139 8 lemma lemma PROPN iajs-779 139 9 is be AUX iajs-779 139 10 needed need VERB iajs-779 139 11 . . PUNCT iajs-779 140 1 1.18 1.18 NUM iajs-779 140 2 lemma lemma PROPN iajs-779 140 3 : : PUNCT iajs-779 140 4 if if SCONJ iajs-779 140 5 m m PROPN iajs-779 140 6 1 1 NUM iajs-779 140 7 , , PUNCT iajs-779 140 8 m m VERB iajs-779 140 9 2 2 NUM iajs-779 140 10 are be AUX iajs-779 140 11 two two NUM iajs-779 140 12 r r NOUN iajs-779 140 13 - - PUNCT iajs-779 140 14 modules module NOUN iajs-779 140 15 and and CCONJ iajs-779 140 16 n1 n1 NOUN iajs-779 140 17 , , PUNCT iajs-779 140 18 n2 n2 NOUN iajs-779 140 19 are be AUX iajs-779 140 20 two two NUM iajs-779 140 21 fully fully ADV iajs-779 140 22 invariant invariant ADJ iajs-779 140 23 submodules submodule NOUN iajs-779 140 24 of of ADP iajs-779 140 25 m m PROPN iajs-779 140 26 1 1 NUM iajs-779 140 27 and and CCONJ iajs-779 140 28 m m PROPN iajs-779 140 29 2 2 NUM iajs-779 140 30 respectively respectively ADV iajs-779 140 31 , , PUNCT iajs-779 140 32 then then ADV iajs-779 140 33 ( ( PUNCT iajs-779 140 34 n1n1)(n2n2)(n1n2 n1n1)(n2n2)(n1n2 NOUN iajs-779 140 35 ) ) PUNCT iajs-779 140 36   PROPN iajs-779 141 1 ( ( PUNCT iajs-779 141 2 n1n2 n1n2 PROPN iajs-779 141 3 ) ) PUNCT iajs-779 141 4 . . PUNCT iajs-779 142 1 proof proof NOUN iajs-779 142 2 : : PUNCT iajs-779 142 3 ( ( PUNCT iajs-779 142 4 n1n1)(n2n2 n1n1)(n2n2 NOUN iajs-779 142 5 ) ) PUNCT iajs-779 142 6 = = SYM iajs-779 143 1 {(f(n1),g(n2 {(f(n1),g(n2 X iajs-779 143 2 ) ) PUNCT iajs-779 143 3 ): ): PUNCT iajs-779 144 1 f f X iajs-779 144 2 : : PUNCT iajs-779 144 3 m1n1 m1n1 NOUN iajs-779 144 4 , , PUNCT iajs-779 144 5 g g NOUN iajs-779 144 6 : : PUNCT iajs-779 144 7 m2n2 m2n2 NOUN iajs-779 144 8 } } PUNCT iajs-779 144 9 . . PUNCT iajs-779 145 1 for for ADP iajs-779 145 2 any any DET iajs-779 145 3 f f NOUN iajs-779 145 4 : : PUNCT iajs-779 145 5 m1n1 m1n1 NOUN iajs-779 145 6 , , PUNCT iajs-779 145 7 g g NOUN iajs-779 145 8 : : PUNCT iajs-779 145 9 m2n2 m2n2 NOUN iajs-779 145 10 , , PUNCT iajs-779 145 11 define define VERB iajs-779 145 12 h h NOUN iajs-779 145 13 : : PUNCT iajs-779 145 14 m m VERB iajs-779 145 15 1m 1m NUM iajs-779 145 16 2 2 NUM iajs-779 145 17   NOUN iajs-779 145 18 n1n2 n1n2 NOUN iajs-779 145 19 by by ADP iajs-779 145 20 h(x h(x PROPN iajs-779 145 21 , , PUNCT iajs-779 145 22 y)=(f(x),g(y y)=(f(x),g(y PROPN iajs-779 145 23 ) ) PUNCT iajs-779 145 24 ) ) PUNCT iajs-779 145 25 for for ADP iajs-779 145 26 all all DET iajs-779 145 27 ( ( PUNCT iajs-779 145 28 x x NOUN iajs-779 145 29 , , PUNCT iajs-779 145 30 y) y) NUM iajs-779 145 31 m m PROPN iajs-779 145 32 1m 1m NUM iajs-779 145 33 2 2 NUM iajs-779 145 34 . . PUNCT iajs-779 146 1 it it PRON iajs-779 146 2 is be AUX iajs-779 146 3 clear clear ADJ iajs-779 146 4 that that SCONJ iajs-779 146 5 h h NOUN iajs-779 146 6 is be AUX iajs-779 146 7 well well ADV iajs-779 146 8 defined define VERB iajs-779 146 9 homomorphism homomorphism NOUN iajs-779 146 10 . . PUNCT iajs-779 147 1 now now ADV iajs-779 147 2 , , PUNCT iajs-779 147 3 h(n1n2)=(f(n1),g(n2 h(n1n2)=(f(n1),g(n2 NOUN iajs-779 147 4 ) ) PUNCT iajs-779 147 5 ) ) PUNCT iajs-779 147 6 . . PUNCT iajs-779 148 1 but but CCONJ iajs-779 148 2 ( ( PUNCT iajs-779 148 3 n1n2 n1n2 PROPN iajs-779 148 4 ) ) PUNCT iajs-779 148 5   PROPN iajs-779 148 6 ( ( PUNCT iajs-779 148 7 n1n2)={(n1n2): n1n2)={(n1n2): NOUN iajs-779 148 8 : : PUNCT iajs-779 148 9 m m VERB iajs-779 148 10 1m 1m NUM iajs-779 148 11 2 2 PROPN iajs-779 148 12 n1n2}.it n1n2}.it PRON iajs-779 149 1 follows follow VERB iajs-779 149 2 that that SCONJ iajs-779 149 3 h(n1n2)= h(n1n2)= ADJ iajs-779 149 4   X iajs-779 149 5 ( ( PUNCT iajs-779 149 6 f(n1),g(n2 f(n1),g(n2 X iajs-779 149 7 ) ) PUNCT iajs-779 149 8 ) ) PUNCT iajs-779 149 9 be be AUX iajs-779 149 10 in in ADP iajs-779 149 11 ( ( PUNCT iajs-779 149 12 n1n2 n1n2 NOUN iajs-779 149 13 ) ) PUNCT iajs-779 149 14   PROPN iajs-779 149 15 ( ( PUNCT iajs-779 149 16 n1n2 n1n2 PROPN iajs-779 149 17 ) ) PUNCT iajs-779 149 18 . . PUNCT iajs-779 150 1 thus thus ADV iajs-779 150 2 we we PRON iajs-779 150 3 have have VERB iajs-779 150 4 ( ( PUNCT iajs-779 150 5 n1n1)(n2n2)(n1n2)(n1n2 n1n1)(n2n2)(n1n2)(n1n2 NUM iajs-779 150 6 ) ) PUNCT iajs-779 150 7 . . PUNCT iajs-779 151 1 1.19 1.19 NUM iajs-779 151 2 proposition proposition NOUN iajs-779 151 3 : : PUNCT iajs-779 151 4 let let VERB iajs-779 151 5 m m PROPN iajs-779 151 6 1 1 NUM iajs-779 151 7 , , PUNCT iajs-779 151 8 m m VERB iajs-779 151 9 2 2 NUM iajs-779 151 10 be be VERB iajs-779 151 11 two two NUM iajs-779 151 12 r r NOUN iajs-779 151 13 - - PUNCT iajs-779 151 14 modules module NOUN iajs-779 151 15 and and CCONJ iajs-779 151 16 m m NOUN iajs-779 151 17 = = PROPN iajs-779 151 18 m m PROPN iajs-779 151 19 1m 1m NUM iajs-779 151 20 2 2 NUM iajs-779 151 21 such such ADJ iajs-779 151 22 that that DET iajs-779 151 23 annm annm NOUN iajs-779 151 24 1+annm2 1+annm2 ADJ iajs-779 151 25 = = NOUN iajs-779 151 26 r. r. PROPN iajs-779 151 27 if if SCONJ iajs-779 151 28 n1 n1 PROPN iajs-779 151 29 and and CCONJ iajs-779 151 30 n2 n2 NOUN iajs-779 151 31 are be AUX iajs-779 151 32 fully fully ADV iajs-779 151 33 semiprime semiprime NOUN iajs-779 151 34 r r NOUN iajs-779 151 35 - - PUNCT iajs-779 151 36 submodules submodule NOUN iajs-779 151 37 of of ADP iajs-779 151 38 m m PROPN iajs-779 151 39 1 1 NUM iajs-779 151 40 and and CCONJ iajs-779 151 41 m m PROPN iajs-779 151 42 2 2 NUM iajs-779 151 43 respectively respectively ADV iajs-779 151 44 , , PUNCT iajs-779 151 45 then then ADV iajs-779 151 46 n1n2 n1n2 PROPN iajs-779 151 47 is be AUX iajs-779 151 48 also also ADV iajs-779 151 49 fully fully ADV iajs-779 151 50 semiprime semiprime ADJ iajs-779 151 51 . . PUNCT iajs-779 152 1 proof proof NOUN iajs-779 152 2 : : PUNCT iajs-779 152 3 to to PART iajs-779 152 4 prove prove VERB iajs-779 152 5 n1n2 n1n2 NOUN iajs-779 152 6 is be AUX iajs-779 152 7 fully fully ADV iajs-779 152 8 semiprime semiprime NOUN iajs-779 152 9 , , PUNCT iajs-779 152 10 let let VERB iajs-779 152 11 n n PRON iajs-779 152 12 be be AUX iajs-779 152 13 a a DET iajs-779 152 14 fully fully ADV iajs-779 152 15 invariant invariant ADJ iajs-779 152 16 submodule submodule NOUN iajs-779 152 17 of of ADP iajs-779 152 18 m m PRON iajs-779 152 19 such such ADJ iajs-779 152 20 that that SCONJ iajs-779 152 21 nnn1n2 nnn1n2 PROPN iajs-779 152 22 . . PROPN iajs-779 153 1 but but CCONJ iajs-779 153 2 n n CCONJ iajs-779 153 3 = = PROPN iajs-779 153 4 kl kl NOUN iajs-779 153 5 for for ADP iajs-779 153 6 some some DET iajs-779 153 7 km km NOUN iajs-779 153 8 1 1 NUM iajs-779 153 9 , , PUNCT iajs-779 153 10 lm lm X iajs-779 153 11 2 2 NUM iajs-779 153 12 , , PUNCT iajs-779 153 13 by by ADP iajs-779 153 14 [ [ X iajs-779 153 15 4,theorem 4,theorem NUM iajs-779 153 16 ( ( PUNCT iajs-779 153 17 4.2),ch.1 4.2),ch.1 NOUN iajs-779 153 18 ] ] PUNCT iajs-779 153 19 . . PUNCT iajs-779 154 1 then then ADV iajs-779 154 2 ( ( PUNCT iajs-779 154 3 kl) kl) PROPN iajs-779 154 4 kln1n2 kln1n2 PROPN iajs-779 154 5 . . PUNCT iajs-779 155 1 thus thus ADV iajs-779 155 2 by by ADP iajs-779 155 3 lemma lemma PROPN iajs-779 155 4 ( ( PUNCT iajs-779 155 5 1.15 1.15 NUM iajs-779 155 6 ) ) PUNCT iajs-779 155 7 , , PUNCT iajs-779 155 8 we we PRON iajs-779 155 9 get get VERB iajs-779 155 10 ( ( PUNCT iajs-779 155 11 kk kk PROPN iajs-779 155 12 ) ) PUNCT iajs-779 155 13   PROPN iajs-779 155 14 ( ( PUNCT iajs-779 155 15 ll ll NOUN iajs-779 155 16 ) ) PUNCT iajs-779 155 17   PROPN iajs-779 155 18 n1n2 n1n2 PROPN iajs-779 156 1 so so SCONJ iajs-779 156 2 kkn1 kkn1 PROPN iajs-779 156 3 , , PUNCT iajs-779 156 4 lln2 lln2 PROPN iajs-779 156 5 . . PUNCT iajs-779 157 1 but but CCONJ iajs-779 157 2 n1 n1 PROPN iajs-779 157 3 and and CCONJ iajs-779 157 4 n2 n2 NOUN iajs-779 157 5 are be AUX iajs-779 157 6 fully fully ADV iajs-779 157 7 semiprime semiprime NOUN iajs-779 157 8 , , PUNCT iajs-779 157 9 then then ADV iajs-779 157 10 kn1 kn1 PROPN iajs-779 157 11 , , PUNCT iajs-779 157 12 ln2 ln2 PROPN iajs-779 157 13 . . PUNCT iajs-779 158 1 thus thus ADV iajs-779 158 2 kln1n2 kln1n2 PROPN iajs-779 158 3 and and CCONJ iajs-779 158 4 hence hence ADV iajs-779 158 5 n1n2 n1n2 NOUN iajs-779 158 6 is be AUX iajs-779 158 7 fully fully ADV iajs-779 158 8 semiprime semiprime ADJ iajs-779 158 9 . . PUNCT iajs-779 159 1 the the DET iajs-779 159 2 converse converse NOUN iajs-779 159 3 of of ADP iajs-779 159 4 proposition proposition NOUN iajs-779 159 5 ( ( PUNCT iajs-779 159 6 1.16 1.16 NUM iajs-779 159 7 ) ) PUNCT iajs-779 159 8 holds hold VERB iajs-779 159 9 if if SCONJ iajs-779 159 10 ( ( PUNCT iajs-779 159 11 n1n2 n1n2 NOUN iajs-779 159 12 ) ) PUNCT iajs-779 159 13   PROPN iajs-779 160 1 ( ( PUNCT iajs-779 160 2 n1n2)= n1n2)= INTJ iajs-779 160 3 ( ( PUNCT iajs-779 160 4 n1n1)(n2n2 n1n1)(n2n2 NOUN iajs-779 160 5 ) ) PUNCT iajs-779 160 6 . . PUNCT iajs-779 161 1 1.20 1.20 NUM iajs-779 161 2 proposition proposition NOUN iajs-779 161 3 : : PUNCT iajs-779 161 4 let let VERB iajs-779 161 5 m m PROPN iajs-779 161 6 1 1 NUM iajs-779 161 7 , , PUNCT iajs-779 161 8 m m VERB iajs-779 161 9 2 2 NUM iajs-779 161 10 be be VERB iajs-779 161 11 two two NUM iajs-779 161 12 r r NOUN iajs-779 161 13 - - PUNCT iajs-779 161 14 modules module NOUN iajs-779 161 15 , , PUNCT iajs-779 161 16 let let VERB iajs-779 161 17 ( ( PUNCT iajs-779 161 18 n1n2 n1n2 NOUN iajs-779 161 19 ) ) PUNCT iajs-779 161 20   PROPN iajs-779 161 21 ( ( PUNCT iajs-779 161 22 n1n2)= n1n2)= INTJ iajs-779 161 23 ( ( PUNCT iajs-779 161 24 n1n1)(n2n2 n1n1)(n2n2 NOUN iajs-779 161 25 ) ) PUNCT iajs-779 161 26 for for ADP iajs-779 161 27 each each DET iajs-779 161 28 n1m n1m PROPN iajs-779 161 29 1 1 NUM iajs-779 161 30 , , PUNCT iajs-779 161 31 n2m n2m NOUN iajs-779 161 32 2 2 NUM iajs-779 161 33 . . PUNCT iajs-779 161 34 then then ADV iajs-779 161 35 n1n2 n1n2 PROPN iajs-779 161 36 is be AUX iajs-779 161 37 fully fully ADV iajs-779 161 38 semiprime semiprime NOUN iajs-779 161 39 implies imply VERB iajs-779 161 40 , , PUNCT iajs-779 161 41 n1 n1 NOUN iajs-779 161 42 and and CCONJ iajs-779 161 43 n2 n2 NOUN iajs-779 161 44 are be AUX iajs-779 161 45 fully fully ADV iajs-779 161 46 semiprime semiprime ADJ iajs-779 161 47 . . PUNCT iajs-779 162 1 proof proof NOUN iajs-779 162 2 : : PUNCT iajs-779 162 3 let let VERB iajs-779 162 4 km km PROPN iajs-779 162 5 1 1 NUM iajs-779 162 6 , , PUNCT iajs-779 162 7 lm lm X iajs-779 162 8 2 2 NUM iajs-779 162 9 such such ADJ iajs-779 162 10 that that SCONJ iajs-779 162 11 kkn1 kkn1 PROPN iajs-779 162 12 and and CCONJ iajs-779 162 13 lln2 lln2 PROPN iajs-779 162 14 . . PUNCT iajs-779 163 1 hence hence ADV iajs-779 163 2 ( ( PUNCT iajs-779 163 3 kk)(ll)n1n2 kk)(ll)n1n2 PROPN iajs-779 163 4 . . PUNCT iajs-779 164 1 it it PRON iajs-779 164 2 follows follow VERB iajs-779 164 3 that that PRON iajs-779 164 4 ( ( PUNCT iajs-779 164 5 kl)(kl)n1n2 kl)(kl)n1n2 PROPN iajs-779 164 6 . . PUNCT iajs-779 165 1 since since SCONJ iajs-779 165 2 n1n2 n1n2 PROPN iajs-779 165 3 is be AUX iajs-779 165 4 fully fully ADV iajs-779 165 5 semiprime semiprime NOUN iajs-779 165 6 , , PUNCT iajs-779 165 7 kln1n2 kln1n2 PROPN iajs-779 165 8 . . PUNCT iajs-779 166 1 hence hence ADV iajs-779 166 2 kn1 kn1 PROPN iajs-779 166 3 and and CCONJ iajs-779 166 4 ln2 ln2 PROPN iajs-779 166 5 . . PUNCT iajs-779 167 1 thus thus ADV iajs-779 167 2 k k PROPN iajs-779 167 3 and and CCONJ iajs-779 167 4 l l NOUN iajs-779 167 5 are be AUX iajs-779 167 6 fully fully ADV iajs-779 167 7 semiprime semiprime ADJ iajs-779 167 8 . . PUNCT iajs-779 168 1 1.21 1.21 NUM iajs-779 168 2 proposition proposition NOUN iajs-779 168 3 : : PUNCT iajs-779 168 4 let let VERB iajs-779 168 5 m m PROPN iajs-779 168 6 1 1 NUM iajs-779 168 7 , , PUNCT iajs-779 168 8 m m VERB iajs-779 168 9 2 2 NUM iajs-779 168 10 be be VERB iajs-779 168 11 two two NUM iajs-779 168 12 r r NOUN iajs-779 168 13 - - PUNCT iajs-779 168 14 modules module NOUN iajs-779 168 15 such such ADJ iajs-779 168 16 that that DET iajs-779 168 17 annm1+annm2 annm1+annm2 PROPN iajs-779 168 18 = = NOUN iajs-779 168 19 r. r. PROPN iajs-779 168 20 then then ADV iajs-779 168 21 m m VERB iajs-779 168 22 1m 1m NUM iajs-779 168 23 2 2 NUM iajs-779 168 24 is be AUX iajs-779 168 25 fully fully ADV iajs-779 168 26 semiprime semiprime NOUN iajs-779 168 27 module module NOUN iajs-779 168 28 if if SCONJ iajs-779 168 29 and and CCONJ iajs-779 168 30 only only ADV iajs-779 168 31 if if SCONJ iajs-779 168 32 m1 m1 PROPN iajs-779 168 33 and and CCONJ iajs-779 168 34 m2 m2 PROPN iajs-779 168 35 are be AUX iajs-779 168 36 fully fully ADV iajs-779 168 37 semiprime semiprime NOUN iajs-779 168 38 modules module NOUN iajs-779 168 39 . . PUNCT iajs-779 169 1 proof proof NOUN iajs-779 169 2 : : PUNCT iajs-779 169 3 let let VERB iajs-779 169 4 n n PRON iajs-779 169 5 be be AUX iajs-779 169 6 a a DET iajs-779 169 7 fully fully ADV iajs-779 169 8 invariant invariant ADJ iajs-779 169 9 submodule submodule NOUN iajs-779 169 10 of of ADP iajs-779 169 11 m m PROPN iajs-779 169 12 1m 1m PROPN iajs-779 169 13 2.if 2.if NUM iajs-779 169 14 nn=0 nn=0 PROPN iajs-779 169 15 , , PUNCT iajs-779 169 16 to to PART iajs-779 169 17 prove prove VERB iajs-779 169 18 n=0 n=0 NUM iajs-779 169 19 . . PUNCT iajs-779 170 1 since since SCONJ iajs-779 170 2 n= n= ADJ iajs-779 170 3 n1n2 n1n2 PROPN iajs-779 171 1 [ [ X iajs-779 171 2 4,theorem 4,theorem NUM iajs-779 171 3 ( ( PUNCT iajs-779 171 4 2.4 2.4 NUM iajs-779 171 5 ) ) PUNCT iajs-779 171 6 ] ] PUNCT iajs-779 171 7 , , PUNCT iajs-779 171 8 then then ADV iajs-779 171 9 nn=(n1n2 nn=(n1n2 NOUN iajs-779 171 10 ) ) PUNCT iajs-779 171 11   PROPN iajs-779 171 12 ( ( PUNCT iajs-779 171 13 n1n2 n1n2 PROPN iajs-779 171 14 ) ) PUNCT iajs-779 171 15 . . PUNCT iajs-779 172 1 by by ADP iajs-779 172 2 lemma lemma PROPN iajs-779 172 3 ( ( PUNCT iajs-779 172 4 1.15 1.15 NUM iajs-779 172 5 ) ) PUNCT iajs-779 172 6 . . PUNCT iajs-779 173 1 ( ( PUNCT iajs-779 173 2 n1n1)(n2n2)(n1n2 n1n1)(n2n2)(n1n2 NOUN iajs-779 173 3 ) ) PUNCT iajs-779 173 4   PROPN iajs-779 173 5 ( ( PUNCT iajs-779 173 6 n1n2)=(0)(0 n1n2)=(0)(0 PROPN iajs-779 173 7 ) ) PUNCT iajs-779 173 8 . . PUNCT iajs-779 174 1 then then ADV iajs-779 174 2 ( ( PUNCT iajs-779 174 3 n1n1)(n2n2)=(0)(0 n1n1)(n2n2)=(0)(0 ADV iajs-779 174 4 ) ) PUNCT iajs-779 174 5 , , PUNCT iajs-779 174 6 implies imply VERB iajs-779 174 7 n1n1=(0 n1n1=(0 PROPN iajs-779 174 8 ) ) PUNCT iajs-779 174 9 and and CCONJ iajs-779 174 10 n2n2=(0 n2n2=(0 PROPN iajs-779 174 11 ) ) PUNCT iajs-779 174 12 . . PUNCT iajs-779 175 1 therefore therefore ADV iajs-779 175 2 n1=(0 n1=(0 ADJ iajs-779 175 3 ) ) PUNCT iajs-779 175 4 and and CCONJ iajs-779 175 5 n2=(0 n2=(0 ADJ iajs-779 175 6 ) ) PUNCT iajs-779 175 7 ( ( PUNCT iajs-779 175 8 because because SCONJ iajs-779 175 9 m m PROPN iajs-779 175 10 1 1 NUM iajs-779 175 11 , , PUNCT iajs-779 175 12 m m VERB iajs-779 175 13 2 2 NUM iajs-779 175 14 are be AUX iajs-779 175 15 fully fully ADV iajs-779 175 16 semiprime semiprime NOUN iajs-779 175 17 ) ) PUNCT iajs-779 175 18 . . PUNCT iajs-779 176 1 thus thus ADV iajs-779 176 2 n1n2=(0)(0 n1n2=(0)(0 NOUN iajs-779 176 3 ) ) PUNCT iajs-779 176 4 and and CCONJ iajs-779 176 5 hence hence ADV iajs-779 176 6 m m VERB iajs-779 176 7 is be AUX iajs-779 176 8 fully fully ADV iajs-779 176 9 semiprime semiprime NOUN iajs-779 176 10 . . PUNCT iajs-779 177 1 conversely conversely ADV iajs-779 177 2 , , PUNCT iajs-779 177 3 suppose suppose VERB iajs-779 177 4 that that SCONJ iajs-779 177 5 m m VERB iajs-779 177 6 1m 1m NUM iajs-779 177 7 2 2 NUM iajs-779 177 8 is be AUX iajs-779 177 9 a a DET iajs-779 177 10 fully fully ADV iajs-779 177 11 semiprime semiprime NOUN iajs-779 177 12 module module NOUN iajs-779 177 13 , , PUNCT iajs-779 177 14 let let VERB iajs-779 177 15 n1 n1 PROPN iajs-779 177 16 be be AUX iajs-779 177 17 a a DET iajs-779 177 18 fully fully ADV iajs-779 177 19 invariant invariant ADJ iajs-779 177 20 submodule submodule NOUN iajs-779 177 21 of of ADP iajs-779 177 22 m m PROPN iajs-779 177 23 1 1 NUM iajs-779 177 24 such such ADJ iajs-779 177 25 that that SCONJ iajs-779 177 26 n1n1=(0 n1n1=(0 PROPN iajs-779 177 27 ) ) PUNCT iajs-779 177 28 . . PUNCT iajs-779 178 1 we we PRON iajs-779 178 2 can can AUX iajs-779 178 3 show show VERB iajs-779 178 4 that that SCONJ iajs-779 178 5 there there PRON iajs-779 178 6 is be VERB iajs-779 178 7 one one NUM iajs-779 178 8 to to ADP iajs-779 178 9 one one NUM iajs-779 178 10 correspondence correspondence NOUN iajs-779 178 11 between between ADP iajs-779 178 12 n1n1 n1n1 PROPN iajs-779 178 13 and and CCONJ iajs-779 178 14 ( ( PUNCT iajs-779 178 15 n1(0))(n1(0 n1(0))(n1(0 ADJ iajs-779 178 16 ) ) PUNCT iajs-779 178 17 ) ) PUNCT iajs-779 178 18 as as SCONJ iajs-779 178 19 follows follow VERB iajs-779 178 20 . . PUNCT iajs-779 179 1 for for ADP iajs-779 179 2 any any DET iajs-779 179 3 f f NOUN iajs-779 179 4 : : PUNCT iajs-779 179 5 m1n1 m1n1 X iajs-779 179 6 , , PUNCT iajs-779 179 7 f(n1) f(n1) PROPN iajs-779 179 8 n1n1 n1n1 PROPN iajs-779 179 9 , , PUNCT iajs-779 179 10 f f PROPN iajs-779 179 11 can can AUX iajs-779 179 12 be be AUX iajs-779 179 13 extended extend VERB iajs-779 179 14 to to ADP iajs-779 179 15 f f PROPN iajs-779 179 16 : : PUNCT iajs-779 179 17 m1m m1m PROPN iajs-779 179 18 2 2 NUM iajs-779 179 19 n1(0 n1(0 PROPN iajs-779 179 20 ) ) PUNCT iajs-779 179 21 by by ADP iajs-779 179 22 f f PROPN iajs-779 179 23 ( ( PUNCT iajs-779 179 24 m1,m2)=(f(m1),0 m1,m2)=(f(m1),0 NOUN iajs-779 179 25 ) ) PUNCT iajs-779 179 26 for for ADP iajs-779 179 27 all all DET iajs-779 179 28 ( ( PUNCT iajs-779 179 29 m1,m2) m1,m2) PROPN iajs-779 179 30 m m NOUN iajs-779 179 31 1m 1m NUM iajs-779 179 32 2 2 NUM iajs-779 180 1 it it PRON iajs-779 180 2 is be AUX iajs-779 180 3 clear clear ADJ iajs-779 180 4 that that SCONJ iajs-779 180 5 f f PROPN iajs-779 180 6 ( ( PUNCT iajs-779 180 7 n1(0))=f(n1)(0 n1(0))=f(n1)(0 NUM iajs-779 180 8 ) ) PUNCT iajs-779 180 9 , , PUNCT iajs-779 180 10 hence hence ADV iajs-779 180 11 if if SCONJ iajs-779 180 12 f(n1)=0 f(n1)=0 PROPN iajs-779 180 13 , , PUNCT iajs-779 180 14 then then ADV iajs-779 180 15 f f PUNCT iajs-779 180 16 ( ( PUNCT iajs-779 180 17 n1(0))=(0 n1(0))=(0 ADJ iajs-779 180 18 ) ) PUNCT iajs-779 180 19 . . PUNCT iajs-779 181 1 similarly similarly ADV iajs-779 181 2 , , PUNCT iajs-779 181 3 if if SCONJ iajs-779 181 4 g g NOUN iajs-779 181 5 : : PUNCT iajs-779 181 6 m m VERB iajs-779 181 7 1m 1m NUM iajs-779 181 8 2 2 NUM iajs-779 181 9 n1(0 n1(0 ADJ iajs-779 181 10 ) ) PUNCT iajs-779 181 11 , , PUNCT iajs-779 181 12 g(n1(0))(n1(0))(n1(0 g(n1(0))(n1(0))(n1(0 ADJ iajs-779 181 13 ) ) PUNCT iajs-779 181 14 ) ) PUNCT iajs-779 181 15 , , PUNCT iajs-779 181 16 then then ADV iajs-779 181 17 we we PRON iajs-779 181 18 define define VERB iajs-779 181 19 g g PROPN iajs-779 181 20 : : PUNCT iajs-779 181 21 m1n1 m1n1 NOUN iajs-779 181 22 by by ADP iajs-779 181 23 g g PROPN iajs-779 181 24 ( ( PUNCT iajs-779 181 25 m1)=g(m1,0 m1)=g(m1,0 NUM iajs-779 181 26 ) ) PUNCT iajs-779 181 27 , , PUNCT iajs-779 181 28   NOUN iajs-779 181 29 mm mm NOUN iajs-779 181 30 1 1 NUM iajs-779 181 31 , , PUNCT iajs-779 181 32 hence hence ADV iajs-779 181 33 g g NUM iajs-779 181 34 ( ( PUNCT iajs-779 181 35 n1)=g(n1(0 n1)=g(n1(0 NOUN iajs-779 181 36 ) ) PUNCT iajs-779 181 37 ) ) PUNCT iajs-779 182 1 and and CCONJ iajs-779 182 2 so so ADV iajs-779 182 3 g g PROPN iajs-779 182 4 ( ( PUNCT iajs-779 182 5 n1) n1) PROPN iajs-779 182 6 n1n1 n1n1 PROPN iajs-779 182 7 . . PUNCT iajs-779 183 1 thus thus ADV iajs-779 183 2 n1n1=0 n1n1=0 ADJ iajs-779 183 3   X iajs-779 183 4 ( ( PUNCT iajs-779 183 5 n1(0))(n1(0))=0 n1(0))(n1(0))=0 PROPN iajs-779 183 6 . . PUNCT iajs-779 184 1 it it PRON iajs-779 184 2 follows follow VERB iajs-779 184 3 that that SCONJ iajs-779 184 4 ( ( PUNCT iajs-779 184 5 n1(0))(n1(0))=0 n1(0))(n1(0))=0 PROPN iajs-779 184 6 , , PUNCT iajs-779 184 7 and and CCONJ iajs-779 184 8 hence hence ADV iajs-779 184 9 n1(0)=0 n1(0)=0 VERB iajs-779 184 10 . . PUNCT iajs-779 185 1 thus thus ADV iajs-779 185 2 n1=0 n1=0 PROPN iajs-779 185 3 . . PUNCT iajs-779 186 1 similarly similarly ADV iajs-779 186 2 if if SCONJ iajs-779 186 3 n2 n2 ADJ iajs-779 186 4 is be AUX iajs-779 186 5 an an DET iajs-779 186 6 invariant invariant ADJ iajs-779 186 7 submodule submodule NOUN iajs-779 186 8 of of ADP iajs-779 186 9 m m PROPN iajs-779 186 10 2 2 NUM iajs-779 186 11 such such ADJ iajs-779 186 12 that that SCONJ iajs-779 186 13 , , PUNCT iajs-779 186 14 n2n2=0 n2n2=0 PROPN iajs-779 186 15 , , PUNCT iajs-779 186 16 implies imply VERB iajs-779 186 17 n2=(0 n2=(0 NOUN iajs-779 186 18 ) ) PUNCT iajs-779 186 19 and and CCONJ iajs-779 186 20 hence hence ADV iajs-779 186 21 m2 m2 PROPN iajs-779 186 22 is be AUX iajs-779 186 23 fully fully ADV iajs-779 186 24 semiprime semiprime ADJ iajs-779 186 25 . . PUNCT iajs-779 187 1 next next ADV iajs-779 187 2 , , PUNCT iajs-779 187 3 we we PRON iajs-779 187 4 prove prove VERB iajs-779 187 5 the the DET iajs-779 187 6 following following NOUN iajs-779 187 7 . . PUNCT iajs-779 188 1 ibn ibn PROPN iajs-779 188 2 alhaitham alhaitham PROPN iajs-779 188 3 j. j. PROPN iajs-779 188 4 for for ADP iajs-779 188 5 pure pure ADJ iajs-779 188 6 & & CCONJ iajs-779 188 7 appl appl PROPN iajs-779 188 8 . . PUNCT iajs-779 189 1 sci sci PROPN iajs-779 189 2 . . PUNCT iajs-779 189 3 vol.24 vol.24 NOUN iajs-779 189 4 ( ( PUNCT iajs-779 189 5 2 2 NUM iajs-779 189 6 ) ) PUNCT iajs-779 189 7 2011 2011 NUM iajs-779 189 8 1.22 1.22 NUM iajs-779 189 9 proposition proposition NOUN iajs-779 189 10 : : PUNCT iajs-779 189 11 let let VERB iajs-779 189 12 m m VERB iajs-779 189 13 = = NOUN iajs-779 189 14 m m PROPN iajs-779 189 15 1m 1m NUM iajs-779 189 16 2 2 NUM iajs-779 189 17 be be AUX iajs-779 189 18 a a DET iajs-779 189 19 direct direct ADJ iajs-779 189 20 sum sum NOUN iajs-779 189 21 of of ADP iajs-779 189 22 two two NUM iajs-779 189 23 r r NOUN iajs-779 189 24 - - PUNCT iajs-779 189 25 modules module NOUN iajs-779 189 26 m m NOUN iajs-779 189 27 1 1 NUM iajs-779 189 28 and and CCONJ iajs-779 189 29 m m PROPN iajs-779 189 30 2 2 NUM iajs-779 189 31 such such ADJ iajs-779 189 32 that that DET iajs-779 189 33 annm1+annm2 annm1+annm2 PROPN iajs-779 189 34 = = NOUN iajs-779 189 35 r. r. PROPN iajs-779 189 36 if if SCONJ iajs-779 189 37 l1 l1 PROPN iajs-779 189 38 is be AUX iajs-779 189 39 a a DET iajs-779 189 40 fully fully ADV iajs-779 189 41 semiprime semiprime NOUN iajs-779 189 42 submodule submodule NOUN iajs-779 189 43 of of ADP iajs-779 189 44 m m PROPN iajs-779 189 45 1 1 NUM iajs-779 189 46 . . PUNCT iajs-779 190 1 then then ADV iajs-779 190 2 l1m l1m PROPN iajs-779 190 3 2 2 NUM iajs-779 190 4 is be AUX iajs-779 190 5 a a DET iajs-779 190 6 fully fully ADV iajs-779 190 7 semiprime semiprime NOUN iajs-779 190 8 submodule submodule NOUN iajs-779 190 9 of of ADP iajs-779 190 10 m. m. NOUN iajs-779 190 11 proof proof NOUN iajs-779 190 12 : : PUNCT iajs-779 190 13 let let VERB iajs-779 190 14 n n PRON iajs-779 190 15 be be AUX iajs-779 190 16 a a DET iajs-779 190 17 fully fully ADV iajs-779 190 18 invariant invariant ADJ iajs-779 190 19 submodule submodule NOUN iajs-779 190 20 of of ADP iajs-779 190 21 m= m= X iajs-779 190 22 m m PROPN iajs-779 190 23 1m 1m NUM iajs-779 190 24 2 2 NUM iajs-779 190 25 such such ADJ iajs-779 190 26 that that PRON iajs-779 190 27 nnlm nnlm PROPN iajs-779 190 28 2 2 NUM iajs-779 190 29 . . PUNCT iajs-779 190 30 by by ADP iajs-779 190 31 [ [ X iajs-779 190 32 4,theorem 4,theorem NUM iajs-779 190 33 ( ( PUNCT iajs-779 190 34 4.2),ch.1 4.2),ch.1 NOUN iajs-779 190 35 ] ] PUNCT iajs-779 190 36 , , PUNCT iajs-779 190 37 there there PRON iajs-779 190 38 exists exist VERB iajs-779 190 39 n1m n1m PROPN iajs-779 190 40 1 1 NUM iajs-779 190 41 , , PUNCT iajs-779 190 42 n2m n2m PROPN iajs-779 190 43 2 2 NUM iajs-779 190 44 such such ADJ iajs-779 190 45 that that SCONJ iajs-779 190 46 n n NOUN iajs-779 190 47 = = NOUN iajs-779 190 48 n1n2 n1n2 NOUN iajs-779 190 49 . . PUNCT iajs-779 191 1 hence hence ADV iajs-779 191 2 ( ( PUNCT iajs-779 191 3 n1n2 n1n2 PROPN iajs-779 191 4 ) ) PUNCT iajs-779 191 5   PROPN iajs-779 192 1 ( ( PUNCT iajs-779 192 2 n1n2)l1m n1n2)l1m NOUN iajs-779 192 3 2 2 NUM iajs-779 192 4 . . PUNCT iajs-779 193 1 but but CCONJ iajs-779 193 2 ( ( PUNCT iajs-779 193 3 n1n1)(n2n2)(n1n2 n1n1)(n2n2)(n1n2 NOUN iajs-779 193 4 ) ) PUNCT iajs-779 193 5   PROPN iajs-779 193 6 ( ( PUNCT iajs-779 193 7 n1n2 n1n2 PROPN iajs-779 193 8 ) ) PUNCT iajs-779 193 9 by by ADP iajs-779 193 10 lemma lemma PROPN iajs-779 193 11 ( ( PUNCT iajs-779 193 12 1.15 1.15 NUM iajs-779 193 13 ) ) PUNCT iajs-779 193 14 . . PUNCT iajs-779 194 1 therefore therefore ADV iajs-779 194 2 ( ( PUNCT iajs-779 194 3 n1n1)(n2n2)l1m n1n1)(n2n2)l1m PROPN iajs-779 194 4 2 2 NUM iajs-779 194 5 . . PUNCT iajs-779 195 1 it it PRON iajs-779 195 2 is be AUX iajs-779 195 3 clear clear ADJ iajs-779 195 4 that that SCONJ iajs-779 195 5 n1n1l1 n1n1l1 NOUN iajs-779 195 6 , , PUNCT iajs-779 195 7 but but CCONJ iajs-779 195 8 l1 l1 PROPN iajs-779 195 9 is be AUX iajs-779 195 10 fully fully ADV iajs-779 195 11 semiprime semiprime NOUN iajs-779 195 12 submodule submodule NOUN iajs-779 195 13 , , PUNCT iajs-779 195 14 then then ADV iajs-779 195 15 n1l1 n1l1 NOUN iajs-779 195 16 . . PUNCT iajs-779 196 1 thus thus ADV iajs-779 196 2 n1n2l1m n1n2l1m ADP iajs-779 196 3 2 2 NUM iajs-779 196 4 . . PUNCT iajs-779 197 1 therefore therefore ADV iajs-779 197 2 l1m l1m PROPN iajs-779 197 3 2 2 NUM iajs-779 197 4 is be AUX iajs-779 197 5 fully fully ADV iajs-779 197 6 semiprime semiprime NOUN iajs-779 197 7 submodule submodule NOUN iajs-779 197 8 of of ADP iajs-779 197 9 m. m. NOUN iajs-779 198 1 2the 2the NUM iajs-779 198 2 relationships relationship NOUN iajs-779 198 3 between between ADP iajs-779 198 4 fully fully ADV iajs-779 198 5 semiprime semiprime NOUN iajs-779 198 6 modules module NOUN iajs-779 198 7 and and CCONJ iajs-779 198 8 certain certain ADJ iajs-779 198 9 types type NOUN iajs-779 198 10 of of ADP iajs-779 198 11 modules module NOUN iajs-779 198 12 in in ADP iajs-779 198 13 this this DET iajs-779 198 14 section section NOUN iajs-779 198 15 , , PUNCT iajs-779 198 16 we we PRON iajs-779 198 17 establishe establishe VERB iajs-779 198 18 some some DET iajs-779 198 19 relationships relationship NOUN iajs-779 198 20 between between ADP iajs-779 198 21 fully fully ADV iajs-779 198 22 semiprime semiprime NOUN iajs-779 198 23 submodules submodule NOUN iajs-779 198 24 and and CCONJ iajs-779 198 25 some some DET iajs-779 198 26 type type NOUN iajs-779 198 27 of of ADP iajs-779 198 28 modules module NOUN iajs-779 198 29 . . PUNCT iajs-779 199 1 recall recall VERB iajs-779 199 2 that that SCONJ iajs-779 199 3 an an DET iajs-779 199 4 r r NOUN iajs-779 199 5 - - PUNCT iajs-779 199 6 module module NOUN iajs-779 199 7 m m NOUN iajs-779 199 8 is be AUX iajs-779 199 9 said say VERB iajs-779 199 10 to to PART iajs-779 199 11 be be AUX iajs-779 199 12 uniform uniform ADJ iajs-779 199 13 module module NOUN iajs-779 199 14 if if SCONJ iajs-779 199 15 every every DET iajs-779 199 16 non non ADJ iajs-779 199 17 - - ADJ iajs-779 199 18 zero zero NUM iajs-779 199 19 submodule submodule NOUN iajs-779 199 20 of of ADP iajs-779 199 21 m m PROPN iajs-779 199 22 is be AUX iajs-779 199 23 essential essential ADJ iajs-779 199 24 see see NOUN iajs-779 199 25 [ [ X iajs-779 199 26 7 7 NUM iajs-779 199 27 ] ] PUNCT iajs-779 199 28 , , PUNCT iajs-779 199 29 where where SCONJ iajs-779 199 30 a a DET iajs-779 199 31 submodule submodule NOUN iajs-779 199 32 n n PROPN iajs-779 199 33 of of ADP iajs-779 199 34 an an DET iajs-779 199 35 r r NOUN iajs-779 199 36 - - PUNCT iajs-779 199 37 module module NOUN iajs-779 199 38 m m NOUN iajs-779 199 39 is be AUX iajs-779 199 40 essential essential ADJ iajs-779 199 41 provided provide VERB iajs-779 199 42 that that SCONJ iajs-779 199 43 nk0 nk0 PROPN iajs-779 199 44 for for ADP iajs-779 199 45 every every DET iajs-779 199 46 non non ADJ iajs-779 199 47 - - ADJ iajs-779 199 48 zero zero NUM iajs-779 199 49 submodule submodule NOUN iajs-779 199 50 k k PROPN iajs-779 199 51 of of ADP iajs-779 199 52 m m PROPN iajs-779 199 53 , , PUNCT iajs-779 199 54 see see VERB iajs-779 199 55 [ [ X iajs-779 199 56 7 7 NUM iajs-779 199 57 ] ] PUNCT iajs-779 199 58 . . PUNCT iajs-779 200 1 hence hence ADV iajs-779 200 2 , , PUNCT iajs-779 200 3 we we PRON iajs-779 200 4 have have VERB iajs-779 200 5 the the DET iajs-779 200 6 following follow VERB iajs-779 200 7 proposition proposition NOUN iajs-779 200 8 . . PUNCT iajs-779 201 1 2.1 2.1 NUM iajs-779 201 2 proposition proposition NOUN iajs-779 201 3 : : PUNCT iajs-779 201 4 let let VERB iajs-779 201 5 m m PRON iajs-779 201 6 be be AUX iajs-779 201 7 a a DET iajs-779 201 8 uniform uniform ADJ iajs-779 201 9 r r NOUN iajs-779 201 10 - - PUNCT iajs-779 201 11 module module NOUN iajs-779 201 12 . . PUNCT iajs-779 202 1 then then ADV iajs-779 202 2 m m PROPN iajs-779 202 3 is be AUX iajs-779 202 4 a a DET iajs-779 202 5 fully fully ADV iajs-779 202 6 semiprime semiprime NOUN iajs-779 202 7 module module NOUN iajs-779 202 8 if if SCONJ iajs-779 202 9 and and CCONJ iajs-779 202 10 only only ADV iajs-779 202 11 if if SCONJ iajs-779 202 12 m m NOUN iajs-779 202 13 is be AUX iajs-779 202 14 a a DET iajs-779 202 15 fully fully ADV iajs-779 202 16 prime prime ADJ iajs-779 202 17 module module NOUN iajs-779 202 18 . . PUNCT iajs-779 203 1 proof proof NOUN iajs-779 203 2 : : PUNCT iajs-779 203 3 suppose suppose VERB iajs-779 203 4 that that SCONJ iajs-779 203 5 m m PROPN iajs-779 203 6 is be AUX iajs-779 203 7 a a DET iajs-779 203 8 fully fully ADV iajs-779 203 9 semiprime semiprime NOUN iajs-779 203 10 r r NOUN iajs-779 203 11 - - NOUN iajs-779 203 12 module module NOUN iajs-779 203 13 , , PUNCT iajs-779 203 14 let let VERB iajs-779 203 15 k k PRON iajs-779 203 16 , , PUNCT iajs-779 203 17 l l NOUN iajs-779 203 18 be be VERB iajs-779 203 19 two two NUM iajs-779 203 20 fully fully ADV iajs-779 203 21 invariant invariant ADJ iajs-779 203 22 submodules submodule NOUN iajs-779 203 23 of of ADP iajs-779 203 24 m m NOUN iajs-779 203 25 such such ADJ iajs-779 203 26 that that SCONJ iajs-779 203 27 kl=(0 kl=(0 PROPN iajs-779 203 28 ) ) PUNCT iajs-779 203 29 . . PUNCT iajs-779 204 1 assume assume VERB iajs-779 204 2 that that SCONJ iajs-779 204 3 k(0 k(0 X iajs-779 204 4 ) ) PUNCT iajs-779 204 5 , , PUNCT iajs-779 204 6 l(0 l(0 PROPN iajs-779 204 7 ) ) PUNCT iajs-779 204 8 . . PUNCT iajs-779 205 1 then then ADV iajs-779 205 2 kl(0 kl(0 PROPN iajs-779 205 3 ) ) PUNCT iajs-779 205 4 ( ( PUNCT iajs-779 205 5 since since SCONJ iajs-779 205 6 m m PROPN iajs-779 205 7 is be AUX iajs-779 205 8 uniform uniform ADJ iajs-779 205 9 ) ) PUNCT iajs-779 205 10 . . PUNCT iajs-779 206 1 on on ADP iajs-779 206 2 the the DET iajs-779 206 3 other other ADJ iajs-779 206 4 hand hand NOUN iajs-779 206 5 klk klk PROPN iajs-779 206 6 , , PUNCT iajs-779 206 7 kll kll PROPN iajs-779 206 8 . . PUNCT iajs-779 207 1 also also ADV iajs-779 207 2 , , PUNCT iajs-779 207 3 note note VERB iajs-779 207 4 that that SCONJ iajs-779 207 5 k k NOUN iajs-779 207 6 , , PUNCT iajs-779 207 7 l l NOUN iajs-779 207 8 are be AUX iajs-779 207 9 fully fully ADV iajs-779 207 10 invariant invariant ADJ iajs-779 207 11 , , PUNCT iajs-779 207 12 implies imply VERB iajs-779 207 13 kl kl PROPN iajs-779 207 14 is be AUX iajs-779 207 15 fully fully ADV iajs-779 207 16 invariant invariant ADJ iajs-779 207 17 . . PUNCT iajs-779 208 1 then then ADV iajs-779 208 2 ( ( PUNCT iajs-779 208 3 kl)(kl)kl=(0 kl)(kl)kl=(0 NOUN iajs-779 208 4 ) ) PUNCT iajs-779 208 5 . . PUNCT iajs-779 209 1 thus thus ADV iajs-779 209 2 ( ( PUNCT iajs-779 209 3 kl)(kl)(0 kl)(kl)(0 X iajs-779 209 4 ) ) PUNCT iajs-779 209 5 . . PUNCT iajs-779 210 1 but but CCONJ iajs-779 210 2 m m PROPN iajs-779 210 3 is be AUX iajs-779 210 4 fully fully ADV iajs-779 210 5 semiprime semiprime ADJ iajs-779 210 6 so so ADV iajs-779 210 7 kl=(0 kl=(0 NOUN iajs-779 210 8 ) ) PUNCT iajs-779 210 9 which which PRON iajs-779 210 10 is be AUX iajs-779 210 11 a a DET iajs-779 210 12 contradiction contradiction NOUN iajs-779 210 13 . . PUNCT iajs-779 211 1 thus thus ADV iajs-779 211 2 either either CCONJ iajs-779 211 3 k=(0 k=(0 NOUN iajs-779 211 4 ) ) PUNCT iajs-779 211 5 or or CCONJ iajs-779 211 6 l=(0 l=(0 ADJ iajs-779 211 7 ) ) PUNCT iajs-779 211 8 . . PUNCT iajs-779 212 1 then then ADV iajs-779 212 2 m m PROPN iajs-779 212 3 is be AUX iajs-779 212 4 fully fully ADV iajs-779 212 5 prime prime ADJ iajs-779 212 6 . . PUNCT iajs-779 213 1 the the DET iajs-779 213 2 converse converse NOUN iajs-779 213 3 is be AUX iajs-779 213 4 obvious obvious ADJ iajs-779 213 5 . . PUNCT iajs-779 214 1 the the DET iajs-779 214 2 following follow VERB iajs-779 214 3 results result NOUN iajs-779 214 4 are be AUX iajs-779 214 5 consequences consequence NOUN iajs-779 214 6 of of ADP iajs-779 214 7 proposition proposition NOUN iajs-779 214 8 ( ( PUNCT iajs-779 214 9 2.1 2.1 NUM iajs-779 214 10 ) ) PUNCT iajs-779 214 11 , , PUNCT iajs-779 214 12 but but CCONJ iajs-779 214 13 first first ADV iajs-779 214 14 we we PRON iajs-779 214 15 need need VERB iajs-779 214 16 to to PART iajs-779 214 17 recall recall VERB iajs-779 214 18 the the DET iajs-779 214 19 following follow VERB iajs-779 214 20 definition definition NOUN iajs-779 214 21 . . PUNCT iajs-779 215 1 an an DET iajs-779 215 2 r r NOUN iajs-779 215 3 - - PUNCT iajs-779 215 4 module module NOUN iajs-779 215 5 m m NOUN iajs-779 215 6 is be AUX iajs-779 215 7 said say VERB iajs-779 215 8 to to PART iajs-779 215 9 be be AUX iajs-779 215 10 chained chain VERB iajs-779 215 11 module module NOUN iajs-779 215 12 if if SCONJ iajs-779 215 13 and and CCONJ iajs-779 215 14 only only ADV iajs-779 215 15 if if SCONJ iajs-779 215 16 every every DET iajs-779 215 17 non non ADJ iajs-779 215 18 - - ADJ iajs-779 215 19 empty empty ADJ iajs-779 215 20 set set NOUN iajs-779 215 21 of of ADP iajs-779 215 22 submodule submodule NOUN iajs-779 215 23 of of ADP iajs-779 215 24 m m PROPN iajs-779 215 25 is be AUX iajs-779 215 26 ordered order VERB iajs-779 215 27 by by ADP iajs-779 215 28 inclusion inclusion NOUN iajs-779 215 29 , , PUNCT iajs-779 216 1 [ [ X iajs-779 216 2 8 8 NUM iajs-779 216 3 ] ] PUNCT iajs-779 216 4 . . PUNCT iajs-779 217 1 hence hence ADV iajs-779 217 2 , , PUNCT iajs-779 217 3 we we PRON iajs-779 217 4 have have VERB iajs-779 217 5 the the DET iajs-779 217 6 following follow VERB iajs-779 217 7 consequence consequence NOUN iajs-779 217 8 of of ADP iajs-779 217 9 ( ( PUNCT iajs-779 217 10 2.1 2.1 NUM iajs-779 217 11 ) ) PUNCT iajs-779 217 12 . . PUNCT iajs-779 218 1 2.2 2.2 NUM iajs-779 218 2 corollary corollary NOUN iajs-779 218 3 : : PUNCT iajs-779 218 4 let let VERB iajs-779 218 5 m m PRON iajs-779 218 6 be be AUX iajs-779 218 7 a a DET iajs-779 218 8 chained chain VERB iajs-779 218 9 r r NOUN iajs-779 218 10 - - PUNCT iajs-779 218 11 module module NOUN iajs-779 218 12 . . PUNCT iajs-779 219 1 then then ADV iajs-779 219 2 m m PROPN iajs-779 219 3 is be AUX iajs-779 219 4 fully fully ADV iajs-779 219 5 semiprime semiprime ADJ iajs-779 219 6 if if SCONJ iajs-779 220 1 and and CCONJ iajs-779 220 2 only only ADV iajs-779 220 3 if if SCONJ iajs-779 220 4 m m NOUN iajs-779 220 5 is be AUX iajs-779 220 6 fully fully ADV iajs-779 220 7 prime prime ADJ iajs-779 220 8 . . PUNCT iajs-779 221 1 proof proof NOUN iajs-779 221 2 : : PUNCT iajs-779 221 3 it it PRON iajs-779 221 4 is be AUX iajs-779 221 5 known know VERB iajs-779 221 6 that that SCONJ iajs-779 221 7 every every DET iajs-779 221 8 chained chain VERB iajs-779 221 9 r r NOUN iajs-779 221 10 - - PUNCT iajs-779 221 11 module module NOUN iajs-779 221 12 m m NOUN iajs-779 221 13 is be AUX iajs-779 221 14 uniform uniform ADJ iajs-779 221 15 , , PUNCT iajs-779 221 16 then then ADV iajs-779 221 17 the the DET iajs-779 221 18 result result NOUN iajs-779 221 19 follows follow VERB iajs-779 221 20 from from ADP iajs-779 221 21 proposition proposition NOUN iajs-779 221 22 ( ( PUNCT iajs-779 221 23 2.1 2.1 NUM iajs-779 221 24 ) ) PUNCT iajs-779 221 25 . . PUNCT iajs-779 222 1 an an DET iajs-779 222 2 r r NOUN iajs-779 222 3 - - PUNCT iajs-779 222 4 module module NOUN iajs-779 222 5 m m NOUN iajs-779 222 6 is be AUX iajs-779 222 7 called call VERB iajs-779 222 8 quasi quasi ADJ iajs-779 222 9 - - NOUN iajs-779 222 10 dedekind dedekind ADJ iajs-779 222 11 if if SCONJ iajs-779 222 12 every every DET iajs-779 222 13 submodule submodule NOUN iajs-779 222 14 n n PROPN iajs-779 222 15 of of ADP iajs-779 222 16 m m PROPN iajs-779 222 17 is be AUX iajs-779 222 18 quasi quasi ADJ iajs-779 222 19 - - ADJ iajs-779 222 20 invertible invertible ADJ iajs-779 222 21 [ [ X iajs-779 222 22 9,definition 9,definition NOUN iajs-779 222 23 ( ( PUNCT iajs-779 222 24 1.1 1.1 NUM iajs-779 222 25 ) ) PUNCT iajs-779 222 26 , , PUNCT iajs-779 222 27 ch.2 ch.2 PROPN iajs-779 222 28 ] ] PUNCT iajs-779 222 29 , , PUNCT iajs-779 222 30 where where SCONJ iajs-779 222 31 a a DET iajs-779 222 32 submodule submodule NOUN iajs-779 222 33 n n PROPN iajs-779 222 34 of of ADP iajs-779 222 35 m m PROPN iajs-779 222 36 is be AUX iajs-779 222 37 called call VERB iajs-779 222 38 quasi quasi ADJ iajs-779 222 39 - - ADJ iajs-779 222 40 invertible invertible ADJ iajs-779 222 41 if if SCONJ iajs-779 222 42 hom(m hom(m PROPN iajs-779 222 43 / / SYM iajs-779 222 44 n n CCONJ iajs-779 222 45 , , PUNCT iajs-779 222 46 m)=0 m)=0 PROPN iajs-779 223 1 [ [ X iajs-779 223 2 9,definition 9,definition NUM iajs-779 223 3 ( ( PUNCT iajs-779 223 4 1.1),ch.1 1.1),ch.1 NUM iajs-779 223 5 ] ] PUNCT iajs-779 223 6 . . PUNCT iajs-779 224 1 2.3 2.3 NUM iajs-779 224 2 corollary corollary NOUN iajs-779 224 3 : : PUNCT iajs-779 224 4 if if SCONJ iajs-779 224 5 m m NOUN iajs-779 224 6 is be AUX iajs-779 224 7 a a DET iajs-779 224 8 uniform uniform NOUN iajs-779 224 9 fully fully ADV iajs-779 224 10 semiprime semiprime NOUN iajs-779 224 11 r r NOUN iajs-779 224 12 - - NOUN iajs-779 224 13 module module NOUN iajs-779 224 14 , , PUNCT iajs-779 224 15 then then ADV iajs-779 224 16 m m NOUN iajs-779 224 17 is be AUX iajs-779 224 18 a a DET iajs-779 224 19 quasi quasi ADJ iajs-779 224 20 - - ADJ iajs-779 224 21 dedekind dedekind ADJ iajs-779 224 22 r r NOUN iajs-779 224 23 - - PUNCT iajs-779 224 24 module module NOUN iajs-779 224 25 . . PUNCT iajs-779 225 1 proof proof NOUN iajs-779 225 2 : : PUNCT iajs-779 225 3 from from ADP iajs-779 225 4 proposition proposition NOUN iajs-779 225 5 ( ( PUNCT iajs-779 225 6 2.1 2.1 NUM iajs-779 225 7 ) ) PUNCT iajs-779 225 8 and and CCONJ iajs-779 225 9 remark remark NOUN iajs-779 225 10 ( ( PUNCT iajs-779 225 11 1.4 1.4 NUM iajs-779 225 12 ) ) PUNCT iajs-779 225 13 , , PUNCT iajs-779 225 14 we we PRON iajs-779 225 15 get get VERB iajs-779 225 16 m m NOUN iajs-779 225 17 is be AUX iajs-779 225 18 a a DET iajs-779 225 19 prime prime ADJ iajs-779 225 20 module module NOUN iajs-779 225 21 . . PUNCT iajs-779 226 1 thus thus ADV iajs-779 226 2 by by ADP iajs-779 226 3 [ [ X iajs-779 226 4 6,theorem 6,theorem NUM iajs-779 226 5 ( ( PUNCT iajs-779 226 6 3.11),ch.3 3.11),ch.3 NOUN iajs-779 226 7 ] ] X iajs-779 226 8 , , PUNCT iajs-779 226 9 m m VERB iajs-779 226 10 is be AUX iajs-779 226 11 quasi quasi ADJ iajs-779 226 12 - - ADJ iajs-779 226 13 dedekind dedekind ADJ iajs-779 226 14 . . PUNCT iajs-779 227 1 as as ADP iajs-779 227 2 an an DET iajs-779 227 3 application application NOUN iajs-779 227 4 of of ADP iajs-779 227 5 corollary corollary ADJ iajs-779 227 6 ( ( PUNCT iajs-779 227 7 2.3 2.3 NUM iajs-779 227 8 ) ) PUNCT iajs-779 227 9 , , PUNCT iajs-779 227 10 we we PRON iajs-779 227 11 give give VERB iajs-779 227 12 the the DET iajs-779 227 13 following follow VERB iajs-779 227 14 example example NOUN iajs-779 227 15 . . PUNCT iajs-779 228 1 2.4 2.4 NUM iajs-779 228 2 example example NOUN iajs-779 228 3 : : PUNCT iajs-779 228 4 z z NOUN iajs-779 228 5 as as SCONJ iajs-779 228 6 a a DET iajs-779 228 7 z z NOUN iajs-779 228 8 - - PUNCT iajs-779 228 9 module module NOUN iajs-779 228 10 is be AUX iajs-779 228 11 uniform uniform ADJ iajs-779 228 12 and and CCONJ iajs-779 228 13 fully fully ADV iajs-779 228 14 prime prime ADJ iajs-779 228 15 module module NOUN iajs-779 228 16 , , PUNCT iajs-779 228 17 also also ADV iajs-779 228 18 it it PRON iajs-779 228 19 is be AUX iajs-779 228 20 quasi quasi ADJ iajs-779 228 21 - - ADJ iajs-779 228 22 dedekind dedekind ADJ iajs-779 228 23 . . PUNCT iajs-779 229 1 recall recall VERB iajs-779 229 2 that that SCONJ iajs-779 229 3 an an DET iajs-779 229 4 r r NOUN iajs-779 229 5 - - PUNCT iajs-779 229 6 module module NOUN iajs-779 229 7 m m NOUN iajs-779 229 8 is be AUX iajs-779 229 9 called call VERB iajs-779 229 10 z z NOUN iajs-779 229 11 - - NOUN iajs-779 229 12 regular regular ADJ iajs-779 229 13 if if SCONJ iajs-779 230 1 and and CCONJ iajs-779 230 2 only only ADV iajs-779 230 3 if if SCONJ iajs-779 230 4 each each DET iajs-779 230 5 cyclic cyclic ADJ iajs-779 230 6 submodule submodule NOUN iajs-779 230 7 of of ADP iajs-779 230 8 m m PROPN iajs-779 230 9 is be AUX iajs-779 230 10 projective projective ADJ iajs-779 230 11 direct direct ADJ iajs-779 230 12 summand summand NOUN iajs-779 230 13 of of ADP iajs-779 230 14 m. m. NOUN iajs-779 230 15 equivalently equivalently ADV iajs-779 230 16 if if SCONJ iajs-779 230 17 for for ADP iajs-779 230 18 each each DET iajs-779 230 19 am am PROPN iajs-779 230 20 , , PUNCT iajs-779 230 21 fm*=hom(m fm*=hom(m NOUN iajs-779 230 22 , , PUNCT iajs-779 230 23 r r NOUN iajs-779 230 24 ) ) PUNCT iajs-779 230 25 such such ADJ iajs-779 230 26 that that SCONJ iajs-779 230 27 a a DET iajs-779 230 28 = = PROPN iajs-779 230 29 f(a)a f(a)a NOUN iajs-779 230 30 , , PUNCT iajs-779 230 31 [ [ X iajs-779 230 32 10 10 NUM iajs-779 230 33 ] ] PUNCT iajs-779 230 34 . . PUNCT iajs-779 231 1 by by ADP iajs-779 231 2 using use VERB iajs-779 231 3 this this DET iajs-779 231 4 concept concept NOUN iajs-779 231 5 , , PUNCT iajs-779 231 6 we we PRON iajs-779 231 7 give give VERB iajs-779 231 8 the the DET iajs-779 231 9 following follow VERB iajs-779 231 10 proposition proposition NOUN iajs-779 231 11 . . PUNCT iajs-779 232 1 ibn ibn PROPN iajs-779 232 2 alhaitham alhaitham PROPN iajs-779 232 3 j. j. PROPN iajs-779 232 4 for for ADP iajs-779 232 5 pure pure ADJ iajs-779 232 6 & & CCONJ iajs-779 232 7 appl appl PROPN iajs-779 232 8 . . PUNCT iajs-779 233 1 sci sci PROPN iajs-779 233 2 . . PUNCT iajs-779 233 3 vol.24 vol.24 NOUN iajs-779 233 4 ( ( PUNCT iajs-779 233 5 2 2 NUM iajs-779 233 6 ) ) PUNCT iajs-779 233 7 2011 2011 NUM iajs-779 233 8 2.5 2.5 NUM iajs-779 233 9 proposition proposition NOUN iajs-779 233 10 : : PUNCT iajs-779 233 11 if if SCONJ iajs-779 233 12 m m NOUN iajs-779 233 13 is be AUX iajs-779 233 14 a a DET iajs-779 233 15 z z NOUN iajs-779 233 16 - - ADJ iajs-779 233 17 regular regular ADJ iajs-779 233 18 r r NOUN iajs-779 233 19 - - PUNCT iajs-779 233 20 module module NOUN iajs-779 233 21 , , PUNCT iajs-779 233 22 then then ADV iajs-779 233 23 m m NOUN iajs-779 233 24 is be AUX iajs-779 233 25 fully fully ADV iajs-779 233 26 semiprime semiprime NOUN iajs-779 233 27 module module NOUN iajs-779 233 28 . . PUNCT iajs-779 234 1 proof proof NOUN iajs-779 234 2 : : PUNCT iajs-779 234 3 let let VERB iajs-779 234 4 k k PRON iajs-779 234 5 be be AUX iajs-779 234 6 a a DET iajs-779 234 7 fully fully ADV iajs-779 234 8 invariant invariant ADJ iajs-779 234 9 submodule submodule NOUN iajs-779 234 10 of of ADP iajs-779 234 11 m m PRON iajs-779 234 12 such such ADJ iajs-779 234 13 that that SCONJ iajs-779 234 14 kk=(0 kk=(0 NOUN iajs-779 234 15 ) ) PUNCT iajs-779 234 16 . . PUNCT iajs-779 235 1 suppose suppose VERB iajs-779 235 2 k0 k0 PROPN iajs-779 235 3 . . PUNCT iajs-779 236 1 then then ADV iajs-779 236 2 there there PRON iajs-779 236 3 exists exist VERB iajs-779 236 4 xk xk PROPN iajs-779 236 5 , , PUNCT iajs-779 236 6 x0 x0 PROPN iajs-779 236 7 . . PUNCT iajs-779 237 1 since since SCONJ iajs-779 237 2 m m PROPN iajs-779 237 3 is be AUX iajs-779 237 4 z z NOUN iajs-779 237 5 - - NOUN iajs-779 237 6 regular regular ADJ iajs-779 237 7 , , PUNCT iajs-779 237 8 then then ADV iajs-779 237 9 there there PRON iajs-779 237 10 exists exist VERB iajs-779 237 11 f f X iajs-779 237 12 : : PUNCT iajs-779 237 13 mr mr VERB iajs-779 237 14 such such ADJ iajs-779 237 15 that that SCONJ iajs-779 237 16 x x NOUN iajs-779 237 17 = = NOUN iajs-779 237 18 f(x)x f(x)x X iajs-779 237 19 . . PUNCT iajs-779 238 1 define define VERB iajs-779 238 2 g g NOUN iajs-779 238 3 : : PUNCT iajs-779 238 4 rk rk NOUN iajs-779 238 5 by by ADP iajs-779 238 6 g(r)=rx g(r)=rx NOUN iajs-779 238 7 for for ADP iajs-779 238 8 each each DET iajs-779 238 9 rr rr NOUN iajs-779 238 10 . . PUNCT iajs-779 239 1 then then ADV iajs-779 239 2 f f X iajs-779 239 3 gm gm PROPN iajs-779 239 4 r r PROPN iajs-779 239 5 k k PROPN iajs-779 239 6   NOUN iajs-779 239 7 , , PUNCT iajs-779 239 8 so so ADV iajs-779 239 9 g g ADJ iajs-779 239 10 f f X iajs-779 239 11 : : PUNCT iajs-779 239 12 mk mk NOUN iajs-779 239 13 such such ADJ iajs-779 239 14 that that SCONJ iajs-779 239 15 ( ( PUNCT iajs-779 239 16 g g NUM iajs-779 239 17 f)(x)=g(f(x))=f(x)x0 f)(x)=g(f(x))=f(x)x0 NOUN iajs-779 239 18 . . PUNCT iajs-779 240 1 thus thus ADV iajs-779 240 2 0g 0g ADJ iajs-779 240 3 f f PROPN iajs-779 240 4 and and CCONJ iajs-779 240 5 hence hence ADV iajs-779 240 6 kk0 kk0 PROPN iajs-779 240 7 which which PRON iajs-779 240 8 is be AUX iajs-779 240 9 a a DET iajs-779 240 10 contradiction contradiction NOUN iajs-779 240 11 . . PUNCT iajs-779 241 1 thus thus ADV iajs-779 241 2 k=0 k=0 X iajs-779 241 3 . . PUNCT iajs-779 242 1 now now ADV iajs-779 242 2 , , PUNCT iajs-779 242 3 we we PRON iajs-779 242 4 have have VERB iajs-779 242 5 the the DET iajs-779 242 6 following follow VERB iajs-779 242 7 proposition proposition NOUN iajs-779 242 8 . . PUNCT iajs-779 243 1 2.6 2.6 NUM iajs-779 243 2 proposition proposition NOUN iajs-779 243 3 : : PUNCT iajs-779 243 4 let let VERB iajs-779 243 5 i i PRON iajs-779 243 6 be be AUX iajs-779 243 7 an an DET iajs-779 243 8 ideal ideal NOUN iajs-779 243 9 of of ADP iajs-779 243 10 r. r. PROPN iajs-779 243 11 then then ADV iajs-779 243 12 the the DET iajs-779 243 13 following follow VERB iajs-779 243 14 statements statement NOUN iajs-779 243 15 are be AUX iajs-779 243 16 equivalent equivalent ADJ iajs-779 243 17 : : PUNCT iajs-779 243 18 1 1 X iajs-779 243 19 . . X iajs-779 244 1 i i PRON iajs-779 244 2 is be AUX iajs-779 244 3 a a DET iajs-779 244 4 fully fully ADV iajs-779 244 5 semiprime semiprime NOUN iajs-779 244 6 submodule submodule NOUN iajs-779 244 7 in in ADP iajs-779 244 8 r. r. PROPN iajs-779 244 9 2 2 NUM iajs-779 244 10 . . PUNCT iajs-779 245 1 i i PRON iajs-779 245 2 is be AUX iajs-779 245 3 a a DET iajs-779 245 4 semiprime semiprime NOUN iajs-779 245 5 submodule submodule NOUN iajs-779 245 6 of of ADP iajs-779 245 7 r. r. PROPN iajs-779 245 8 3 3 NUM iajs-779 245 9 . . PUNCT iajs-779 246 1 r r X iajs-779 246 2 / / SYM iajs-779 246 3 i i PRON iajs-779 246 4 is be AUX iajs-779 246 5 a a DET iajs-779 246 6 semiprime semiprime NOUN iajs-779 246 7 ring ring NOUN iajs-779 246 8 . . PUNCT iajs-779 247 1 proof proof NOUN iajs-779 247 2 : : PUNCT iajs-779 247 3 12 12 NUM iajs-779 247 4 , , PUNCT iajs-779 247 5 let let VERB iajs-779 247 6 j j PROPN iajs-779 247 7 be be AUX iajs-779 247 8 an an DET iajs-779 247 9 ideal ideal NOUN iajs-779 247 10 of of ADP iajs-779 247 11 r r NOUN iajs-779 247 12 such such ADJ iajs-779 247 13 that that SCONJ iajs-779 247 14 j j PROPN iajs-779 247 15 2i 2i PROPN iajs-779 247 16 . . PUNCT iajs-779 248 1 then then ADV iajs-779 248 2 jji jji PROPN iajs-779 248 3 ( ( PUNCT iajs-779 248 4 by by ADP iajs-779 248 5 note note NOUN iajs-779 248 6 ( ( PUNCT iajs-779 248 7 1.4 1.4 NUM iajs-779 248 8 ) ) PUNCT iajs-779 248 9 ) ) PUNCT iajs-779 248 10 . . PUNCT iajs-779 249 1 thus thus ADV iajs-779 249 2 ji ji PROPN iajs-779 249 3 , , PUNCT iajs-779 249 4 by by ADP iajs-779 249 5 ( ( PUNCT iajs-779 249 6 1 1 NUM iajs-779 249 7 ) ) PUNCT iajs-779 249 8 . . PUNCT iajs-779 250 1 21 21 NUM iajs-779 250 2 , , PUNCT iajs-779 250 3 let let VERB iajs-779 250 4 jji jji NOUN iajs-779 250 5 . . PUNCT iajs-779 251 1 then then ADV iajs-779 251 2 j j PROPN iajs-779 251 3 2 2 NUM iajs-779 251 4 i i NOUN iajs-779 251 5 ( ( PUNCT iajs-779 251 6 by by ADP iajs-779 251 7 note note NOUN iajs-779 251 8 ( ( PUNCT iajs-779 251 9 1.4 1.4 NUM iajs-779 251 10 ) ) PUNCT iajs-779 251 11 ) ) PUNCT iajs-779 251 12 implies imply VERB iajs-779 251 13 ji ji PROPN iajs-779 251 14 . . PUNCT iajs-779 252 1 23 23 PROPN iajs-779 252 2 , , PUNCT iajs-779 252 3 it it PRON iajs-779 252 4 is be AUX iajs-779 252 5 obvious obvious ADJ iajs-779 252 6 . . PUNCT iajs-779 253 1 2.7 2.7 NUM iajs-779 253 2 notes note NOUN iajs-779 253 3 , , PUNCT iajs-779 253 4 [ [ X iajs-779 253 5 11 11 NUM iajs-779 253 6 ] ] SYM iajs-779 253 7 : : PUNCT iajs-779 253 8 1 1 X iajs-779 253 9 . . X iajs-779 253 10 for for ADP iajs-779 253 11 any any DET iajs-779 253 12 r r NOUN iajs-779 253 13 - - PUNCT iajs-779 253 14 module module NOUN iajs-779 253 15 m m NOUN iajs-779 253 16 and and CCONJ iajs-779 253 17 for for ADP iajs-779 253 18 any any DET iajs-779 253 19 ideals ideal NOUN iajs-779 253 20 i i PRON iajs-779 253 21 , , PUNCT iajs-779 253 22 j j PROPN iajs-779 253 23 of of ADP iajs-779 253 24 r. r. PROPN iajs-779 253 25 then then ADV iajs-779 253 26 ( ( PUNCT iajs-779 253 27 im)(jm im)(jm X iajs-779 253 28 ) ) PUNCT iajs-779 253 29 (ij)m (ij)m NOUN iajs-779 253 30 . . PUNCT iajs-779 254 1 and and CCONJ iajs-779 254 2 the the DET iajs-779 254 3 reverse reverse ADJ iajs-779 254 4 inclusion inclusion NOUN iajs-779 254 5 is be AUX iajs-779 254 6 also also ADV iajs-779 254 7 easily easily ADV iajs-779 254 8 established establish VERB iajs-779 254 9 provided provide VERB iajs-779 254 10 m m PROPN iajs-779 254 11 is be AUX iajs-779 254 12 self self NOUN iajs-779 254 13 generator generator NOUN iajs-779 254 14 , , PUNCT iajs-779 254 15 that that PRON iajs-779 254 16 is be AUX iajs-779 254 17 trac(m trac(m NOUN iajs-779 254 18 , , PUNCT iajs-779 254 19 jm jm NOUN iajs-779 254 20 ) ) PUNCT iajs-779 255 1 = = SYM iajs-779 255 2 jm jm PROPN iajs-779 255 3 . . PROPN iajs-779 255 4 2 2 NUM iajs-779 255 5 . . X iajs-779 256 1 the the DET iajs-779 256 2 multiplication multiplication NOUN iajs-779 256 3 modules module NOUN iajs-779 256 4 over over ADP iajs-779 256 5 commutative commutative ADJ iajs-779 256 6 ring ring NOUN iajs-779 256 7 are be AUX iajs-779 256 8 self self NOUN iajs-779 256 9 generator generator NOUN iajs-779 256 10 whose whose DET iajs-779 256 11 submodules submodule NOUN iajs-779 256 12 are be AUX iajs-779 256 13 fully fully ADV iajs-779 256 14 invariant invariant ADJ iajs-779 256 15 . . PUNCT iajs-779 257 1 recall recall VERB iajs-779 257 2 that that SCONJ iajs-779 257 3 an an DET iajs-779 257 4 r r NOUN iajs-779 257 5 - - PUNCT iajs-779 257 6 module module NOUN iajs-779 257 7 m m NOUN iajs-779 257 8 is be AUX iajs-779 257 9 called call VERB iajs-779 257 10 multiplication multiplication NOUN iajs-779 257 11 if if SCONJ iajs-779 257 12 for for ADP iajs-779 257 13 every every DET iajs-779 257 14 submodule submodule NOUN iajs-779 257 15 n n PROPN iajs-779 257 16 of of ADP iajs-779 257 17 m m PRON iajs-779 257 18 , , PUNCT iajs-779 257 19 there there PRON iajs-779 257 20 exists exist VERB iajs-779 257 21 an an DET iajs-779 257 22 ideal ideal NOUN iajs-779 257 23 i i PRON iajs-779 257 24 of of ADP iajs-779 257 25 r r NOUN iajs-779 257 26 such such ADJ iajs-779 257 27 that that SCONJ iajs-779 257 28 n n CCONJ iajs-779 257 29 = = NOUN iajs-779 257 30 im im ADJ iajs-779 257 31 , , PUNCT iajs-779 257 32 equivalently equivalently ADV iajs-779 257 33 for for ADP iajs-779 257 34 every every DET iajs-779 257 35 submodule submodule NOUN iajs-779 257 36 n n PROPN iajs-779 257 37 of of ADP iajs-779 257 38 m m PRON iajs-779 257 39 , , PUNCT iajs-779 257 40 n=[n n=[n PROPN iajs-779 257 41 r r NOUN iajs-779 257 42 : : PUNCT iajs-779 257 43 m]m m]m NOUN iajs-779 257 44 , , PUNCT iajs-779 258 1 [ [ X iajs-779 258 2 7 7 NUM iajs-779 258 3 ] ] PUNCT iajs-779 258 4 . . PUNCT iajs-779 259 1 2.9 2.9 NUM iajs-779 259 2 corollary corollary NOUN iajs-779 259 3 : : PUNCT iajs-779 259 4 let let VERB iajs-779 259 5 m m PRON iajs-779 259 6 be be AUX iajs-779 259 7 a a DET iajs-779 259 8 multiplication multiplication NOUN iajs-779 259 9 r r NOUN iajs-779 259 10 - - NOUN iajs-779 259 11 module module NOUN iajs-779 259 12 . . PUNCT iajs-779 260 1 then then ADV iajs-779 260 2 ( ( PUNCT iajs-779 260 3 im)(jm im)(jm NUM iajs-779 260 4 ) ) PUNCT iajs-779 260 5 =( =( NOUN iajs-779 260 6 ij)m ij)m PROPN iajs-779 260 7 . . PUNCT iajs-779 261 1 2.10 2.10 NUM iajs-779 261 2 proposition proposition NOUN iajs-779 261 3 : : PUNCT iajs-779 261 4 let let VERB iajs-779 261 5 m m PRON iajs-779 261 6 be be AUX iajs-779 261 7 a a DET iajs-779 261 8 faithful faithful ADJ iajs-779 261 9 multiplication multiplication NOUN iajs-779 261 10 r r NOUN iajs-779 261 11 - - PUNCT iajs-779 261 12 module module NOUN iajs-779 261 13 m. m. NOUN iajs-779 261 14 then then ADV iajs-779 261 15 m m VERB iajs-779 261 16 is be AUX iajs-779 261 17 fully fully ADV iajs-779 261 18 semiprime semiprime ADJ iajs-779 261 19 if if SCONJ iajs-779 262 1 and and CCONJ iajs-779 262 2 only only ADV iajs-779 262 3 if if SCONJ iajs-779 262 4 r r NOUN iajs-779 262 5 is be AUX iajs-779 262 6 semiprime semiprime NOUN iajs-779 262 7 ring ring NOUN iajs-779 262 8 . . PUNCT iajs-779 263 1 proof proof NOUN iajs-779 263 2 : : PUNCT iajs-779 263 3 let let VERB iajs-779 263 4 i i PRON iajs-779 263 5 be be AUX iajs-779 263 6 a a DET iajs-779 263 7 proper proper ADJ iajs-779 263 8 ideal ideal NOUN iajs-779 263 9 of of ADP iajs-779 263 10 r r NOUN iajs-779 263 11 such such ADJ iajs-779 263 12 that that SCONJ iajs-779 263 13 i i PRON iajs-779 263 14 2=0 2=0 NUM iajs-779 263 15 , , PUNCT iajs-779 263 16 let let VERB iajs-779 263 17 n n CCONJ iajs-779 263 18 = = NOUN iajs-779 263 19 im im X iajs-779 263 20 , , PUNCT iajs-779 263 21 n n PRON iajs-779 263 22 is be AUX iajs-779 263 23 a a DET iajs-779 263 24 fully fully ADV iajs-779 263 25 invariant invariant ADJ iajs-779 263 26 submodule submodule NOUN iajs-779 263 27 of of ADP iajs-779 263 28 m. m. NOUN iajs-779 263 29 then then ADV iajs-779 263 30 nn nn PROPN iajs-779 263 31 = = SYM iajs-779 263 32 i2m=0 i2m=0 PROPN iajs-779 263 33 by by ADP iajs-779 263 34 corollary corollary ADJ iajs-779 263 35 ( ( PUNCT iajs-779 263 36 2.9 2.9 NUM iajs-779 263 37 ) ) PUNCT iajs-779 263 38 , , PUNCT iajs-779 263 39 so so ADV iajs-779 263 40 nn=0 nn=0 PROPN iajs-779 263 41 . . PUNCT iajs-779 264 1 but but CCONJ iajs-779 264 2 m m PROPN iajs-779 264 3 is be AUX iajs-779 264 4 fully fully ADV iajs-779 264 5 semiprime semiprime NOUN iajs-779 264 6 , , PUNCT iajs-779 264 7 implies imply VERB iajs-779 264 8 n=0 n=0 NUM iajs-779 264 9 and and CCONJ iajs-779 264 10 so so ADV iajs-779 264 11 im=0 im=0 PROPN iajs-779 264 12 . . PROPN iajs-779 265 1 then then ADV iajs-779 265 2 iann(m)=0 iann(m)=0 PROPN iajs-779 265 3 , , PUNCT iajs-779 265 4 since since SCONJ iajs-779 265 5 m m PROPN iajs-779 265 6 is be AUX iajs-779 265 7 faithful faithful ADJ iajs-779 265 8 . . PUNCT iajs-779 266 1 thus thus ADV iajs-779 266 2 i=0 i=0 X iajs-779 266 3 . . PUNCT iajs-779 267 1 conversely conversely ADV iajs-779 267 2 , , PUNCT iajs-779 267 3 let let VERB iajs-779 267 4 n n PRON iajs-779 267 5 be be AUX iajs-779 267 6 a a DET iajs-779 267 7 fully fully ADV iajs-779 267 8 invariant invariant ADJ iajs-779 267 9 submodule submodule NOUN iajs-779 267 10 and and CCONJ iajs-779 267 11 nn=0 nn=0 PROPN iajs-779 267 12 , , PUNCT iajs-779 267 13 since since ADV iajs-779 267 14 , , PUNCT iajs-779 267 15 n n CCONJ iajs-779 267 16 = = NOUN iajs-779 267 17 im im NOUN iajs-779 267 18 for for ADP iajs-779 267 19 some some DET iajs-779 267 20 ideal ideal NOUN iajs-779 267 21 i i PRON iajs-779 267 22 of of ADP iajs-779 267 23 r. r. PROPN iajs-779 267 24 implies imply VERB iajs-779 267 25 that that SCONJ iajs-779 267 26 nn nn PROPN iajs-779 267 27 = = SYM iajs-779 267 28 i i PROPN iajs-779 267 29 2 2 NUM iajs-779 267 30 m m NOUN iajs-779 267 31 , , PUNCT iajs-779 267 32 so so SCONJ iajs-779 267 33 i i PRON iajs-779 267 34 2 2 NUM iajs-779 267 35 m=0 m=0 PROPN iajs-779 267 36 . . PUNCT iajs-779 268 1 thus thus ADV iajs-779 268 2 i i PRON iajs-779 268 3 2 2 NUM iajs-779 268 4 ann(m)=0 ann(m)=0 NOUN iajs-779 268 5 . . PUNCT iajs-779 269 1 therefore therefore ADV iajs-779 269 2 i=0 i=0 PROPN iajs-779 269 3 ( ( PUNCT iajs-779 269 4 since since SCONJ iajs-779 269 5 r r NOUN iajs-779 269 6 is be AUX iajs-779 269 7 semiprime semiprime NOUN iajs-779 269 8 ) ) PUNCT iajs-779 269 9 , , PUNCT iajs-779 269 10 and and CCONJ iajs-779 269 11 hence hence ADV iajs-779 269 12 n n CCONJ iajs-779 269 13 = = NOUN iajs-779 269 14 im=(0 im=(0 ADJ iajs-779 269 15 ) ) PUNCT iajs-779 269 16 . . PUNCT iajs-779 270 1 now now ADV iajs-779 270 2 , , PUNCT iajs-779 270 3 we we PRON iajs-779 270 4 have have VERB iajs-779 270 5 the the DET iajs-779 270 6 following follow VERB iajs-779 270 7 proposition proposition NOUN iajs-779 270 8 . . PUNCT iajs-779 271 1 2.11 2.11 NUM iajs-779 271 2 proposition proposition NOUN iajs-779 271 3 : : PUNCT iajs-779 271 4 let let VERB iajs-779 271 5 m m PRON iajs-779 271 6 be be AUX iajs-779 271 7 a a DET iajs-779 271 8 multiplication multiplication NOUN iajs-779 271 9 r r NOUN iajs-779 271 10 - - PUNCT iajs-779 271 11 module module NOUN iajs-779 271 12 with with ADP iajs-779 271 13 annr(m annr(m NOUN iajs-779 271 14 ) ) PUNCT iajs-779 271 15 is be AUX iajs-779 271 16 semiprime semiprime NOUN iajs-779 271 17 . . PUNCT iajs-779 272 1 then then ADV iajs-779 272 2 m m PROPN iajs-779 272 3 is be AUX iajs-779 272 4 fully fully ADV iajs-779 272 5 semiprime semiprime NOUN iajs-779 272 6 module module NOUN iajs-779 272 7 . . PUNCT iajs-779 273 1 proof proof NOUN iajs-779 273 2 : : PUNCT iajs-779 273 3 let let VERB iajs-779 273 4 n n PRON iajs-779 273 5 be be AUX iajs-779 273 6 a a DET iajs-779 273 7 fully fully ADV iajs-779 273 8 invariant invariant ADJ iajs-779 273 9 submodule submodule NOUN iajs-779 273 10 of of ADP iajs-779 273 11 m m PRON iajs-779 273 12 such such ADJ iajs-779 273 13 that that DET iajs-779 273 14 nn=0 nn=0 PROPN iajs-779 273 15 . . PUNCT iajs-779 274 1 but but CCONJ iajs-779 274 2 n n CCONJ iajs-779 274 3 = = NOUN iajs-779 274 4 im im NOUN iajs-779 274 5 for for ADP iajs-779 274 6 some some DET iajs-779 274 7 ideal ideal ADJ iajs-779 274 8 i i PRON iajs-779 274 9 of of ADP iajs-779 274 10 m m PROPN iajs-779 274 11 , , PUNCT iajs-779 274 12 since since SCONJ iajs-779 274 13 m m PROPN iajs-779 274 14 is be AUX iajs-779 274 15 multiplication multiplication NOUN iajs-779 274 16 module module NOUN iajs-779 274 17 . . PUNCT iajs-779 275 1 hence hence ADV iajs-779 275 2 imim=0 imim=0 PROPN iajs-779 275 3 which which PRON iajs-779 275 4 implies imply VERB iajs-779 275 5 that that SCONJ iajs-779 275 6 i i PRON iajs-779 275 7 2m=0 2m=0 NOUN iajs-779 275 8 . . PUNCT iajs-779 276 1 thus thus ADV iajs-779 276 2 i2annr(m i2annr(m NUM iajs-779 276 3 ) ) PUNCT iajs-779 276 4 and and CCONJ iajs-779 276 5 hence hence ADV iajs-779 276 6 iannr(m).then iannr(m).then ADV iajs-779 276 7 im=0 im=0 PROPN iajs-779 276 8 . . PUNCT iajs-779 276 9 therefore therefore PROPN iajs-779 276 10 n=(0 n=(0 NUM iajs-779 276 11 ) ) PUNCT iajs-779 276 12 . . PUNCT iajs-779 277 1 this this PRON iajs-779 277 2 completes complete VERB iajs-779 277 3 the the DET iajs-779 277 4 proof proof NOUN iajs-779 277 5 . . PUNCT iajs-779 278 1 the the DET iajs-779 278 2 following follow VERB iajs-779 278 3 is be AUX iajs-779 278 4 an an DET iajs-779 278 5 immediate immediate ADJ iajs-779 278 6 consequence consequence NOUN iajs-779 278 7 of of ADP iajs-779 278 8 proposition proposition NOUN iajs-779 278 9 ( ( PUNCT iajs-779 278 10 2.11 2.11 NUM iajs-779 278 11 ) ) PUNCT iajs-779 278 12 . . PUNCT iajs-779 279 1 2.12 2.12 NUM iajs-779 279 2 corollary corollary NOUN iajs-779 279 3 : : PUNCT iajs-779 279 4 let let VERB iajs-779 279 5 m m PRON iajs-779 279 6 be be AUX iajs-779 279 7 a a DET iajs-779 279 8 multiplication multiplication NOUN iajs-779 279 9 r r NOUN iajs-779 279 10 - - NOUN iajs-779 279 11 module module NOUN iajs-779 279 12 . . PUNCT iajs-779 280 1 then then ADV iajs-779 280 2 m m PROPN iajs-779 280 3 is be AUX iajs-779 280 4 a a DET iajs-779 280 5 semiprime semiprime NOUN iajs-779 280 6 r r NOUN iajs-779 280 7 - - PUNCT iajs-779 280 8 module module NOUN iajs-779 280 9 if if SCONJ iajs-779 280 10 and and CCONJ iajs-779 280 11 only only ADV iajs-779 281 1 if if SCONJ iajs-779 281 2 annr(m annr(m NOUN iajs-779 281 3 ) ) PUNCT iajs-779 281 4 is be AUX iajs-779 281 5 semiprime semiprime NOUN iajs-779 281 6 ideal ideal ADJ iajs-779 281 7 . . PUNCT iajs-779 282 1 proof proof NOUN iajs-779 282 2 : : PUNCT iajs-779 282 3 assume assume VERB iajs-779 282 4 that that SCONJ iajs-779 282 5 m m PROPN iajs-779 282 6 is be AUX iajs-779 282 7 semiprime semiprime NOUN iajs-779 282 8 r r NOUN iajs-779 282 9 - - PUNCT iajs-779 282 10 module module NOUN iajs-779 282 11 . . PUNCT iajs-779 283 1 then then ADV iajs-779 283 2 ( ( PUNCT iajs-779 283 3 0 0 X iajs-779 283 4 ) ) PUNCT iajs-779 283 5 is be AUX iajs-779 283 6 a a DET iajs-779 283 7 semiprime semiprime NOUN iajs-779 283 8 submodule submodule NOUN iajs-779 283 9 , , PUNCT iajs-779 283 10 so so CCONJ iajs-779 283 11 (o: (o: PROPN iajs-779 283 12 m)=annr(m m)=annr(m X iajs-779 283 13 ) ) PUNCT iajs-779 283 14 is be AUX iajs-779 283 15 semiprime semiprime NOUN iajs-779 283 16 ideal ideal ADJ iajs-779 283 17 . . PUNCT iajs-779 284 1 conversely conversely ADV iajs-779 284 2 , , PUNCT iajs-779 284 3 let let VERB iajs-779 284 4 annr(m annr(m NOUN iajs-779 284 5 ) ) PUNCT iajs-779 284 6 be be VERB iajs-779 284 7 a a DET iajs-779 284 8 semiprime semiprime NOUN iajs-779 284 9 ideal ideal NOUN iajs-779 284 10 . . PUNCT iajs-779 285 1 then then ADV iajs-779 285 2 by by ADP iajs-779 285 3 proposition proposition NOUN iajs-779 285 4 ( ( PUNCT iajs-779 285 5 2.11 2.11 NUM iajs-779 285 6 ) ) PUNCT iajs-779 285 7 , , PUNCT iajs-779 285 8 we we PRON iajs-779 285 9 get get VERB iajs-779 285 10 m m VERB iajs-779 285 11 is be AUX iajs-779 285 12 fully fully ADV iajs-779 285 13 semiprime semiprime NOUN iajs-779 285 14 and and CCONJ iajs-779 285 15 hence hence ADV iajs-779 285 16 m m VERB iajs-779 285 17 is be AUX iajs-779 285 18 semiprime semiprime NOUN iajs-779 285 19 by by ADP iajs-779 285 20 remarks remark NOUN iajs-779 285 21 ( ( PUNCT iajs-779 285 22 1.4 1.4 NUM iajs-779 285 23 ) ) PUNCT iajs-779 285 24 , , PUNCT iajs-779 285 25 see see VERB iajs-779 285 26 [ [ X iajs-779 285 27 4 4 NUM iajs-779 285 28 ] ] PUNCT iajs-779 285 29 . . PUNCT iajs-779 286 1 ibn ibn PROPN iajs-779 286 2 alhaitham alhaitham PROPN iajs-779 286 3 j. j. PROPN iajs-779 286 4 for for ADP iajs-779 286 5 pure pure ADJ iajs-779 286 6 & & CCONJ iajs-779 286 7 appl appl PROPN iajs-779 286 8 . . PUNCT iajs-779 287 1 sci sci PROPN iajs-779 287 2 . . PUNCT iajs-779 287 3 vol.24 vol.24 NOUN iajs-779 287 4 ( ( PUNCT iajs-779 287 5 2 2 NUM iajs-779 287 6 ) ) PUNCT iajs-779 287 7 2011 2011 NUM iajs-779 287 8 by by ADP iajs-779 287 9 proposition proposition NOUN iajs-779 287 10 ( ( PUNCT iajs-779 287 11 2.11 2.11 NUM iajs-779 287 12 ) ) PUNCT iajs-779 287 13 , , PUNCT iajs-779 287 14 we we PRON iajs-779 287 15 have have VERB iajs-779 287 16 the the DET iajs-779 287 17 following following NOUN iajs-779 287 18 . . PUNCT iajs-779 288 1 2.13 2.13 NUM iajs-779 288 2 corollary corollary NOUN iajs-779 288 3 : : PUNCT iajs-779 288 4 let let VERB iajs-779 288 5 m m PRON iajs-779 288 6 be be AUX iajs-779 288 7 a a DET iajs-779 288 8 faithful faithful ADJ iajs-779 288 9 multiplication multiplication NOUN iajs-779 288 10 r r NOUN iajs-779 288 11 - - NOUN iajs-779 288 12 module module NOUN iajs-779 288 13 . . PUNCT iajs-779 289 1 the the DET iajs-779 289 2 following follow VERB iajs-779 289 3 are be AUX iajs-779 289 4 equivalent equivalent ADJ iajs-779 289 5 : : PUNCT iajs-779 289 6 1 1 X iajs-779 289 7 . . X iajs-779 289 8 m m PROPN iajs-779 289 9 is be AUX iajs-779 289 10 fully fully ADV iajs-779 289 11 semiprime semiprime ADJ iajs-779 289 12 . . PUNCT iajs-779 290 1 2 2 X iajs-779 290 2 . . X iajs-779 290 3 r r NOUN iajs-779 290 4 is be AUX iajs-779 290 5 semiprime semiprime NOUN iajs-779 290 6 ring ring NOUN iajs-779 290 7 . . PUNCT iajs-779 291 1 3 3 X iajs-779 291 2 . . X iajs-779 291 3 m m PROPN iajs-779 291 4 is be AUX iajs-779 291 5 semiprime semiprime NOUN iajs-779 291 6 module module NOUN iajs-779 291 7 . . PUNCT iajs-779 292 1 proof proof NOUN iajs-779 292 2 : : PUNCT iajs-779 292 3 12 12 NUM iajs-779 292 4 it it PRON iajs-779 292 5 is be AUX iajs-779 292 6 obvious obvious ADJ iajs-779 292 7 . . PUNCT iajs-779 293 1 23 23 NUM iajs-779 293 2 r r NOUN iajs-779 293 3 is be AUX iajs-779 293 4 semiprime semiprime NOUN iajs-779 293 5 ring ring NOUN iajs-779 293 6   X iajs-779 293 7 ( ( PUNCT iajs-779 293 8 0 0 NUM iajs-779 293 9 ) ) PUNCT iajs-779 293 10 is be AUX iajs-779 293 11 semiprime semiprime NOUN iajs-779 293 12 ideal ideal PROPN iajs-779 293 13   PROPN iajs-779 293 14 annr(m annr(m PROPN iajs-779 293 15 ) ) PUNCT iajs-779 293 16 is be AUX iajs-779 293 17 semiprime semiprime NOUN iajs-779 293 18 ideal ideal NOUN iajs-779 294 1   SCONJ iajs-779 294 2 m m VERB iajs-779 294 3 is be AUX iajs-779 294 4 semiprime semiprime NOUN iajs-779 294 5 module module NOUN iajs-779 294 6 ( ( PUNCT iajs-779 294 7 by by ADP iajs-779 294 8 proposition proposition NOUN iajs-779 294 9 ( ( PUNCT iajs-779 294 10 2.11 2.11 NUM iajs-779 294 11 ) ) PUNCT iajs-779 294 12 ) ) PUNCT iajs-779 294 13 . . PUNCT iajs-779 295 1 an an DET iajs-779 295 2 r r NOUN iajs-779 295 3 - - PUNCT iajs-779 295 4 module module NOUN iajs-779 295 5 m m NOUN iajs-779 295 6 is be AUX iajs-779 295 7 called call VERB iajs-779 295 8 coprime coprime ADV iajs-779 295 9 if if SCONJ iajs-779 295 10 for for ADP iajs-779 295 11 every every DET iajs-779 295 12 proper proper ADJ iajs-779 295 13 submodule submodule NOUN iajs-779 295 14 n n PROPN iajs-779 295 15 of of ADP iajs-779 295 16 m m PROPN iajs-779 295 17 , , PUNCT iajs-779 295 18 annr(m)=annr(m annr(m)=annr(m PROPN iajs-779 295 19 / / SYM iajs-779 295 20 n n CCONJ iajs-779 295 21 ) ) PUNCT iajs-779 295 22 , , PUNCT iajs-779 296 1 [ [ X iajs-779 296 2 2 2 NUM iajs-779 296 3 ] ] PUNCT iajs-779 296 4 . . PUNCT iajs-779 297 1 recall recall VERB iajs-779 297 2 that that SCONJ iajs-779 297 3 an an DET iajs-779 297 4 r r NOUN iajs-779 297 5 - - PUNCT iajs-779 297 6 module module NOUN iajs-779 297 7 m m NOUN iajs-779 297 8 is be AUX iajs-779 297 9 called call VERB iajs-779 297 10 a a DET iajs-779 297 11 scalar scalar ADJ iajs-779 297 12 module module NOUN iajs-779 297 13 if if SCONJ iajs-779 297 14 for for ADP iajs-779 297 15 all all DET iajs-779 297 16 fendr(m);f0 fendr(m);f0 NOUN iajs-779 297 17 , , PUNCT iajs-779 297 18 there there PRON iajs-779 297 19 exists exist VERB iajs-779 297 20 rr rr NOUN iajs-779 297 21 , , PUNCT iajs-779 297 22 r0 r0 NOUN iajs-779 297 23 such such ADJ iajs-779 297 24 that that DET iajs-779 297 25 f(m)=rm f(m)=rm NOUN iajs-779 297 26 , , PUNCT iajs-779 297 27 see see VERB iajs-779 297 28 [ [ X iajs-779 297 29 12 12 NUM iajs-779 297 30 ] ] PUNCT iajs-779 297 31 . . PUNCT iajs-779 298 1 by by ADP iajs-779 298 2 using use VERB iajs-779 298 3 these these DET iajs-779 298 4 concepts concept NOUN iajs-779 298 5 , , PUNCT iajs-779 298 6 we we PRON iajs-779 298 7 can can AUX iajs-779 298 8 prove prove VERB iajs-779 298 9 the the DET iajs-779 298 10 following following NOUN iajs-779 298 11 . . PUNCT iajs-779 299 1 2.14 2.14 NUM iajs-779 299 2 proposition proposition NOUN iajs-779 299 3 : : PUNCT iajs-779 299 4 let let VERB iajs-779 299 5 m m PRON iajs-779 299 6 be be AUX iajs-779 299 7 a a DET iajs-779 299 8 coprime coprime NOUN iajs-779 299 9 scalar scalar ADJ iajs-779 299 10 and and CCONJ iajs-779 299 11 fully fully ADV iajs-779 299 12 semiprime semiprime NOUN iajs-779 299 13 r r NOUN iajs-779 299 14 - - NOUN iajs-779 299 15 module module NOUN iajs-779 299 16 . . PUNCT iajs-779 300 1 then then ADV iajs-779 300 2 m m VERB iajs-779 300 3 is be AUX iajs-779 300 4 simple simple ADJ iajs-779 300 5 . . PUNCT iajs-779 301 1 proof proof NOUN iajs-779 301 2 : : PUNCT iajs-779 301 3 assume assume VERB iajs-779 301 4 that that SCONJ iajs-779 301 5 n n PRON iajs-779 301 6 be be VERB iajs-779 301 7 a a DET iajs-779 301 8 proper proper ADJ iajs-779 301 9 r r NOUN iajs-779 301 10 - - PUNCT iajs-779 301 11 submodule submodule NOUN iajs-779 301 12 of of ADP iajs-779 301 13 m m PRON iajs-779 301 14 , , PUNCT iajs-779 301 15 let let VERB iajs-779 301 16 f f X iajs-779 301 17 : : PUNCT iajs-779 301 18 mn mn PROPN iajs-779 301 19 . . PUNCT iajs-779 302 1 then then ADV iajs-779 302 2 there there PRON iajs-779 302 3 exists exist VERB iajs-779 302 4 rr rr PUNCT iajs-779 302 5 such such ADJ iajs-779 302 6 that that DET iajs-779 302 7 f(m)=rm f(m)=rm NOUN iajs-779 302 8 for for ADP iajs-779 302 9 all all DET iajs-779 302 10 mm mm PROPN iajs-779 302 11 ( ( PUNCT iajs-779 302 12 since since SCONJ iajs-779 302 13 m m PROPN iajs-779 302 14 is be AUX iajs-779 302 15 a a DET iajs-779 302 16 scalar scalar ADJ iajs-779 302 17 module module NOUN iajs-779 302 18 ) ) PUNCT iajs-779 302 19 . . PUNCT iajs-779 303 1 therefore therefore ADV iajs-779 303 2 f(m)=rm f(m)=rm NOUN iajs-779 303 3   PROPN iajs-779 303 4 n n CCONJ iajs-779 303 5 , , PUNCT iajs-779 303 6 implies imply VERB iajs-779 303 7 r[n r[n PROPN iajs-779 303 8 r r NOUN iajs-779 303 9 : : PUNCT iajs-779 303 10 m m VERB iajs-779 303 11 ] ] X iajs-779 303 12 . . PUNCT iajs-779 304 1 but but CCONJ iajs-779 304 2 m m PROPN iajs-779 304 3 is be AUX iajs-779 304 4 a a DET iajs-779 304 5 coprime coprime NOUN iajs-779 304 6 module module NOUN iajs-779 304 7 , , PUNCT iajs-779 304 8 so so ADV iajs-779 304 9 annrm=[n annrm=[n ADP iajs-779 304 10 r r NOUN iajs-779 304 11 : : PUNCT iajs-779 304 12 m m VERB iajs-779 304 13 ] ] X iajs-779 304 14 . . PUNCT iajs-779 305 1 thus thus ADV iajs-779 305 2 rannrm rannrm PROPN iajs-779 305 3 . . PROPN iajs-779 306 1 then then ADV iajs-779 306 2 rm=0 rm=0 NOUN iajs-779 306 3 . . PUNCT iajs-779 307 1 thus thus ADV iajs-779 307 2 f(n)=rn=0 f(n)=rn=0 VERB iajs-779 307 3 . . PUNCT iajs-779 308 1 hence hence ADV iajs-779 308 2 {f(n):f {f(n):f PROPN iajs-779 308 3 : : PUNCT iajs-779 308 4 mn}=0 mn}=0 PROPN iajs-779 308 5 . . PUNCT iajs-779 309 1 then then ADV iajs-779 309 2 nn=0 nn=0 PROPN iajs-779 309 3 and and CCONJ iajs-779 309 4 since since SCONJ iajs-779 309 5 n n PRON iajs-779 309 6 is be AUX iajs-779 309 7 a a DET iajs-779 309 8 fully fully ADV iajs-779 309 9 semiprime semiprime NOUN iajs-779 309 10 , , PUNCT iajs-779 309 11 we we PRON iajs-779 309 12 get get VERB iajs-779 309 13 n=0 n=0 NUM iajs-779 309 14 . . PUNCT iajs-779 310 1 recall recall VERB iajs-779 310 2 that that SCONJ iajs-779 310 3 an an DET iajs-779 310 4 r r NOUN iajs-779 310 5 - - PUNCT iajs-779 310 6 module module NOUN iajs-779 310 7 m m NOUN iajs-779 310 8 is be AUX iajs-779 310 9 called call VERB iajs-779 310 10 retractable retractable ADJ iajs-779 310 11 if if SCONJ iajs-779 310 12 hom(m hom(m PROPN iajs-779 310 13 , , PUNCT iajs-779 310 14 n)0 n)0 ADJ iajs-779 310 15 , , PUNCT iajs-779 310 16 for for ADP iajs-779 310 17 every every DET iajs-779 310 18 non non ADJ iajs-779 310 19 - - ADJ iajs-779 310 20 zero zero NUM iajs-779 310 21 submodule submodule NOUN iajs-779 310 22 n n PROPN iajs-779 310 23 of of ADP iajs-779 310 24 m m PRON iajs-779 310 25 , , PUNCT iajs-779 310 26 see see VERB iajs-779 310 27 [ [ X iajs-779 310 28 13 13 NUM iajs-779 310 29 ] ] PUNCT iajs-779 310 30 . . PUNCT iajs-779 311 1 now now ADV iajs-779 311 2 , , PUNCT iajs-779 311 3 we we PRON iajs-779 311 4 state state VERB iajs-779 311 5 and and CCONJ iajs-779 311 6 prove prove VERB iajs-779 311 7 the the DET iajs-779 311 8 following follow VERB iajs-779 311 9 result result NOUN iajs-779 311 10 . . PUNCT iajs-779 312 1 2.15 2.15 NUM iajs-779 312 2 theorem theorem NOUN iajs-779 312 3 : : PUNCT iajs-779 312 4 let let VERB iajs-779 312 5 m m PRON iajs-779 312 6 be be AUX iajs-779 312 7 an an DET iajs-779 312 8 r r NOUN iajs-779 312 9 - - PUNCT iajs-779 312 10 module module NOUN iajs-779 312 11 , , PUNCT iajs-779 312 12 if if SCONJ iajs-779 312 13 m m NOUN iajs-779 312 14 is be AUX iajs-779 312 15 fully fully ADV iajs-779 312 16 semiprime semiprime NOUN iajs-779 312 17 , , PUNCT iajs-779 312 18 then then ADV iajs-779 312 19 m m VERB iajs-779 312 20 is be AUX iajs-779 312 21 retractable retractable ADJ iajs-779 312 22 and and CCONJ iajs-779 312 23 the the DET iajs-779 312 24 converse converse NOUN iajs-779 312 25 is be AUX iajs-779 312 26 true true ADJ iajs-779 312 27 if if SCONJ iajs-779 312 28 endr(m endr(m PROPN iajs-779 312 29 ) ) PUNCT iajs-779 312 30 is be AUX iajs-779 312 31 semiprime semiprime NOUN iajs-779 312 32 . . PUNCT iajs-779 313 1 proof proof NOUN iajs-779 313 2 : : PUNCT iajs-779 313 3 assume assume VERB iajs-779 313 4 that that SCONJ iajs-779 313 5 m m NOUN iajs-779 313 6 is be AUX iajs-779 313 7 fully fully ADV iajs-779 313 8 semiprime semiprime NOUN iajs-779 313 9 and and CCONJ iajs-779 313 10 let let VERB iajs-779 313 11 n n PRON iajs-779 313 12 be be AUX iajs-779 313 13 a a DET iajs-779 313 14 submodule submodule NOUN iajs-779 313 15 of of ADP iajs-779 313 16 m m PROPN iajs-779 313 17 , , PUNCT iajs-779 313 18 n0 n0 PROPN iajs-779 313 19 . . PUNCT iajs-779 314 1 assume assume VERB iajs-779 314 2 m m NOUN iajs-779 314 3 is be AUX iajs-779 314 4 not not PART iajs-779 314 5 retractable retractable ADJ iajs-779 314 6 , , PUNCT iajs-779 314 7 that that PRON iajs-779 314 8 is be AUX iajs-779 314 9 hom(m hom(m PROPN iajs-779 314 10 , , PUNCT iajs-779 314 11 n)=0 n)=0 PRON iajs-779 314 12 , , PUNCT iajs-779 314 13 then then ADV iajs-779 314 14 nn=0 nn=0 PROPN iajs-779 315 1 and and CCONJ iajs-779 316 1 so so ADV iajs-779 316 2 n=0 n=0 NUM iajs-779 316 3 which which PRON iajs-779 316 4 is be AUX iajs-779 316 5 a a DET iajs-779 316 6 contradiction contradiction NOUN iajs-779 316 7 . . PUNCT iajs-779 317 1 hence hence ADV iajs-779 317 2 m m NOUN iajs-779 317 3 is be AUX iajs-779 317 4 retractable retractable ADJ iajs-779 317 5 . . PUNCT iajs-779 318 1 conversely conversely ADV iajs-779 318 2 , , PUNCT iajs-779 318 3 if if SCONJ iajs-779 318 4 m m NOUN iajs-779 318 5 is be AUX iajs-779 318 6 retractable retractable ADJ iajs-779 318 7 and and CCONJ iajs-779 318 8 endr(m endr(m VERB iajs-779 318 9 ) ) PUNCT iajs-779 318 10 is be AUX iajs-779 318 11 semiprime semiprime NOUN iajs-779 318 12 , , PUNCT iajs-779 318 13 let let VERB iajs-779 318 14 n n PRON iajs-779 318 15 be be AUX iajs-779 318 16 a a DET iajs-779 318 17 fully fully ADV iajs-779 318 18 invariant invariant ADJ iajs-779 318 19 submodule submodule NOUN iajs-779 318 20 of of ADP iajs-779 318 21 m m PRON iajs-779 318 22 such such ADJ iajs-779 318 23 that that DET iajs-779 318 24 nn=0 nn=0 PROPN iajs-779 318 25 . . PUNCT iajs-779 319 1 suppose suppose VERB iajs-779 320 1 n0 n0 PROPN iajs-779 320 2 . . PUNCT iajs-779 321 1 since since SCONJ iajs-779 321 2 m m PROPN iajs-779 321 3 is be AUX iajs-779 321 4 retractable retractable ADJ iajs-779 321 5 , , PUNCT iajs-779 321 6 then then ADV iajs-779 321 7 there there PRON iajs-779 321 8 exists exist VERB iajs-779 321 9 f f X iajs-779 321 10 : : PUNCT iajs-779 321 11 mn mn PROPN iajs-779 321 12 , , PUNCT iajs-779 321 13 f0 f0 PROPN iajs-779 321 14 . . PUNCT iajs-779 322 1 but but CCONJ iajs-779 322 2 0= 0= PRON iajs-779 323 1 nn={f(n):f nn={f(n):f ADJ iajs-779 323 2 : : PUNCT iajs-779 323 3 mn}=0 mn}=0 NOUN iajs-779 323 4 , , PUNCT iajs-779 323 5 hence hence ADV iajs-779 323 6 f(n)=0 f(n)=0 NUM iajs-779 323 7 . . PUNCT iajs-779 324 1 then then ADV iajs-779 324 2 for for ADP iajs-779 324 3 any any DET iajs-779 324 4 mm mm PROPN iajs-779 324 5 , , PUNCT iajs-779 324 6 f f PROPN iajs-779 324 7 2 2 NUM iajs-779 324 8 ( ( PUNCT iajs-779 324 9 m)=f(f(m))=0 m)=f(f(m))=0 PROPN iajs-779 324 10 . . PUNCT iajs-779 325 1 then then ADV iajs-779 325 2 f f PROPN iajs-779 325 3 2 2 NUM iajs-779 325 4 = = SYM iajs-779 325 5 0 0 NUM iajs-779 325 6 and and CCONJ iajs-779 325 7 hence hence ADV iajs-779 325 8 f=0 f=0 X iajs-779 325 9 which which PRON iajs-779 325 10 is be AUX iajs-779 325 11 a a DET iajs-779 325 12 contradiction contradiction NOUN iajs-779 325 13 . . PUNCT iajs-779 326 1 thus thus ADV iajs-779 326 2 n=0 n=0 NUM iajs-779 326 3 . . PUNCT iajs-779 327 1 therefore therefore ADV iajs-779 327 2 m m PROPN iajs-779 327 3 is be AUX iajs-779 327 4 a a DET iajs-779 327 5 fully fully ADV iajs-779 327 6 semiprime semiprime NOUN iajs-779 327 7 r r NOUN iajs-779 327 8 - - NOUN iajs-779 327 9 module module NOUN iajs-779 327 10 . . PUNCT iajs-779 328 1 we we PRON iajs-779 328 2 end end VERB iajs-779 328 3 this this DET iajs-779 328 4 section section NOUN iajs-779 328 5 by by ADP iajs-779 328 6 the the DET iajs-779 328 7 following follow VERB iajs-779 328 8 corollary corollary NOUN iajs-779 328 9 . . PUNCT iajs-779 329 1 2.16 2.16 NUM iajs-779 329 2 corollary corollary NOUN iajs-779 329 3 : : PUNCT iajs-779 329 4 let let VERB iajs-779 329 5 m m PRON iajs-779 329 6 be be AUX iajs-779 329 7 a a DET iajs-779 329 8 finitely finitely ADV iajs-779 329 9 generated generate VERB iajs-779 329 10 multiplication multiplication NOUN iajs-779 329 11 r r NOUN iajs-779 329 12 - - NOUN iajs-779 329 13 module module NOUN iajs-779 329 14 . . PUNCT iajs-779 330 1 the the DET iajs-779 330 2 following follow VERB iajs-779 330 3 are be AUX iajs-779 330 4 equivalent equivalent ADJ iajs-779 330 5 : : PUNCT iajs-779 330 6 1 1 X iajs-779 330 7 . . X iajs-779 330 8 m m PROPN iajs-779 330 9 is be AUX iajs-779 330 10 retractable retractable ADJ iajs-779 330 11 and and CCONJ iajs-779 330 12 endr(m endr(m VERB iajs-779 330 13 ) ) PUNCT iajs-779 330 14 is be AUX iajs-779 330 15 semiprime semiprime NOUN iajs-779 330 16 . . PUNCT iajs-779 331 1 2 2 X iajs-779 331 2 . . X iajs-779 331 3 m m PROPN iajs-779 331 4 is be AUX iajs-779 331 5 fully fully ADV iajs-779 331 6 semiprime semiprime ADJ iajs-779 331 7 . . PUNCT iajs-779 332 1 3 3 X iajs-779 332 2 . . X iajs-779 332 3 m m PROPN iajs-779 332 4 is be AUX iajs-779 332 5 semiprime semiprime NOUN iajs-779 332 6 . . PUNCT iajs-779 333 1 4 4 X iajs-779 333 2 . . NUM iajs-779 333 3 annr(m annr(m NOUN iajs-779 333 4 ) ) PUNCT iajs-779 333 5 is be AUX iajs-779 333 6 semiprime semiprime NOUN iajs-779 333 7 . . PUNCT iajs-779 334 1 proof proof NOUN iajs-779 334 2 : : PUNCT iajs-779 334 3 it it PRON iajs-779 334 4 is be AUX iajs-779 334 5 obvious obvious ADJ iajs-779 334 6 . . PUNCT iajs-779 335 1 references reference NOUN iajs-779 335 2 1 1 NUM iajs-779 335 3 . . PUNCT iajs-779 335 4 abuihlail abuihlail PROPN iajs-779 335 5 , , PUNCT iajs-779 335 6 j. j. PROPN iajs-779 335 7 ( ( PUNCT iajs-779 335 8 2007 2007 NUM iajs-779 335 9 ) ) PUNCT iajs-779 335 10 , , PUNCT iajs-779 335 11 zariski zariski NOUN iajs-779 335 12 - - PUNCT iajs-779 335 13 like like ADJ iajs-779 335 14 topologies topology NOUN iajs-779 335 15 for for ADP iajs-779 335 16 modules module NOUN iajs-779 335 17 over over ADP iajs-779 335 18 commutative commutative ADJ iajs-779 335 19 rings ring NOUN iajs-779 335 20 , , PUNCT iajs-779 335 21 research research NOUN iajs-779 335 22 project project NOUN iajs-779 335 23 submitted submit VERB iajs-779 335 24 to to ADP iajs-779 335 25 king king PROPN iajs-779 335 26 fahad fahad PROPN iajs-779 335 27 univ univ PROPN iajs-779 335 28 . . PROPN iajs-779 335 29 of of ADP iajs-779 335 30 petroleum petroleum NOUN iajs-779 335 31 and and CCONJ iajs-779 335 32 minerals mineral NOUN iajs-779 335 33 , , PUNCT iajs-779 335 34 deanship deanship NOUN iajs-779 335 35 of of ADP iajs-779 335 36 scientific scientific ADJ iajs-779 335 37 research research NOUN iajs-779 335 38 , , PUNCT iajs-779 335 39 from from ADP iajs-779 335 40 internet internet NOUN iajs-779 335 41 . . PUNCT iajs-779 336 1 2 2 X iajs-779 336 2 . . X iajs-779 336 3 bican bican PROPN iajs-779 336 4 , , PUNCT iajs-779 336 5 l. l. PROPN iajs-779 336 6 ; ; PUNCT iajs-779 336 7 jambor jambor PROPN iajs-779 336 8 , , PUNCT iajs-779 336 9 p. p. NOUN iajs-779 336 10 ; ; PUNCT iajs-779 336 11 kepka kepka NOUN iajs-779 336 12 , , PUNCT iajs-779 336 13 t. t. NOUN iajs-779 336 14 and and CCONJ iajs-779 336 15 nëmec nëmec PROPN iajs-779 336 16 , , PUNCT iajs-779 336 17 p. p. NOUN iajs-779 336 18 ( ( PUNCT iajs-779 336 19 1980 1980 NUM iajs-779 336 20 ) ) PUNCT iajs-779 336 21 , , PUNCT iajs-779 336 22 prime prime ADJ iajs-779 336 23 and and CCONJ iajs-779 336 24 coprime coprime NOUN iajs-779 336 25 modules module NOUN iajs-779 336 26 , , PUNCT iajs-779 336 27 fund fund NOUN iajs-779 336 28 . . PUNCT iajs-779 337 1 m m AUX iajs-779 337 2 ath ath NOUN iajs-779 337 3 . . PUNCT iajs-779 338 1 107(1 107(1 NUM iajs-779 338 2 ) ) PUNCT iajs-779 338 3 , , PUNCT iajs-779 338 4 33 33 NUM iajs-779 338 5 - - SYM iajs-779 338 6 45 45 NUM iajs-779 338 7 . . PUNCT iajs-779 339 1 ibn ibn PROPN iajs-779 339 2 alhaitham alhaitham PROPN iajs-779 339 3 j. j. PROPN iajs-779 339 4 for for ADP iajs-779 339 5 pure pure ADJ iajs-779 339 6 & & CCONJ iajs-779 339 7 appl appl PROPN iajs-779 339 8 . . PUNCT iajs-779 340 1 sci sci PROPN iajs-779 340 2 . . PUNCT iajs-779 340 3 vol.24 vol.24 NOUN iajs-779 340 4 ( ( PUNCT iajs-779 340 5 2 2 NUM iajs-779 340 6 ) ) PUNCT iajs-779 340 7 2011 2011 NUM iajs-779 340 8 3 3 X iajs-779 340 9 . . X iajs-779 340 10 wijayanti wijayanti PROPN iajs-779 340 11 , , PUNCT iajs-779 340 12 e.i e.i PROPN iajs-779 340 13 . . PROPN iajs-779 341 1 ( ( PUNCT iajs-779 341 2 2006 2006 NUM iajs-779 341 3 ) ) PUNCT iajs-779 341 4 , , PUNCT iajs-779 341 5 coprime coprime NOUN iajs-779 341 6 modules module NOUN iajs-779 341 7 and and CCONJ iajs-779 341 8 comodules comodule NOUN iajs-779 341 9 , , PUNCT iajs-779 341 10 ph.d ph.d PROPN iajs-779 341 11 . . PUNCT iajs-779 342 1 thesis thesis PROPN iajs-779 342 2 heinrich heinrich PROPN iajs-779 342 3 - - PUNCT iajs-779 342 4 heine heine PROPN iajs-779 342 5 universität universität PROPN iajs-779 342 6 , , PUNCT iajs-779 342 7 düsseldorf düsseldorf PROPN iajs-779 342 8 . . PUNCT iajs-779 343 1 4 4 X iajs-779 343 2 . . X iajs-779 343 3 abass abass PROPN iajs-779 343 4 , , PUNCT iajs-779 343 5 m. m. NOUN iajs-779 343 6 s. s. PROPN iajs-779 343 7 ( ( PUNCT iajs-779 343 8 1999 1999 NUM iajs-779 343 9 ) ) PUNCT iajs-779 343 10 , , PUNCT iajs-779 343 11 on on ADP iajs-779 343 12 fully fully ADV iajs-779 343 13 stable stable ADJ iajs-779 343 14 modules module NOUN iajs-779 343 15 , , PUNCT iajs-779 343 16 ph.d.thesis ph.d.thesis PROPN iajs-779 343 17 , , PUNCT iajs-779 343 18 univ univ PROPN iajs-779 343 19 . . PROPN iajs-779 343 20 of of ADP iajs-779 343 21 baghdad baghdad PROPN iajs-779 343 22 . . PUNCT iajs-779 344 1 5 5 X iajs-779 344 2 . . X iajs-779 344 3 desale desale NOUN iajs-779 344 4 , , PUNCT iajs-779 344 5 g. g. PROPN iajs-779 344 6 and and CCONJ iajs-779 344 7 nicholson nicholson PROPN iajs-779 344 8 , , PUNCT iajs-779 344 9 k.w k.w PROPN iajs-779 344 10 . . PROPN iajs-779 344 11 ( ( PUNCT iajs-779 344 12 1981 1981 NUM iajs-779 344 13 ) ) PUNCT iajs-779 344 14 , , PUNCT iajs-779 344 15 endoprimtive endoprimtive ADJ iajs-779 344 16 ings ing NOUN iajs-779 344 17 , , PUNCT iajs-779 344 18 j.algebra j.algebra PROPN iajs-779 344 19 , , PUNCT iajs-779 344 20 70 70 NUM iajs-779 344 21 , , PUNCT iajs-779 344 22 548 548 NUM iajs-779 344 23 - - SYM iajs-779 344 24 560 560 NUM iajs-779 344 25 . . NOUN iajs-779 344 26 6 6 NUM iajs-779 344 27 . . X iajs-779 345 1 al al PROPN iajs-779 345 2 - - PUNCT iajs-779 345 3 sharide sharide NOUN iajs-779 345 4 , , PUNCT iajs-779 345 5 f. f. PROPN iajs-779 345 6 a. a. PROPN iajs-779 345 7 f. f. PROPN iajs-779 345 8 ( ( PUNCT iajs-779 345 9 2008 2008 NUM iajs-779 345 10 ) ) PUNCT iajs-779 345 11 , , PUNCT iajs-779 345 12 s s X iajs-779 345 13 - - PUNCT iajs-779 345 14 compactly compactly ADV iajs-779 345 15 packed pack VERB iajs-779 345 16 submodules submodule NOUN iajs-779 345 17 and and CCONJ iajs-779 345 18 semiprime semiprime NOUN iajs-779 345 19 modules module NOUN iajs-779 345 20 , , PUNCT iajs-779 345 21 ms.c.thesis ms.c.thesis PROPN iajs-779 345 22 , , PUNCT iajs-779 345 23 univ univ PROPN iajs-779 345 24 . . PUNCT iajs-779 345 25 of of ADP iajs-779 345 26 tikrit tikrit NOUN iajs-779 345 27 . . PUNCT iajs-779 346 1 7 7 X iajs-779 346 2 . . X iajs-779 346 3 abdul abdul PROPN iajs-779 346 4 - - PUNCT iajs-779 346 5 rahman rahman PROPN iajs-779 346 6 , , PUNCT iajs-779 346 7 a. a. NOUN iajs-779 346 8 and and CCONJ iajs-779 346 9 al al PROPN iajs-779 346 10 - - PUNCT iajs-779 346 11 hashimi hashimi PROPN iajs-779 346 12 , , PUNCT iajs-779 346 13 b. b. 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