id sid tid token lemma pos iajs-700 1 1 مجلة مجلة VERB iajs-700 1 2 إبن إبن VERB iajs-700 1 3 الھیثم الھیثم NOUN iajs-700 1 4 للعلوم للعلوم NOUN iajs-700 1 5 الصرفة الصرفة NOUN iajs-700 1 6 و و PRON iajs-700 1 7 التطبیقیة التطبیقیة PROPN iajs-700 1 8 2012 2012 NUM iajs-700 1 9 السنة السنة NOUN iajs-700 1 10 25 25 NUM iajs-700 1 11 المجلد المجلد NOUN iajs-700 1 12 1 1 NUM iajs-700 1 13 العدد العدد PROPN iajs-700 1 14 ibn ibn PROPN iajs-700 1 15 al al PROPN iajs-700 1 16 - - PUNCT iajs-700 1 17 haitham haitham PROPN iajs-700 1 18 journal journal PROPN iajs-700 1 19 for for ADP iajs-700 1 20 pure pure ADJ iajs-700 1 21 and and CCONJ iajs-700 1 22 applied apply VERB iajs-700 1 23 science science NOUN iajs-700 1 24 no no NOUN iajs-700 1 25 . . NOUN iajs-700 1 26 1 1 NUM iajs-700 1 27 vol vol NOUN iajs-700 1 28 . . PUNCT iajs-700 2 1 25 25 NUM iajs-700 2 2 year year NOUN iajs-700 2 3 2012 2012 NUM iajs-700 2 4 semiprime semiprime NOUN iajs-700 2 5 fuzzy fuzzy ADJ iajs-700 2 6 modules module NOUN iajs-700 2 7 m. m. NOUN iajs-700 2 8 a.hamil a.hamil PROPN iajs-700 2 9 department department PROPN iajs-700 2 10 of of ADP iajs-700 2 11 mathematics mathematics PROPN iajs-700 2 12 , , PUNCT iajs-700 2 13 college college NOUN iajs-700 2 14 of of ADP iajs-700 2 15 education education NOUN iajs-700 2 16 -ibn -ibn ADV iajs-700 2 17 - - PUNCT iajs-700 2 18 al al PROPN iajs-700 2 19 - - PUNCT iajs-700 2 20 haitham haitham PROPN iajs-700 2 21 university university PROPN iajs-700 2 22 of of ADP iajs-700 2 23 baghdad baghdad PROPN iajs-700 2 24 received receive VERB iajs-700 2 25 in in ADP iajs-700 2 26 : : PUNCT iajs-700 2 27 19 19 NUM iajs-700 2 28 june june PROPN iajs-700 2 29 2011 2011 NUM iajs-700 2 30 , , PUNCT iajs-700 2 31 accepted accept VERB iajs-700 2 32 in in ADP iajs-700 2 33 : : PUNCT iajs-700 2 34 20 20 NUM iajs-700 2 35 september september PROPN iajs-700 2 36 2011 2011 NUM iajs-700 2 37 abstract abstract NOUN iajs-700 2 38 in in ADP iajs-700 2 39 this this DET iajs-700 2 40 paper paper NOUN iajs-700 2 41 we we PRON iajs-700 2 42 introduce introduce VERB iajs-700 2 43 the the DET iajs-700 2 44 notion notion NOUN iajs-700 2 45 of of ADP iajs-700 2 46 semiprime semiprime NOUN iajs-700 2 47 fuzzy fuzzy ADJ iajs-700 2 48 module module NOUN iajs-700 2 49 as as ADP iajs-700 2 50 a a DET iajs-700 2 51 generalization generalization NOUN iajs-700 2 52 of of ADP iajs-700 2 53 semiprime semiprime NOUN iajs-700 2 54 module module NOUN iajs-700 2 55 . . PUNCT iajs-700 3 1 we we PRON iajs-700 3 2 investigate investigate VERB iajs-700 3 3 several several ADJ iajs-700 3 4 characterizations characterization NOUN iajs-700 3 5 and and CCONJ iajs-700 3 6 properties property NOUN iajs-700 3 7 of of ADP iajs-700 3 8 this this DET iajs-700 3 9 concept concept NOUN iajs-700 3 10 . . PUNCT iajs-700 4 1 key key ADJ iajs-700 4 2 words word NOUN iajs-700 4 3 : : PUNCT iajs-700 4 4 prime prime ADJ iajs-700 4 5 fuzzy fuzzy ADJ iajs-700 4 6 module module NOUN iajs-700 4 7 , , PUNCT iajs-700 4 8 semiprime semiprime NOUN iajs-700 4 9 module module NOUN iajs-700 4 10 , , PUNCT iajs-700 4 11 semiprime semiprime NOUN iajs-700 4 12 fuzzy fuzzy ADJ iajs-700 4 13 module module NOUN iajs-700 4 14 . . PUNCT iajs-700 5 1 introduction introduction NOUN iajs-700 5 2 the the DET iajs-700 5 3 notion notion NOUN iajs-700 5 4 of of ADP iajs-700 5 5 fuzzy fuzzy ADJ iajs-700 5 6 subsets subset NOUN iajs-700 5 7 of of ADP iajs-700 5 8 a a DET iajs-700 5 9 set set NOUN iajs-700 5 10 s s VERB iajs-700 5 11   ADJ iajs-700 5 12 as as ADP iajs-700 5 13 a a DET iajs-700 5 14 function function NOUN iajs-700 5 15 from from ADP iajs-700 5 16 s s PRON iajs-700 5 17 into into ADP iajs-700 5 18 [ [ X iajs-700 5 19 0,1 0,1 NUM iajs-700 5 20 ] ] PUNCT iajs-700 5 21 was be AUX iajs-700 5 22 first first ADV iajs-700 5 23 developed develop VERB iajs-700 5 24 by by ADP iajs-700 5 25 zadeh zadeh PROPN iajs-700 6 1 [ [ X iajs-700 6 2 1 1 NUM iajs-700 6 3 ] ] PUNCT iajs-700 6 4 . . PUNCT iajs-700 7 1 the the DET iajs-700 7 2 concept concept NOUN iajs-700 7 3 of of ADP iajs-700 7 4 fuzzy fuzzy ADJ iajs-700 7 5 modules module NOUN iajs-700 7 6 was be AUX iajs-700 7 7 introduced introduce VERB iajs-700 7 8 by by ADP iajs-700 7 9 negoita negoita NOUN iajs-700 7 10 and and CCONJ iajs-700 7 11 ralescu ralescu NOUN iajs-700 7 12 in in ADP iajs-700 7 13 [ [ X iajs-700 7 14 2 2 NUM iajs-700 7 15 ] ] PUNCT iajs-700 7 16 . . PUNCT iajs-700 8 1 the the DET iajs-700 8 2 concept concept NOUN iajs-700 8 3 of of ADP iajs-700 8 4 fuzzy fuzzy ADJ iajs-700 8 5 submodule submodule NOUN iajs-700 8 6 was be AUX iajs-700 8 7 introduced introduce VERB iajs-700 8 8 by by ADP iajs-700 8 9 mashinch mashinch NOUN iajs-700 8 10 and and CCONJ iajs-700 8 11 zahedi zahedi PROPN iajs-700 9 1 [ [ X iajs-700 9 2 3 3 NUM iajs-700 9 3 ] ] PUNCT iajs-700 9 4 . . PUNCT iajs-700 10 1 the the DET iajs-700 10 2 concept concept NOUN iajs-700 10 3 of of ADP iajs-700 10 4 fuzzy fuzzy ADJ iajs-700 10 5 ideal ideal NOUN iajs-700 10 6 of of ADP iajs-700 10 7 a a DET iajs-700 10 8 ring ring NOUN iajs-700 10 9 by by ADP iajs-700 10 10 liu liu PROPN iajs-700 10 11 in in ADP iajs-700 10 12 [ [ X iajs-700 10 13 4 4 NUM iajs-700 10 14 ] ] PUNCT iajs-700 10 15 . . PUNCT iajs-700 11 1 dauns daun NOUN iajs-700 11 2 in in ADP iajs-700 11 3 [ [ X iajs-700 11 4 5 5 NUM iajs-700 11 5 ] ] PUNCT iajs-700 11 6 introduced introduce VERB iajs-700 11 7 the the DET iajs-700 11 8 notion notion NOUN iajs-700 11 9 of of ADP iajs-700 11 10 semiprime semiprime NOUN iajs-700 11 11 submodules submodule NOUN iajs-700 11 12 as as ADP iajs-700 11 13 a a DET iajs-700 11 14 generalization generalization NOUN iajs-700 11 15 of of ADP iajs-700 11 16 semiprime semiprime NOUN iajs-700 11 17 ideals ideal NOUN iajs-700 11 18 of of ADP iajs-700 11 19 a a DET iajs-700 11 20 ring ring NOUN iajs-700 11 21 . . PUNCT iajs-700 12 1 eman eman NOUN iajs-700 12 2 in in ADP iajs-700 12 3 [ [ X iajs-700 12 4 6 6 NUM iajs-700 12 5 ] ] PUNCT iajs-700 12 6 studied study VERB iajs-700 12 7 semiprime semiprime NOUN iajs-700 12 8 submodules submodule NOUN iajs-700 12 9 . . PUNCT iajs-700 13 1 i.m.hadi i.m.hadi NOUN iajs-700 13 2 in in ADP iajs-700 13 3 [ [ X iajs-700 13 4 7 7 NUM iajs-700 13 5 ] ] PUNCT iajs-700 13 6 introduced introduce VERB iajs-700 13 7 the the DET iajs-700 13 8 notion notion NOUN iajs-700 13 9 of of ADP iajs-700 13 10 semiprime semiprime NOUN iajs-700 13 11 fuzzy fuzzy ADJ iajs-700 13 12 ideals ideal NOUN iajs-700 13 13 of of ADP iajs-700 13 14 a a DET iajs-700 13 15 ring ring NOUN iajs-700 13 16 also also ADV iajs-700 13 17 introduced introduce VERB iajs-700 13 18 semiprime semiprime NOUN iajs-700 13 19 fuzzy fuzzy ADJ iajs-700 13 20 submodules submodule NOUN iajs-700 13 21 of of ADP iajs-700 13 22 fuzzy fuzzy ADJ iajs-700 13 23 module module NOUN iajs-700 13 24 in in ADP iajs-700 13 25 [ [ X iajs-700 13 26 8 8 NUM iajs-700 13 27 ] ] PUNCT iajs-700 13 28 . . PUNCT iajs-700 14 1 frias frias PROPN iajs-700 14 2 in in ADP iajs-700 14 3 [ [ X iajs-700 14 4 9 9 NUM iajs-700 14 5 ] ] PUNCT iajs-700 14 6 studies study NOUN iajs-700 14 7 semiprime semiprime NOUN iajs-700 14 8 module module NOUN iajs-700 14 9 . . PUNCT iajs-700 15 1 in in ADP iajs-700 15 2 this this DET iajs-700 15 3 paper paper NOUN iajs-700 15 4 we we PRON iajs-700 15 5 introduce introduce VERB iajs-700 15 6 the the DET iajs-700 15 7 notion notion NOUN iajs-700 15 8 of of ADP iajs-700 15 9 semiprime semiprime NOUN iajs-700 15 10 fuzzy fuzzy ADJ iajs-700 15 11 modules module NOUN iajs-700 15 12 as as ADP iajs-700 15 13 a a DET iajs-700 15 14 generalization generalization NOUN iajs-700 15 15 of of ADP iajs-700 15 16 r r NOUN iajs-700 15 17 - - PUNCT iajs-700 15 18 semiprime semiprime NOUN iajs-700 15 19 modules module NOUN iajs-700 15 20 and and CCONJ iajs-700 15 21 give give VERB iajs-700 15 22 many many ADJ iajs-700 15 23 properties property NOUN iajs-700 15 24 of of ADP iajs-700 15 25 this this DET iajs-700 15 26 concept concept NOUN iajs-700 15 27 . . PUNCT iajs-700 16 1 throughout throughout ADP iajs-700 16 2 this this DET iajs-700 16 3 paper paper NOUN iajs-700 16 4 r r NOUN iajs-700 16 5 is be AUX iajs-700 16 6 commutative commutative ADJ iajs-700 16 7 ring ring NOUN iajs-700 16 8 with with ADP iajs-700 16 9 unity unity NOUN iajs-700 16 10 , , PUNCT iajs-700 16 11 m m VERB iajs-700 16 12 is be AUX iajs-700 16 13 an an DET iajs-700 16 14 r r NOUN iajs-700 16 15 - - PUNCT iajs-700 16 16 module module NOUN iajs-700 16 17 and and CCONJ iajs-700 16 18 x x NOUN iajs-700 16 19 is be AUX iajs-700 16 20 a a DET iajs-700 16 21 fuzzy fuzzy ADJ iajs-700 16 22 module module NOUN iajs-700 16 23 of of ADP iajs-700 16 24 an an DET iajs-700 16 25 r r NOUN iajs-700 16 26 - - PUNCT iajs-700 16 27 module module NOUN iajs-700 16 28 m. m. NOUN iajs-700 16 29 1preliminaries 1preliminaries NUM iajs-700 16 30 in in ADP iajs-700 16 31 this this DET iajs-700 16 32 section section NOUN iajs-700 16 33 , , PUNCT iajs-700 16 34 we we PRON iajs-700 16 35 shall shall AUX iajs-700 16 36 formulate formulate VERB iajs-700 16 37 the the DET iajs-700 16 38 preliminary preliminary ADJ iajs-700 16 39 definitions definition NOUN iajs-700 16 40 and and CCONJ iajs-700 16 41 results result NOUN iajs-700 16 42 that that PRON iajs-700 16 43 are be AUX iajs-700 16 44 required require VERB iajs-700 16 45 later later ADV iajs-700 16 46 in in ADP iajs-700 16 47 this this DET iajs-700 16 48 paper paper NOUN iajs-700 16 49 . . PUNCT iajs-700 17 1 1.1 1.1 NUM iajs-700 17 2 definition definition NOUN iajs-700 17 3 : : PUNCT iajs-700 17 4 [ [ X iajs-700 17 5 1 1 X iajs-700 17 6 ] ] X iajs-700 17 7 let let VERB iajs-700 17 8 s s PRON iajs-700 17 9 be be AUX iajs-700 17 10 a a DET iajs-700 17 11 non non ADJ iajs-700 17 12 - - ADJ iajs-700 17 13 empty empty ADJ iajs-700 17 14 set set NOUN iajs-700 17 15 . . PUNCT iajs-700 18 1 a a DET iajs-700 18 2 fuzzy fuzzy ADJ iajs-700 18 3 set set VERB iajs-700 18 4 a a PRON iajs-700 18 5 in in ADP iajs-700 18 6 s s PROPN iajs-700 18 7 ( ( PUNCT iajs-700 18 8 a a DET iajs-700 18 9 fuzzy fuzzy ADJ iajs-700 18 10 subset subset NOUN iajs-700 18 11 of of ADP iajs-700 18 12 s s NOUN iajs-700 18 13 ) ) PUNCT iajs-700 18 14 is be AUX iajs-700 18 15 a a DET iajs-700 18 16 function function NOUN iajs-700 18 17 from from ADP iajs-700 18 18 s s PRON iajs-700 18 19 into into ADP iajs-700 18 20 [ [ X iajs-700 18 21 0,1 0,1 NUM iajs-700 18 22 ] ] PUNCT iajs-700 18 23 . . PUNCT iajs-700 19 1 1.2 1.2 NUM iajs-700 19 2 definition definition NOUN iajs-700 19 3 : : PUNCT iajs-700 19 4 [ [ X iajs-700 19 5 2 2 X iajs-700 19 6 ] ] X iajs-700 19 7 let let VERB iajs-700 19 8 xt xt NUM iajs-700 19 9 : : PUNCT iajs-700 19 10 s s AUX iajs-700 19 11 [0,1 [0,1 NOUN iajs-700 19 12 ] ] PUNCT iajs-700 19 13 be be AUX iajs-700 19 14 a a DET iajs-700 19 15 fuzzy fuzzy ADJ iajs-700 19 16 set set NOUN iajs-700 19 17 in in ADP iajs-700 19 18 s s PROPN iajs-700 19 19 , , PUNCT iajs-700 19 20 where where SCONJ iajs-700 19 21 xs xs PROPN iajs-700 19 22 , , PUNCT iajs-700 19 23 t[0,1 t[0,1 NOUN iajs-700 19 24 ] ] PUNCT iajs-700 19 25 defined define VERB iajs-700 19 26 by by ADP iajs-700 19 27 : : PUNCT iajs-700 19 28 t t PROPN iajs-700 19 29 t t PROPN iajs-700 19 30 if if SCONJ iajs-700 19 31 x x PROPN iajs-700 19 32 y y NOUN iajs-700 19 33 x x X iajs-700 19 34 ( ( PUNCT iajs-700 19 35 y y NOUN iajs-700 19 36 ) ) PUNCT iajs-700 19 37 0 0 PUNCT iajs-700 20 1 if if SCONJ iajs-700 20 2 x x PRON iajs-700 20 3 y y PROPN iajs-700 20 4   NOUN iajs-700 20 5   PROPN iajs-700 20 6   NUM iajs-700 20 7   VERB iajs-700 20 8 for for ADP iajs-700 20 9 all all DET iajs-700 20 10 y y PROPN iajs-700 20 11   PROPN iajs-700 20 12 s. s. PROPN iajs-700 20 13 xt xt PROPN iajs-700 20 14 is be AUX iajs-700 20 15 called call VERB iajs-700 20 16 a a DET iajs-700 20 17 fuzzy fuzzy ADJ iajs-700 20 18 singleton singleton NOUN iajs-700 20 19 . . PUNCT iajs-700 21 1 1.3 1.3 NUM iajs-700 21 2 proposition proposition NOUN iajs-700 21 3 : : PUNCT iajs-700 22 1 [ [ X iajs-700 22 2 3 3 X iajs-700 22 3 ] ] X iajs-700 22 4 let let VERB iajs-700 22 5 at at ADP iajs-700 22 6 , , PUNCT iajs-700 22 7 bk bk NOUN iajs-700 22 8 be be AUX iajs-700 22 9 two two NUM iajs-700 22 10 fuzzy fuzzy ADJ iajs-700 22 11 singletons singleton NOUN iajs-700 22 12 of of ADP iajs-700 22 13 a a DET iajs-700 22 14 set set NOUN iajs-700 22 15 s. s. PROPN iajs-700 22 16 if if SCONJ iajs-700 22 17 at at ADP iajs-700 22 18 = = SYM iajs-700 22 19 bk bk PROPN iajs-700 22 20 , , PUNCT iajs-700 22 21 then then ADV iajs-700 22 22 a a DET iajs-700 22 23 = = SYM iajs-700 22 24 b b PROPN iajs-700 22 25 and and CCONJ iajs-700 22 26 t t NOUN iajs-700 22 27 = = SYM iajs-700 22 28 k k PROPN iajs-700 22 29 , , PUNCT iajs-700 22 30 where where SCONJ iajs-700 22 31 t t PROPN iajs-700 22 32 , , PUNCT iajs-700 22 33 k k PROPN iajs-700 22 34   PROPN iajs-700 23 1 [ [ X iajs-700 23 2 0,1 0,1 NUM iajs-700 23 3 ] ] PUNCT iajs-700 23 4 . . PUNCT iajs-700 24 1 مجلة مجلة PROPN iajs-700 24 2 إبن إبن VERB iajs-700 24 3 الھیثم الھیثم NOUN iajs-700 24 4 للعلوم للعلوم NOUN iajs-700 24 5 الصرفة الصرفة NOUN iajs-700 24 6 و و PRON iajs-700 24 7 التطبیقیة التطبیقیة PROPN iajs-700 24 8 2012 2012 NUM iajs-700 24 9 السنة السنة NOUN iajs-700 25 1 25 25 NUM iajs-700 25 2 المجلد المجلد NOUN iajs-700 25 3 1 1 NUM iajs-700 25 4 العدد العدد PROPN iajs-700 25 5 ibn ibn PROPN iajs-700 25 6 al al PROPN iajs-700 25 7 - - PUNCT iajs-700 25 8 haitham haitham PROPN iajs-700 25 9 journal journal PROPN iajs-700 25 10 for for ADP iajs-700 25 11 pure pure ADJ iajs-700 25 12 and and CCONJ iajs-700 25 13 applied apply VERB iajs-700 25 14 science science NOUN iajs-700 25 15 no no NOUN iajs-700 25 16 . . NOUN iajs-700 25 17 1 1 NUM iajs-700 25 18 vol vol NOUN iajs-700 25 19 . . PUNCT iajs-700 26 1 25 25 NUM iajs-700 26 2 year year NOUN iajs-700 26 3 2012 2012 NUM iajs-700 26 4 1.4 1.4 NUM iajs-700 26 5 definition definition NOUN iajs-700 26 6 : : PUNCT iajs-700 27 1 [ [ X iajs-700 27 2 4 4 X iajs-700 27 3 ] ] PUNCT iajs-700 27 4 let let VERB iajs-700 27 5 a a PRON iajs-700 27 6 and and CCONJ iajs-700 27 7 b b NOUN iajs-700 27 8 be be AUX iajs-700 27 9 two two NUM iajs-700 27 10 fuzzy fuzzy ADJ iajs-700 27 11 sets set NOUN iajs-700 27 12 in in ADP iajs-700 27 13 s s PROPN iajs-700 27 14 , , PUNCT iajs-700 27 15 then then ADV iajs-700 27 16 : : PUNCT iajs-700 27 17 1a 1a X iajs-700 28 1 = = SYM iajs-700 28 2 b b X iajs-700 28 3 iff iff PROPN iajs-700 28 4 a(x a(x NOUN iajs-700 28 5 ) ) PUNCT iajs-700 28 6 = = SYM iajs-700 28 7 b(x b(x NOUN iajs-700 28 8 ) ) PUNCT iajs-700 28 9 , , PUNCT iajs-700 28 10 for for ADP iajs-700 28 11 all all DET iajs-700 28 12 x x ADJ iajs-700 28 13   PROPN iajs-700 28 14 s. s. PROPN iajs-700 28 15 2a 2a PROPN iajs-700 28 16   PROPN iajs-700 29 1 b b PROPN iajs-700 29 2 iff iff PROPN iajs-700 29 3 a(x a(x NOUN iajs-700 29 4 ) ) PUNCT iajs-700 29 5   NOUN iajs-700 29 6 b(x b(x NOUN iajs-700 29 7 ) ) PUNCT iajs-700 29 8 , , PUNCT iajs-700 29 9 for for ADP iajs-700 29 10 all all DET iajs-700 29 11 x x ADJ iajs-700 29 12   PROPN iajs-700 29 13 s. s. PROPN iajs-700 29 14 3(ab)(x 3(ab)(x PROPN iajs-700 29 15 ) ) PUNCT iajs-700 30 1 = = SYM iajs-700 30 2 min{a(x),b(x min{a(x),b(x PROPN iajs-700 30 3 ) ) PUNCT iajs-700 30 4 } } PUNCT iajs-700 30 5 , , PUNCT iajs-700 30 6 for for ADP iajs-700 30 7 all all DET iajs-700 30 8 x x ADJ iajs-700 30 9   NOUN iajs-700 30 10 s s NOUN iajs-700 30 11 , , PUNCT iajs-700 30 12 [ [ X iajs-700 30 13 2 2 NUM iajs-700 30 14 ] ] PUNCT iajs-700 30 15 . . PUNCT iajs-700 31 1 1.5 1.5 NUM iajs-700 31 2 definition definition NOUN iajs-700 31 3 : : PUNCT iajs-700 31 4 [ [ X iajs-700 31 5 5 5 NUM iajs-700 31 6 ] ] PUNCT iajs-700 31 7 let let VERB iajs-700 31 8 a a DET iajs-700 31 9 be be AUX iajs-700 31 10 any any DET iajs-700 31 11 fuzzy fuzzy ADJ iajs-700 31 12 set set NOUN iajs-700 31 13 in in ADP iajs-700 31 14 s s PRON iajs-700 31 15 for for ADP iajs-700 31 16 all all DET iajs-700 31 17 t t NOUN iajs-700 31 18   NOUN iajs-700 32 1 [ [ X iajs-700 32 2 0,1 0,1 NUM iajs-700 32 3 ] ] PUNCT iajs-700 32 4 , , PUNCT iajs-700 32 5 the the DET iajs-700 32 6 set set NOUN iajs-700 32 7 at at ADP iajs-700 32 8 = = PUNCT iajs-700 32 9 { { PUNCT iajs-700 32 10 x x PROPN iajs-700 32 11   PROPN iajs-700 32 12 s s NOUN iajs-700 32 13 , , PUNCT iajs-700 32 14 a(x a(x PROPN iajs-700 32 15 ) ) PUNCT iajs-700 32 16   NUM iajs-700 32 17 t t PROPN iajs-700 32 18 } } PUNCT iajs-700 32 19 is be AUX iajs-700 32 20 called call VERB iajs-700 32 21 a a DET iajs-700 32 22 level level NOUN iajs-700 32 23 subset subset NOUN iajs-700 32 24 of of ADP iajs-700 32 25 a. a. PROPN iajs-700 32 26 1.6 1.6 NUM iajs-700 32 27 remark remark NOUN iajs-700 32 28 : : PUNCT iajs-700 33 1 [ [ X iajs-700 33 2 1 1 X iajs-700 33 3 ] ] X iajs-700 33 4 the the DET iajs-700 33 5 following follow VERB iajs-700 33 6 properties property NOUN iajs-700 33 7 of of ADP iajs-700 33 8 level level NOUN iajs-700 33 9 subsets subset NOUN iajs-700 33 10 hold hold VERB iajs-700 33 11 for for ADP iajs-700 33 12 each each DET iajs-700 33 13 t t NOUN iajs-700 33 14   NOUN iajs-700 34 1 [ [ X iajs-700 34 2 0,1 0,1 NUM iajs-700 34 3 ] ] PUNCT iajs-700 34 4 . . PUNCT iajs-700 35 1 1(ab)t 1(ab)t PROPN iajs-700 35 2 = = SYM iajs-700 35 3 at at ADP iajs-700 35 4   NOUN iajs-700 35 5 bt bt PROPN iajs-700 35 6 . . PROPN iajs-700 35 7 2a 2a NUM iajs-700 36 1 = = SYM iajs-700 36 2 b b X iajs-700 36 3 iff iff PROPN iajs-700 36 4 at at ADP iajs-700 36 5 = = PROPN iajs-700 36 6 bt bt PROPN iajs-700 36 7 . . PROPN iajs-700 36 8 1.7 1.7 NUM iajs-700 36 9 definition definition NOUN iajs-700 36 10 : : PUNCT iajs-700 36 11 [ [ X iajs-700 36 12 1 1 X iajs-700 36 13 ] ] PUNCT iajs-700 36 14 let let VERB iajs-700 36 15 f f PRON iajs-700 36 16 be be AUX iajs-700 36 17 a a DET iajs-700 36 18 mapping mapping NOUN iajs-700 36 19 from from ADP iajs-700 36 20 a a DET iajs-700 36 21 set set NOUN iajs-700 36 22 m m NOUN iajs-700 36 23 into into ADP iajs-700 36 24 a a DET iajs-700 36 25 set set NOUN iajs-700 36 26 n n CCONJ iajs-700 36 27 , , PUNCT iajs-700 36 28 let let VERB iajs-700 36 29 a a PRON iajs-700 36 30 be be AUX iajs-700 36 31 a a DET iajs-700 36 32 fuzzy fuzzy ADJ iajs-700 36 33 set set NOUN iajs-700 36 34 in in ADP iajs-700 36 35 m m PROPN iajs-700 36 36 and and CCONJ iajs-700 36 37 b b X iajs-700 36 38 be be AUX iajs-700 36 39 a a DET iajs-700 36 40 fuzzy fuzzy ADJ iajs-700 36 41 set set NOUN iajs-700 36 42 in in ADP iajs-700 36 43 n. n. NOUN iajs-700 36 44 the the DET iajs-700 36 45 image image NOUN iajs-700 36 46 of of ADP iajs-700 36 47 a a DET iajs-700 36 48 denoted denote VERB iajs-700 36 49 by by ADP iajs-700 36 50 f(a f(a PROPN iajs-700 36 51 ) ) PUNCT iajs-700 36 52 is be AUX iajs-700 36 53 the the DET iajs-700 36 54 fuzzy fuzzy ADJ iajs-700 36 55 set set NOUN iajs-700 36 56 in in ADP iajs-700 36 57 n n CCONJ iajs-700 36 58 defined define VERB iajs-700 36 59 by by ADP iajs-700 36 60 : : PUNCT iajs-700 36 61 1 1 NUM iajs-700 36 62 1sup{a(z 1sup{a(z NUM iajs-700 36 63 ): ): PUNCT iajs-700 36 64 z z PROPN iajs-700 36 65 f f X iajs-700 36 66 ( ( PUNCT iajs-700 36 67 y y NOUN iajs-700 36 68 ) ) PUNCT iajs-700 36 69 } } PUNCT iajs-700 36 70 if if SCONJ iajs-700 36 71 f f PROPN iajs-700 36 72 ( ( PUNCT iajs-700 36 73 y y NOUN iajs-700 36 74 ) ) PUNCT iajs-700 36 75 for for ADP iajs-700 36 76 all all DET iajs-700 36 77 y y PROPN iajs-700 36 78 n n CCONJ iajs-700 36 79 , , PUNCT iajs-700 36 80 f(a)(y f(a)(y NOUN iajs-700 36 81 ) ) PUNCT iajs-700 36 82 0 0 PUNCT iajs-700 37 1 otherwise otherwise ADV iajs-700 37 2   PROPN iajs-700 37 3   PROPN iajs-700 37 4   NOUN iajs-700 37 5   NOUN iajs-700 37 6   NUM iajs-700 37 7   NUM iajs-700 38 1   NOUN iajs-700 38 2   INTJ iajs-700 38 3 and and CCONJ iajs-700 38 4 the the DET iajs-700 38 5 inverse inverse ADJ iajs-700 38 6 image image NOUN iajs-700 38 7 of of ADP iajs-700 38 8 b b NOUN iajs-700 38 9 , , PUNCT iajs-700 38 10 denoted denote VERB iajs-700 38 11 by by ADP iajs-700 38 12 f f PROPN iajs-700 38 13 – – PUNCT iajs-700 38 14 1(b 1(b NUM iajs-700 38 15 ) ) PUNCT iajs-700 38 16 is be AUX iajs-700 38 17 the the DET iajs-700 38 18 fuzzy fuzzy ADJ iajs-700 38 19 set set NOUN iajs-700 38 20 in in ADP iajs-700 38 21 m m PROPN iajs-700 38 22 defined define VERB iajs-700 38 23 by by ADP iajs-700 38 24 : : PUNCT iajs-700 38 25 f f X iajs-700 38 26 – – PUNCT iajs-700 38 27 1 1 NUM iajs-700 38 28 ( ( PUNCT iajs-700 38 29 b)(x b)(x NOUN iajs-700 38 30 ) ) PUNCT iajs-700 38 31 = = SYM iajs-700 38 32 b(f(x b(f(x PROPN iajs-700 38 33 ) ) PUNCT iajs-700 38 34 ) ) PUNCT iajs-700 38 35 , , PUNCT iajs-700 38 36 for for ADP iajs-700 38 37 all all DET iajs-700 38 38 x x ADJ iajs-700 38 39   PROPN iajs-700 38 40 m. m. NOUN iajs-700 38 41 1.8 1.8 NUM iajs-700 38 42 definition definition NOUN iajs-700 38 43 : : PUNCT iajs-700 39 1 [ [ X iajs-700 39 2 2 2 X iajs-700 39 3 ] ] X iajs-700 39 4 let let VERB iajs-700 39 5 m m PRON iajs-700 39 6 be be AUX iajs-700 39 7 an an DET iajs-700 39 8 r r NOUN iajs-700 39 9 - - PUNCT iajs-700 39 10 module module NOUN iajs-700 39 11 . . PUNCT iajs-700 40 1 a a DET iajs-700 40 2 fuzzy fuzzy ADJ iajs-700 40 3 set set NOUN iajs-700 40 4 x x PUNCT iajs-700 40 5 of of ADP iajs-700 40 6 m m PROPN iajs-700 40 7 is be AUX iajs-700 40 8 called call VERB iajs-700 40 9 fuzzy fuzzy ADJ iajs-700 40 10 module module NOUN iajs-700 40 11 of of ADP iajs-700 40 12 an an DET iajs-700 40 13 r r NOUN iajs-700 40 14 - - PUNCT iajs-700 40 15 module module NOUN iajs-700 40 16 m m NOUN iajs-700 40 17 if if SCONJ iajs-700 40 18 : : PUNCT iajs-700 40 19 1x(x 1x(x X iajs-700 40 20 – – PUNCT iajs-700 40 21 y y NOUN iajs-700 40 22 ) ) PUNCT iajs-700 40 23   NUM iajs-700 40 24 min{x(x min{x(x PROPN iajs-700 40 25 ) ) PUNCT iajs-700 40 26 , , PUNCT iajs-700 40 27 x(y x(y PROPN iajs-700 40 28 ) ) PUNCT iajs-700 40 29 } } PUNCT iajs-700 40 30 for for ADP iajs-700 40 31 all all DET iajs-700 40 32 x x NOUN iajs-700 40 33 , , PUNCT iajs-700 40 34 y y PROPN iajs-700 40 35   PROPN iajs-700 40 36 m. m. NOUN iajs-700 40 37 2x(rx 2x(rx NUM iajs-700 40 38 ) ) PUNCT iajs-700 40 39   NUM iajs-700 40 40 x(x x(x NOUN iajs-700 40 41 ) ) PUNCT iajs-700 40 42 , , PUNCT iajs-700 40 43 for for ADP iajs-700 40 44 all all DET iajs-700 40 45 x x PRON iajs-700 40 46   NOUN iajs-700 40 47 m m NOUN iajs-700 40 48 and and CCONJ iajs-700 40 49 r r PROPN iajs-700 40 50   PROPN iajs-700 40 51 r. r. PROPN iajs-700 40 52 3x(0 3x(0 NUM iajs-700 40 53 ) ) PUNCT iajs-700 41 1 = = PUNCT iajs-700 41 2 1 1 NUM iajs-700 41 3 . . NOUN iajs-700 41 4 1.9 1.9 NUM iajs-700 41 5 definition definition NOUN iajs-700 41 6 : : PUNCT iajs-700 42 1 [ [ X iajs-700 42 2 5 5 NUM iajs-700 42 3 ] ] PUNCT iajs-700 42 4 let let VERB iajs-700 42 5 x x PRON iajs-700 42 6 and and CCONJ iajs-700 42 7 a a DET iajs-700 42 8 be be AUX iajs-700 42 9 two two NUM iajs-700 42 10 fuzzy fuzzy ADJ iajs-700 42 11 modules module NOUN iajs-700 42 12 of of ADP iajs-700 42 13 r r NOUN iajs-700 42 14 - - PUNCT iajs-700 42 15 module module NOUN iajs-700 42 16 m. m. NOUN iajs-700 42 17 a a PRON iajs-700 42 18 is be AUX iajs-700 42 19 called call VERB iajs-700 42 20 a a DET iajs-700 42 21 fuzzy fuzzy ADJ iajs-700 42 22 submodule submodule NOUN iajs-700 42 23 of of ADP iajs-700 42 24 x x PRON iajs-700 42 25 if if SCONJ iajs-700 42 26 a a DET iajs-700 42 27   PROPN iajs-700 42 28 x. x. NOUN iajs-700 42 29 1.10 1.10 NUM iajs-700 42 30 proposition proposition NOUN iajs-700 42 31 : : PUNCT iajs-700 43 1 [ [ X iajs-700 43 2 6 6 NUM iajs-700 43 3 ] ] PUNCT iajs-700 43 4 let let VERB iajs-700 43 5 a a PRON iajs-700 43 6 be be AUX iajs-700 43 7 a a DET iajs-700 43 8 fuzzy fuzzy ADJ iajs-700 43 9 set set NOUN iajs-700 43 10 of of ADP iajs-700 43 11 an an DET iajs-700 43 12 r r NOUN iajs-700 43 13 - - PUNCT iajs-700 43 14 module module NOUN iajs-700 43 15 m. m. NOUN iajs-700 43 16 then then ADV iajs-700 44 1 the the DET iajs-700 44 2 level level NOUN iajs-700 44 3 subset subset VERB iajs-700 44 4 at at ADP iajs-700 44 5 , , PUNCT iajs-700 44 6 t t PROPN iajs-700 44 7   NOUN iajs-700 44 8 [ [ X iajs-700 44 9 0,1 0,1 NUM iajs-700 44 10 ] ] PUNCT iajs-700 44 11 is be AUX iajs-700 44 12 a a DET iajs-700 44 13 submodule submodule NOUN iajs-700 44 14 of of ADP iajs-700 44 15 m m PROPN iajs-700 44 16 iff iff PROPN iajs-700 44 17 a a PRON iajs-700 44 18 is be AUX iajs-700 44 19 a a DET iajs-700 44 20 fuzzy fuzzy ADJ iajs-700 44 21 submodule submodule NOUN iajs-700 44 22 of of ADP iajs-700 44 23 x x PROPN iajs-700 44 24 , , PUNCT iajs-700 44 25 where where SCONJ iajs-700 44 26 x x PRON iajs-700 44 27 is be AUX iajs-700 44 28 a a DET iajs-700 44 29 fuzzy fuzzy ADJ iajs-700 44 30 module module NOUN iajs-700 44 31 of of ADP iajs-700 44 32 an an DET iajs-700 44 33 r r NOUN iajs-700 44 34 - - PUNCT iajs-700 44 35 module module NOUN iajs-700 44 36 m. m. NOUN iajs-700 44 37 1.11 1.11 NUM iajs-700 44 38 remark remark NOUN iajs-700 44 39 : : PUNCT iajs-700 45 1 [ [ X iajs-700 45 2 5 5 X iajs-700 45 3 ] ] PUNCT iajs-700 45 4 if if SCONJ iajs-700 45 5 x x PRON iajs-700 45 6 is be AUX iajs-700 45 7 a a DET iajs-700 45 8 fuzzy fuzzy ADJ iajs-700 45 9 module module NOUN iajs-700 45 10 of of ADP iajs-700 45 11 an an DET iajs-700 45 12 r r NOUN iajs-700 45 13 - - PUNCT iajs-700 45 14 module module NOUN iajs-700 45 15 m m NOUN iajs-700 45 16 and and CCONJ iajs-700 45 17 xt xt X iajs-700 45 18   PROPN iajs-700 45 19 x x X iajs-700 45 20 then then ADV iajs-700 45 21 for for ADP iajs-700 45 22 all all DET iajs-700 45 23 fuzzy fuzzy ADJ iajs-700 45 24 singleton singleton NOUN iajs-700 45 25 rk rk NOUN iajs-700 45 26 of of ADP iajs-700 45 27 r r PROPN iajs-700 45 28 , , PUNCT iajs-700 45 29 rk rk NOUN iajs-700 45 30 xt xt PROPN iajs-700 45 31 = = PUNCT iajs-700 45 32 ( ( PUNCT iajs-700 45 33 rx) rx) PROPN iajs-700 45 34 , , PUNCT iajs-700 45 35 where where SCONJ iajs-700 45 36   ADJ iajs-700 45 37 = = SYM iajs-700 45 38 min{k min{k NOUN iajs-700 45 39 , , PUNCT iajs-700 45 40 t t PROPN iajs-700 45 41 } } PUNCT iajs-700 45 42 . . PUNCT iajs-700 46 1 1.12 1.12 NUM iajs-700 46 2 definition definition NOUN iajs-700 46 3 : : PUNCT iajs-700 46 4 [ [ X iajs-700 46 5 7 7 X iajs-700 46 6 ] ] X iajs-700 46 7 a a DET iajs-700 46 8 fuzzy fuzzy ADJ iajs-700 46 9 subset subset NOUN iajs-700 46 10 k k PROPN iajs-700 46 11 of of ADP iajs-700 46 12 a a DET iajs-700 46 13 ring ring NOUN iajs-700 46 14 r r NOUN iajs-700 46 15 is be AUX iajs-700 46 16 called call VERB iajs-700 46 17 a a DET iajs-700 46 18 fuzzy fuzzy ADJ iajs-700 46 19 ideal ideal NOUN iajs-700 46 20 of of ADP iajs-700 46 21 r r NOUN iajs-700 46 22 if if SCONJ iajs-700 46 23 for for ADP iajs-700 46 24 each each DET iajs-700 46 25 x x NOUN iajs-700 46 26 , , PUNCT iajs-700 46 27 y y PROPN iajs-700 46 28   NOUN iajs-700 46 29 r r NOUN iajs-700 46 30 : : PUNCT iajs-700 46 31 مجلة مجلة NOUN iajs-700 46 32 إبن إبن VERB iajs-700 46 33 الھیثم الھیثم NOUN iajs-700 46 34 للعلوم للعلوم NOUN iajs-700 46 35 الصرفة الصرفة NOUN iajs-700 46 36 و و PRON iajs-700 46 37 التطبیقیة التطبیقیة PROPN iajs-700 46 38 2012 2012 NUM iajs-700 46 39 السنة السنة NOUN iajs-700 46 40 25 25 NUM iajs-700 47 1 المجلد المجلد NOUN iajs-700 48 1 1 1 NUM iajs-700 49 1 العدد العدد PROPN iajs-700 49 2 ibn ibn PROPN iajs-700 49 3 al al PROPN iajs-700 49 4 - - PUNCT iajs-700 49 5 haitham haitham PROPN iajs-700 49 6 journal journal PROPN iajs-700 49 7 for for ADP iajs-700 49 8 pure pure ADJ iajs-700 49 9 and and CCONJ iajs-700 49 10 applied apply VERB iajs-700 49 11 science science NOUN iajs-700 49 12 no no NOUN iajs-700 49 13 . . NOUN iajs-700 49 14 1 1 NUM iajs-700 49 15 vol vol NOUN iajs-700 49 16 . . PUNCT iajs-700 50 1 25 25 NUM iajs-700 50 2 year year NOUN iajs-700 50 3 2012 2012 NUM iajs-700 50 4 1k(x 1k(x PROPN iajs-700 50 5 – – PUNCT iajs-700 50 6 y y NOUN iajs-700 50 7 ) ) PUNCT iajs-700 50 8   NUM iajs-700 50 9 min{k(x),k(y min{k(x),k(y PROPN iajs-700 50 10 ) ) PUNCT iajs-700 50 11 } } PUNCT iajs-700 50 12 . . PUNCT iajs-700 51 1 2k(xy 2k(xy NOUN iajs-700 51 2 ) ) PUNCT iajs-700 51 3   NUM iajs-700 51 4 max{k(x),k(y max{k(x),k(y PROPN iajs-700 51 5 ) ) PUNCT iajs-700 51 6 } } PUNCT iajs-700 51 7 . . PUNCT iajs-700 52 1 1.13 1.13 NUM iajs-700 52 2 proposition proposition NOUN iajs-700 52 3 : : PUNCT iajs-700 52 4 [ [ X iajs-700 52 5 7 7 X iajs-700 52 6 ] ] X iajs-700 52 7 a a DET iajs-700 52 8 fuzzy fuzzy ADJ iajs-700 52 9 subset subset NOUN iajs-700 52 10 k k PROPN iajs-700 52 11 of of ADP iajs-700 52 12 a a DET iajs-700 52 13 ring ring NOUN iajs-700 52 14 r r NOUN iajs-700 52 15 is be AUX iajs-700 52 16 a a DET iajs-700 52 17 fuzzy fuzzy ADJ iajs-700 52 18 ideal ideal NOUN iajs-700 52 19 iff iff PROPN iajs-700 52 20 kt kt PROPN iajs-700 52 21 , , PUNCT iajs-700 52 22 t t PROPN iajs-700 52 23 [0,1 [0,1 NOUN iajs-700 52 24 ] ] PUNCT iajs-700 52 25 is be AUX iajs-700 52 26 an an DET iajs-700 52 27 ideal ideal NOUN iajs-700 52 28 of of ADP iajs-700 52 29 r. r. PROPN iajs-700 52 30 1.14 1.14 NUM iajs-700 52 31 definition definition NOUN iajs-700 52 32 : : PUNCT iajs-700 53 1 [ [ X iajs-700 53 2 2 2 X iajs-700 53 3 ] ] PUNCT iajs-700 53 4 let let VERB iajs-700 53 5 a a PRON iajs-700 53 6 and and CCONJ iajs-700 53 7 b b NOUN iajs-700 53 8 be be AUX iajs-700 53 9 two two NUM iajs-700 53 10 fuzzy fuzzy ADJ iajs-700 53 11 submodules submodule NOUN iajs-700 53 12 of of ADP iajs-700 53 13 a a DET iajs-700 53 14 fuzzy fuzzy ADJ iajs-700 53 15 module module NOUN iajs-700 53 16 x x PUNCT iajs-700 53 17 of of ADP iajs-700 53 18 an an DET iajs-700 53 19 r r NOUN iajs-700 53 20 - - PUNCT iajs-700 53 21 module module NOUN iajs-700 53 22 m. m. NOUN iajs-700 53 23 the the DET iajs-700 53 24 residual residual ADJ iajs-700 53 25 quotient quotient NOUN iajs-700 53 26 a a PRON iajs-700 53 27 and and CCONJ iajs-700 53 28 b b NOUN iajs-700 53 29 denoted denote VERB iajs-700 53 30 by by ADP iajs-700 53 31 ( ( PUNCT iajs-700 53 32 a a DET iajs-700 53 33 : : SYM iajs-700 53 34 b b X iajs-700 53 35 ) ) PUNCT iajs-700 53 36 is be AUX iajs-700 53 37 the the DET iajs-700 53 38 fuzzy fuzzy ADJ iajs-700 53 39 subset subset NOUN iajs-700 53 40 of of ADP iajs-700 53 41 r r NOUN iajs-700 53 42 defined define VERB iajs-700 53 43 by by ADP iajs-700 53 44 : : PUNCT iajs-700 53 45 ( ( PUNCT iajs-700 53 46 a a DET iajs-700 53 47 : : PUNCT iajs-700 53 48 b)(r b)(r NOUN iajs-700 53 49 ) ) PUNCT iajs-700 54 1 = = SYM iajs-700 54 2 sup{t[0,1 sup{t[0,1 NOUN iajs-700 54 3 ] ] PUNCT iajs-700 54 4 , , PUNCT iajs-700 54 5 rtb rtb NOUN iajs-700 54 6   PROPN iajs-700 54 7 a a PRON iajs-700 54 8 } } PUNCT iajs-700 54 9 for for ADP iajs-700 54 10 all all DET iajs-700 54 11 r r NOUN iajs-700 54 12   PROPN iajs-700 54 13 r. r. NOUN iajs-700 54 14 that that PRON iajs-700 54 15 is be AUX iajs-700 54 16 ( ( PUNCT iajs-700 54 17 a a DET iajs-700 54 18 : : SYM iajs-700 54 19 b b X iajs-700 54 20 ) ) PUNCT iajs-700 54 21 = = NOUN iajs-700 54 22 { { PUNCT iajs-700 54 23 rt rt NOUN iajs-700 54 24 : : PUNCT iajs-700 54 25 rtb rtb VERB iajs-700 54 26   PROPN iajs-700 54 27 a a PROPN iajs-700 54 28 , , PUNCT iajs-700 54 29 rt rt PROPN iajs-700 54 30 is be AUX iajs-700 54 31 fuzzy fuzzy ADJ iajs-700 54 32 singleton singleton NOUN iajs-700 54 33 of of ADP iajs-700 54 34 r r NOUN iajs-700 54 35 } } PUNCT iajs-700 54 36 . . PUNCT iajs-700 55 1 1.15 1.15 NUM iajs-700 55 2 theorem theorem VERB iajs-700 55 3 : : PUNCT iajs-700 55 4 [ [ X iajs-700 55 5 2 2 NUM iajs-700 55 6 ] ] PUNCT iajs-700 55 7 let let VERB iajs-700 55 8 a a PRON iajs-700 55 9 and and CCONJ iajs-700 55 10 b b NOUN iajs-700 55 11 be be AUX iajs-700 55 12 two two NUM iajs-700 55 13 fuzzy fuzzy ADJ iajs-700 55 14 submodules submodule NOUN iajs-700 55 15 of of ADP iajs-700 55 16 a a DET iajs-700 55 17 fuzzy fuzzy ADJ iajs-700 55 18 module module NOUN iajs-700 55 19 x x PUNCT iajs-700 55 20 of of ADP iajs-700 55 21 an an DET iajs-700 55 22 r r NOUN iajs-700 55 23 - - PUNCT iajs-700 55 24 module module NOUN iajs-700 55 25 m. m. NOUN iajs-700 55 26 then then ADV iajs-700 55 27 the the DET iajs-700 55 28 residual residual ADJ iajs-700 55 29 quotient quotient NOUN iajs-700 55 30 ( ( PUNCT iajs-700 55 31 a a DET iajs-700 55 32 : : SYM iajs-700 55 33 b b NOUN iajs-700 55 34 ) ) PUNCT iajs-700 55 35 of of ADP iajs-700 55 36 a a PRON iajs-700 55 37 and and CCONJ iajs-700 55 38 b b NOUN iajs-700 55 39 is be AUX iajs-700 55 40 a a DET iajs-700 55 41 fuzzy fuzzy ADJ iajs-700 55 42 ideal ideal NOUN iajs-700 55 43 of of ADP iajs-700 55 44 r. r. PROPN iajs-700 55 45 1.16 1.16 NUM iajs-700 55 46 definition definition NOUN iajs-700 55 47 : : PUNCT iajs-700 55 48 [ [ X iajs-700 55 49 8 8 NUM iajs-700 55 50 ] ] PUNCT iajs-700 55 51 let let VERB iajs-700 55 52 a a PRON iajs-700 55 53 be be AUX iajs-700 55 54 fuzzy fuzzy ADJ iajs-700 55 55 submodule submodule NOUN iajs-700 55 56 of of ADP iajs-700 55 57 a a DET iajs-700 55 58 fuzzy fuzzy ADJ iajs-700 55 59 module module NOUN iajs-700 55 60 x. x. NOUN iajs-700 55 61 the the DET iajs-700 55 62 fuzzy fuzzy ADJ iajs-700 55 63 annihilator annihilator NOUN iajs-700 55 64 of of ADP iajs-700 55 65 a a DET iajs-700 55 66 denoted denote VERB iajs-700 55 67 by by ADP iajs-700 55 68 f f PROPN iajs-700 55 69 - - PUNCT iajs-700 55 70 anna anna PROPN iajs-700 55 71 is be AUX iajs-700 55 72 defined define VERB iajs-700 55 73 by by ADP iajs-700 55 74 : : PUNCT iajs-700 55 75 ( ( PUNCT iajs-700 55 76 f f X iajs-700 55 77 - - PUNCT iajs-700 55 78 anna)(r anna)(r NOUN iajs-700 55 79 ) ) PUNCT iajs-700 56 1 = = SYM iajs-700 56 2 sup{t sup{t PROPN iajs-700 56 3 : : PUNCT iajs-700 56 4 t[0,1 t[0,1 NOUN iajs-700 56 5 ] ] PUNCT iajs-700 56 6 , , PUNCT iajs-700 56 7 rta rta PROPN iajs-700 56 8   PROPN iajs-700 56 9 o1 o1 PROPN iajs-700 56 10 } } PUNCT iajs-700 56 11 for for ADP iajs-700 56 12 all all DET iajs-700 56 13 r r NOUN iajs-700 56 14   PROPN iajs-700 56 15 r. r. NOUN iajs-700 56 16 that that PRON iajs-700 56 17 is be AUX iajs-700 56 18 f f NOUN iajs-700 56 19 - - PUNCT iajs-700 56 20 anna anna NOUN iajs-700 56 21 = = SYM iajs-700 56 22 ( ( PUNCT iajs-700 56 23 o1 o1 PROPN iajs-700 56 24 : : PUNCT iajs-700 56 25 a a X iajs-700 56 26 ) ) PUNCT iajs-700 56 27 . . PUNCT iajs-700 57 1 1.17 1.17 NUM iajs-700 57 2 definition definition NOUN iajs-700 57 3 : : PUNCT iajs-700 57 4 [ [ X iajs-700 57 5 9 9 NUM iajs-700 57 6 ] ] PUNCT iajs-700 57 7 let let VERB iajs-700 57 8 x x PRON iajs-700 57 9 and and CCONJ iajs-700 57 10 y y PROPN iajs-700 57 11 be be AUX iajs-700 57 12 two two NUM iajs-700 57 13 fuzzy fuzzy ADJ iajs-700 57 14 modules module NOUN iajs-700 57 15 of of ADP iajs-700 57 16 m m PROPN iajs-700 57 17 1 1 NUM iajs-700 57 18 and and CCONJ iajs-700 57 19 m m PROPN iajs-700 57 20 2 2 NUM iajs-700 57 21 respectively respectively ADV iajs-700 57 22 defined define VERB iajs-700 57 23 xy xy NOUN iajs-700 57 24 : : PUNCT iajs-700 57 25 m1m m1m PROPN iajs-700 57 26 2 2 NUM iajs-700 57 27   NOUN iajs-700 57 28 [ [ X iajs-700 57 29 0,1 0,1 NUM iajs-700 57 30 ] ] PUNCT iajs-700 57 31 by by ADP iajs-700 57 32 ( ( PUNCT iajs-700 57 33 xy)(a xy)(a PROPN iajs-700 57 34 , , PUNCT iajs-700 57 35 b b NOUN iajs-700 57 36 ) ) PUNCT iajs-700 57 37 = = SYM iajs-700 57 38 min{x(a),y(b min{x(a),y(b NOUN iajs-700 57 39 ) ) PUNCT iajs-700 57 40 } } PUNCT iajs-700 57 41 for for ADP iajs-700 57 42 all all DET iajs-700 57 43 ( ( PUNCT iajs-700 57 44 a a DET iajs-700 57 45 , , PUNCT iajs-700 57 46 b b NOUN iajs-700 57 47 ) ) PUNCT iajs-700 57 48   NOUN iajs-700 57 49 xy xy PROPN iajs-700 57 50 . . PUNCT iajs-700 58 1 xy xy PROPN iajs-700 58 2 is be AUX iajs-700 58 3 called call VERB iajs-700 58 4 a a DET iajs-700 58 5 fuzzy fuzzy ADJ iajs-700 58 6 external external ADJ iajs-700 58 7 direct direct ADJ iajs-700 58 8 sum sum NOUN iajs-700 58 9 of of ADP iajs-700 58 10 x x X iajs-700 58 11 and and CCONJ iajs-700 58 12 y. y. PROPN iajs-700 58 13 1.18 1.18 NUM iajs-700 58 14 proposition proposition NOUN iajs-700 58 15 : : PUNCT iajs-700 59 1 [ [ X iajs-700 59 2 9 9 NUM iajs-700 59 3 ] ] PUNCT iajs-700 59 4 let let VERB iajs-700 59 5 x x PRON iajs-700 59 6 and and CCONJ iajs-700 59 7 y y PROPN iajs-700 59 8 be be AUX iajs-700 59 9 fuzzy fuzzy ADJ iajs-700 59 10 modules module NOUN iajs-700 59 11 of of ADP iajs-700 59 12 m m PROPN iajs-700 59 13 1 1 NUM iajs-700 59 14 and and CCONJ iajs-700 59 15 m2 m2 PROPN iajs-700 59 16 respectively respectively ADV iajs-700 59 17 then then ADV iajs-700 59 18 xy xy PROPN iajs-700 59 19 is be AUX iajs-700 59 20 a a DET iajs-700 59 21 fuzzy fuzzy ADJ iajs-700 59 22 module module NOUN iajs-700 59 23 of of ADP iajs-700 59 24 m m PROPN iajs-700 59 25 1m 1m PROPN iajs-700 59 26 2 2 NUM iajs-700 59 27 . . NOUN iajs-700 59 28 1.19 1.19 NUM iajs-700 59 29 remark remark NOUN iajs-700 59 30 : : PUNCT iajs-700 59 31 [ [ X iajs-700 59 32 9 9 NUM iajs-700 59 33 ] ] PUNCT iajs-700 59 34 let let VERB iajs-700 59 35 a a PRON iajs-700 59 36 and and CCONJ iajs-700 59 37 b b NOUN iajs-700 59 38 be be AUX iajs-700 59 39 two two NUM iajs-700 59 40 fuzzy fuzzy ADJ iajs-700 59 41 submodules submodule NOUN iajs-700 59 42 of of ADP iajs-700 59 43 a a DET iajs-700 59 44 fuzzy fuzzy ADJ iajs-700 59 45 module module NOUN iajs-700 59 46 x x ADP iajs-700 60 1 such such ADJ iajs-700 60 2 that that SCONJ iajs-700 60 3 x x X iajs-700 60 4 = = SYM iajs-700 60 5 ab ab PROPN iajs-700 60 6 , , PUNCT iajs-700 60 7 then then ADV iajs-700 60 8 xs xs PROPN iajs-700 60 9 = = SYM iajs-700 60 10 asbs asbs PROPN iajs-700 60 11 , , PUNCT iajs-700 60 12 for for ADP iajs-700 60 13 all all DET iajs-700 60 14 s s PART iajs-700 60 15   NOUN iajs-700 60 16 [ [ X iajs-700 60 17 0,1 0,1 NUM iajs-700 60 18 ] ] PUNCT iajs-700 60 19 . . PUNCT iajs-700 61 1 1.20 1.20 NUM iajs-700 61 2 definition definition NOUN iajs-700 61 3 : : PUNCT iajs-700 62 1 [ [ X iajs-700 62 2 9 9 X iajs-700 62 3 ] ] PUNCT iajs-700 62 4 a a DET iajs-700 62 5 fuzzy fuzzy ADJ iajs-700 62 6 module module NOUN iajs-700 62 7 x x PUNCT iajs-700 62 8 of of ADP iajs-700 62 9 an an DET iajs-700 62 10 r r NOUN iajs-700 62 11 - - PUNCT iajs-700 62 12 module module NOUN iajs-700 62 13 m m NOUN iajs-700 62 14 is be AUX iajs-700 62 15 called call VERB iajs-700 62 16 a a DET iajs-700 62 17 prime prime ADJ iajs-700 62 18 fuzzy fuzzy ADJ iajs-700 62 19 module module NOUN iajs-700 62 20 if if SCONJ iajs-700 62 21 f f NOUN iajs-700 62 22 - - PUNCT iajs-700 62 23 anna anna NOUN iajs-700 62 24 = = SYM iajs-700 62 25 f f PROPN iajs-700 62 26 - - PUNCT iajs-700 62 27 annx annx PROPN iajs-700 62 28 , , PUNCT iajs-700 62 29 for for ADP iajs-700 62 30 any any DET iajs-700 62 31 nontrivial nontrivial ADJ iajs-700 62 32 fuzzy fuzzy ADJ iajs-700 62 33 submodule submodule NOUN iajs-700 62 34 a a PRON iajs-700 62 35 of of ADP iajs-700 62 36 x. x. NOUN iajs-700 62 37 2semiprime 2semiprime NUM iajs-700 62 38 fuzzy fuzzy ADJ iajs-700 62 39 modules module NOUN iajs-700 62 40 firas firas X iajs-700 62 41 in in ADP iajs-700 62 42 [ [ X iajs-700 62 43 9 9 NUM iajs-700 62 44 ] ] PUNCT iajs-700 62 45 introduced introduce VERB iajs-700 62 46 the the DET iajs-700 62 47 concept concept NOUN iajs-700 62 48 of of ADP iajs-700 62 49 semiprime semiprime NOUN iajs-700 62 50 r r NOUN iajs-700 62 51 - - PUNCT iajs-700 62 52 module module NOUN iajs-700 62 53 ( ( PUNCT iajs-700 62 54 where where SCONJ iajs-700 62 55 m m PROPN iajs-700 62 56 is be AUX iajs-700 62 57 called call VERB iajs-700 62 58 a a DET iajs-700 62 59 semiprime semiprime NOUN iajs-700 62 60 module module NOUN iajs-700 62 61 if if SCONJ iajs-700 62 62 for for ADP iajs-700 62 63 each each DET iajs-700 62 64 r r NOUN iajs-700 62 65   NOUN iajs-700 62 66 r r NOUN iajs-700 62 67 , , PUNCT iajs-700 62 68 x x SYM iajs-700 62 69   PROPN iajs-700 62 70 m m NOUN iajs-700 62 71 , , PUNCT iajs-700 62 72 r2x r2x PROPN iajs-700 62 73   PROPN iajs-700 62 74 m m PROPN iajs-700 62 75 implies imply VERB iajs-700 62 76 rx rx VERB iajs-700 62 77   PROPN iajs-700 62 78 m m PROPN iajs-700 62 79 .. .. PUNCT iajs-700 62 80 we we PRON iajs-700 62 81 shall shall AUX iajs-700 62 82 fuzzify fuzzify VERB iajs-700 62 83 this this DET iajs-700 62 84 concept concept NOUN iajs-700 62 85 in in ADP iajs-700 62 86 definition definition NOUN iajs-700 62 87 2.3 2.3 NUM iajs-700 62 88 . . PUNCT iajs-700 63 1 but but CCONJ iajs-700 63 2 first first ADV iajs-700 63 3 we we PRON iajs-700 63 4 give give VERB iajs-700 63 5 the the DET iajs-700 63 6 two two NUM iajs-700 63 7 definitions definition NOUN iajs-700 63 8 : : PUNCT iajs-700 63 9 2.1 2.1 NUM iajs-700 63 10 definition definition NOUN iajs-700 63 11 : : PUNCT iajs-700 64 1 [ [ X iajs-700 64 2 7 7 X iajs-700 64 3 ] ] PUNCT iajs-700 64 4 let let VERB iajs-700 64 5 a a PRON iajs-700 64 6 be be AUX iajs-700 64 7 a a DET iajs-700 64 8 non non ADJ iajs-700 64 9 constant constant ADJ iajs-700 64 10 fuzzy fuzzy ADJ iajs-700 64 11 ideal ideal NOUN iajs-700 64 12 of of ADP iajs-700 64 13 a a DET iajs-700 64 14 ring ring NOUN iajs-700 64 15 r. r. PROPN iajs-700 64 16 a a PRON iajs-700 64 17 is be AUX iajs-700 64 18 called call VERB iajs-700 64 19 semiprime semiprime NOUN iajs-700 64 20 fuzzy fuzzy ADJ iajs-700 64 21 ideal ideal NOUN iajs-700 64 22 if if SCONJ iajs-700 64 23 for for ADP iajs-700 64 24 any any DET iajs-700 64 25 fuzzy fuzzy ADJ iajs-700 64 26 singleton singleton PROPN iajs-700 64 27 xt xt PUNCT iajs-700 65 1   PROPN iajs-700 65 2 r r NOUN iajs-700 65 3 , , PUNCT iajs-700 65 4 2 2 NUM iajs-700 65 5 t t NOUN iajs-700 65 6 x x PROPN iajs-700 65 7   PROPN iajs-700 65 8 a a PROPN iajs-700 65 9 , , PUNCT iajs-700 65 10 implies imply VERB iajs-700 65 11 xt xt ADP iajs-700 65 12   NOUN iajs-700 65 13 a. a. NOUN iajs-700 65 14 مجلة مجلة PROPN iajs-700 65 15 إبن إبن VERB iajs-700 65 16 الھیثم الھیثم NOUN iajs-700 65 17 للعلوم للعلوم NOUN iajs-700 65 18 الصرفة الصرفة NOUN iajs-700 65 19 و و PRON iajs-700 65 20 التطبیقیة التطبیقیة PROPN iajs-700 65 21 2012 2012 NUM iajs-700 65 22 السنة السنة NOUN iajs-700 66 1 25 25 NUM iajs-700 66 2 المجلد المجلد NOUN iajs-700 66 3 1 1 NUM iajs-700 66 4 العدد العدد PROPN iajs-700 66 5 ibn ibn PROPN iajs-700 66 6 al al PROPN iajs-700 66 7 - - PUNCT iajs-700 66 8 haitham haitham PROPN iajs-700 66 9 journal journal PROPN iajs-700 66 10 for for ADP iajs-700 66 11 pure pure ADJ iajs-700 66 12 and and CCONJ iajs-700 66 13 applied apply VERB iajs-700 66 14 science science NOUN iajs-700 66 15 no no NOUN iajs-700 66 16 . . NOUN iajs-700 66 17 1 1 NUM iajs-700 66 18 vol vol NOUN iajs-700 66 19 . . PUNCT iajs-700 67 1 25 25 NUM iajs-700 67 2 year year NOUN iajs-700 67 3 2012 2012 NUM iajs-700 67 4 2.2 2.2 NUM iajs-700 67 5 definition definition NOUN iajs-700 67 6 : : PUNCT iajs-700 68 1 [ [ X iajs-700 68 2 8 8 NUM iajs-700 68 3 ] ] PUNCT iajs-700 68 4 let let VERB iajs-700 68 5 a a PRON iajs-700 68 6 be be AUX iajs-700 68 7 a a DET iajs-700 68 8 fuzzy fuzzy ADJ iajs-700 68 9 submodule submodule NOUN iajs-700 68 10 of of ADP iajs-700 68 11 a a DET iajs-700 68 12 fuzzy fuzzy ADJ iajs-700 68 13 module module NOUN iajs-700 68 14 x x PUNCT iajs-700 68 15 of of ADP iajs-700 68 16 an an DET iajs-700 68 17 r r NOUN iajs-700 68 18 - - PUNCT iajs-700 68 19 module module NOUN iajs-700 68 20 m m NOUN iajs-700 68 21 such such ADJ iajs-700 68 22 that that SCONJ iajs-700 68 23 a a DET iajs-700 68 24   NOUN iajs-700 68 25 x x NOUN iajs-700 68 26 , , PUNCT iajs-700 68 27 a a PRON iajs-700 68 28 is be AUX iajs-700 68 29 called call VERB iajs-700 68 30 semiprime semiprime ADJ iajs-700 68 31 fuzzy fuzzy ADJ iajs-700 68 32 submodule submodule PROPN iajs-700 68 33 if if SCONJ iajs-700 68 34 for for ADP iajs-700 68 35 each each DET iajs-700 68 36 fuzzy fuzzy ADJ iajs-700 68 37 singletone singletone PROPN iajs-700 68 38 rt rt PROPN iajs-700 68 39   PROPN iajs-700 68 40 r r PROPN iajs-700 68 41 , , PUNCT iajs-700 68 42 xs xs PROPN iajs-700 68 43   PROPN iajs-700 68 44 x x PROPN iajs-700 68 45 , , PUNCT iajs-700 68 46 2 2 NUM iajs-700 68 47 t t NOUN iajs-700 68 48 r r NOUN iajs-700 68 49 xs xs PROPN iajs-700 68 50   PROPN iajs-700 68 51 a a DET iajs-700 68 52 implies imply VERB iajs-700 68 53 rtxs rtxs NOUN iajs-700 68 54   PROPN iajs-700 68 55 a. a. NOUN iajs-700 68 56 2.3 2.3 NUM iajs-700 68 57 definition definition NOUN iajs-700 68 58 : : PUNCT iajs-700 68 59 let let VERB iajs-700 68 60 x x PRON iajs-700 68 61 be be AUX iajs-700 68 62 a a DET iajs-700 68 63 fuzzy fuzzy ADJ iajs-700 68 64 module module NOUN iajs-700 68 65 of of ADP iajs-700 68 66 an an DET iajs-700 68 67 r r NOUN iajs-700 68 68 - - PUNCT iajs-700 68 69 module module NOUN iajs-700 68 70 m m NOUN iajs-700 68 71 , , PUNCT iajs-700 68 72 x x VERB iajs-700 68 73 is be AUX iajs-700 68 74 called call VERB iajs-700 68 75 semiprime semiprime ADJ iajs-700 68 76 fuzzy fuzzy ADJ iajs-700 68 77 module module NOUN iajs-700 68 78 if if SCONJ iajs-700 68 79 for for ADP iajs-700 68 80 each each DET iajs-700 68 81 non non ADJ iajs-700 68 82 - - ADJ iajs-700 68 83 zero zero ADJ iajs-700 68 84 fuzzy fuzzy ADJ iajs-700 68 85 submodule submodule NOUN iajs-700 68 86 a a PRON iajs-700 68 87 of of ADP iajs-700 68 88 x x PRON iajs-700 68 89 , , PUNCT iajs-700 68 90 f f X iajs-700 68 91 - - PUNCT iajs-700 68 92 anna anna PROPN iajs-700 68 93 is be AUX iajs-700 68 94 a a DET iajs-700 68 95 semiprime semiprime NOUN iajs-700 68 96 fuzzy fuzzy ADJ iajs-700 68 97 ideal ideal NOUN iajs-700 68 98 of of ADP iajs-700 68 99 r. r. PROPN iajs-700 68 100 2.4 2.4 NUM iajs-700 68 101 remarks remark VERB iajs-700 68 102 : : PUNCT iajs-700 68 103 1every 1every NUM iajs-700 68 104 prime prime ADJ iajs-700 68 105 fuzzy fuzzy ADJ iajs-700 68 106 module module NOUN iajs-700 68 107 x x PRON iajs-700 68 108 is be AUX iajs-700 68 109 a a DET iajs-700 68 110 semiprime semiprime NOUN iajs-700 68 111 fuzzy fuzzy ADJ iajs-700 68 112 module module NOUN iajs-700 68 113 . . PUNCT iajs-700 69 1 proof proof NOUN iajs-700 69 2 : : PUNCT iajs-700 69 3 let let VERB iajs-700 69 4 a a PRON iajs-700 69 5 be be AUX iajs-700 69 6 a a DET iajs-700 69 7 fuzzy fuzzy ADJ iajs-700 69 8 submodule submodule NOUN iajs-700 69 9 of of ADP iajs-700 69 10 x. x. NOUN iajs-700 69 11 since since SCONJ iajs-700 69 12 x x PROPN iajs-700 69 13 is be AUX iajs-700 69 14 prime prime ADJ iajs-700 69 15 , , PUNCT iajs-700 69 16 hence hence ADV iajs-700 69 17 f f NOUN iajs-700 69 18 - - PUNCT iajs-700 69 19 anna anna PROPN iajs-700 69 20 is be AUX iajs-700 69 21 a a DET iajs-700 69 22 prime prime ADJ iajs-700 69 23 fuzzy fuzzy ADJ iajs-700 69 24 ideal ideal NOUN iajs-700 69 25 by by ADP iajs-700 69 26 [ [ X iajs-700 69 27 17 17 NUM iajs-700 69 28 ] ] PUNCT iajs-700 69 29 which which PRON iajs-700 69 30 implies imply VERB iajs-700 69 31 f f X iajs-700 69 32 - - PUNCT iajs-700 69 33 anna anna PROPN iajs-700 69 34 is be AUX iajs-700 69 35 a a DET iajs-700 69 36 semiprime semiprime NOUN iajs-700 69 37 fuzzy fuzzy ADJ iajs-700 69 38 ideal ideal NOUN iajs-700 69 39 by by ADP iajs-700 69 40 [ [ X iajs-700 69 41 7 7 NUM iajs-700 69 42 ] ] PUNCT iajs-700 69 43 . . PUNCT iajs-700 70 1 thus thus ADV iajs-700 70 2 x x X iajs-700 70 3 is be AUX iajs-700 70 4 a a DET iajs-700 70 5 semiprime semiprime NOUN iajs-700 70 6 fuzzy fuzzy ADJ iajs-700 70 7 module module NOUN iajs-700 70 8 . . PUNCT iajs-700 70 9 2if 2if NOUN iajs-700 70 10 x x PUNCT iajs-700 70 11 is be AUX iajs-700 70 12 a a DET iajs-700 70 13 semiprime semiprime NOUN iajs-700 70 14 fuzzy fuzzy ADJ iajs-700 70 15 module module NOUN iajs-700 70 16 , , PUNCT iajs-700 70 17 then then ADV iajs-700 70 18 f f X iajs-700 70 19 - - PUNCT iajs-700 70 20 annx annx PROPN iajs-700 70 21 is be AUX iajs-700 70 22 a a DET iajs-700 70 23 semiprime semiprime NOUN iajs-700 70 24 fuzzy fuzzy ADJ iajs-700 70 25 ideal ideal ADJ iajs-700 70 26 . . PUNCT iajs-700 71 1 proof proof NOUN iajs-700 71 2 : : PUNCT iajs-700 71 3 it it PRON iajs-700 71 4 is be AUX iajs-700 71 5 clear clear ADJ iajs-700 71 6 by by ADP iajs-700 71 7 definition definition NOUN iajs-700 71 8 2.3 2.3 NUM iajs-700 71 9 , , PUNCT iajs-700 71 10 so so ADV iajs-700 71 11 is be AUX iajs-700 71 12 omitted omit VERB iajs-700 71 13 . . PUNCT iajs-700 72 1 the the DET iajs-700 72 2 following follow VERB iajs-700 72 3 is be AUX iajs-700 72 4 a a DET iajs-700 72 5 characterization characterization NOUN iajs-700 72 6 of of ADP iajs-700 72 7 semiprime semiprime NOUN iajs-700 72 8 fuzzy fuzzy ADJ iajs-700 72 9 module module NOUN iajs-700 72 10 . . PUNCT iajs-700 73 1 2.5 2.5 NUM iajs-700 73 2 proposition proposition NOUN iajs-700 73 3 : : PUNCT iajs-700 73 4 let let VERB iajs-700 73 5 x x PRON iajs-700 73 6 be be AUX iajs-700 73 7 a a DET iajs-700 73 8 fuzzy fuzzy ADJ iajs-700 73 9 module module NOUN iajs-700 73 10 of of ADP iajs-700 73 11 an an DET iajs-700 73 12 r r NOUN iajs-700 73 13 - - PUNCT iajs-700 73 14 module module NOUN iajs-700 73 15 m. m. NOUN iajs-700 73 16 then then ADV iajs-700 73 17 x x PRON iajs-700 73 18 is be AUX iajs-700 73 19 a a DET iajs-700 73 20 semiprime semiprime NOUN iajs-700 73 21 fuzzy fuzzy ADJ iajs-700 73 22 module module NOUN iajs-700 73 23 if if SCONJ iajs-700 73 24 and and CCONJ iajs-700 73 25 only only ADV iajs-700 73 26 if if SCONJ iajs-700 73 27 xt xt PROPN iajs-700 73 28 is be AUX iajs-700 73 29 a a DET iajs-700 73 30 semiprime semiprime NOUN iajs-700 73 31 module module NOUN iajs-700 73 32 , , PUNCT iajs-700 73 33   PROPN iajs-700 73 34 t t PROPN iajs-700 73 35   NOUN iajs-700 73 36 [ [ X iajs-700 73 37 0,1 0,1 NUM iajs-700 73 38 ] ] PUNCT iajs-700 73 39 . . PUNCT iajs-700 74 1 proof proof NOUN iajs-700 74 2 : : PUNCT iajs-700 74 3 ( ( PUNCT iajs-700 74 4   NOUN iajs-700 74 5 ) ) PUNCT iajs-700 74 6 let let VERB iajs-700 74 7 n n PRON iajs-700 74 8   PROPN iajs-700 74 9 xt xt PROPN iajs-700 74 10 , , PUNCT iajs-700 74 11 t t PROPN iajs-700 74 12   NOUN iajs-700 75 1 [ [ X iajs-700 75 2 0,1 0,1 NUM iajs-700 75 3 ] ] PUNCT iajs-700 75 4 . . PUNCT iajs-700 76 1 to to PART iajs-700 76 2 prove prove VERB iajs-700 76 3 annrn annrn NOUN iajs-700 76 4 is be AUX iajs-700 76 5 a a DET iajs-700 76 6 semiprime semiprime NOUN iajs-700 76 7 ideal ideal NOUN iajs-700 76 8 of of ADP iajs-700 76 9 r. r. PROPN iajs-700 76 10 let let VERB iajs-700 76 11 a2annrn a2annrn NOUN iajs-700 76 12   PROPN iajs-700 76 13 xt xt PROPN iajs-700 76 14 . . PUNCT iajs-700 77 1 since since SCONJ iajs-700 77 2 a2annrn a2annrn NOUN iajs-700 77 3 , , PUNCT iajs-700 77 4 then then ADV iajs-700 77 5 a2n a2n PROPN iajs-700 77 6 = = SYM iajs-700 77 7 0 0 NUM iajs-700 77 8 , , PUNCT iajs-700 77 9 let let VERB iajs-700 77 10 x x PRON iajs-700 77 11   NOUN iajs-700 77 12 n n CCONJ iajs-700 77 13 , , PUNCT iajs-700 77 14 hence hence ADV iajs-700 77 15 a2x a2x PROPN iajs-700 77 16 = = SYM iajs-700 77 17 0 0 X iajs-700 77 18 . . PUNCT iajs-700 77 19 assume assume VERB iajs-700 77 20 x(x x(x PROPN iajs-700 77 21 ) ) PUNCT iajs-700 78 1 = = PUNCT iajs-700 78 2 k. k. PROPN iajs-700 78 3 hence hence ADV iajs-700 78 4 xk xk PROPN iajs-700 79 1   PROPN iajs-700 79 2 x x SYM iajs-700 79 3 , , PUNCT iajs-700 79 4 so so SCONJ iajs-700 79 5 < < X iajs-700 79 6 xk xk X iajs-700 79 7 > > X iajs-700 79 8   PROPN iajs-700 79 9 x. x. PROPN iajs-700 79 10 but but CCONJ iajs-700 79 11 f f X iajs-700 79 12 - - PUNCT iajs-700 79 13 ann ann PROPN iajs-700 79 14 < < PROPN iajs-700 79 15 xk xk PROPN iajs-700 79 16 > > X iajs-700 79 17 is be AUX iajs-700 79 18 a a DET iajs-700 79 19 semiprime semiprime NOUN iajs-700 79 20 fuzzy fuzzy ADJ iajs-700 79 21 ideal ideal NOUN iajs-700 79 22 . . PUNCT iajs-700 80 1 and and CCONJ iajs-700 80 2 2 2 NUM iajs-700 80 3 ka ka PROPN iajs-700 80 4 xk xk PROPN iajs-700 81 1 = = PRON iajs-700 81 2 ( ( PUNCT iajs-700 81 3 a2x)k a2x)k X iajs-700 81 4   PROPN iajs-700 81 5 ok ok ADJ iajs-700 81 6   PROPN iajs-700 81 7 o1 o1 PROPN iajs-700 81 8 . . PUNCT iajs-700 82 1 thus thus ADV iajs-700 82 2 2 2 NUM iajs-700 82 3 ka ka PROPN iajs-700 82 4   PROPN iajs-700 82 5 f f PROPN iajs-700 82 6 - - PUNCT iajs-700 82 7 ann ann PROPN iajs-700 82 8 < < PROPN iajs-700 82 9 xk xk PROPN iajs-700 82 10 > > PROPN iajs-700 82 11 . . PUNCT iajs-700 83 1 since since SCONJ iajs-700 83 2 f f PROPN iajs-700 83 3 - - PUNCT iajs-700 83 4 ann ann PROPN iajs-700 83 5 < < PROPN iajs-700 83 6 xk xk PROPN iajs-700 83 7 > > X iajs-700 83 8 is be AUX iajs-700 83 9 a a DET iajs-700 83 10 semiprime semiprime NOUN iajs-700 83 11 fuzzy fuzzy ADJ iajs-700 83 12 ideal ideal NOUN iajs-700 83 13 . . PUNCT iajs-700 84 1 thus thus ADV iajs-700 84 2 akf akf X iajs-700 84 3 - - PUNCT iajs-700 84 4 ann ann PROPN iajs-700 84 5 < < PROPN iajs-700 84 6 xk xk PROPN iajs-700 84 7 > > PROPN iajs-700 84 8 , , PUNCT iajs-700 84 9 hence hence ADV iajs-700 84 10 akxk akxk PROPN iajs-700 84 11   PROPN iajs-700 84 12 o1 o1 PROPN iajs-700 84 13 , , PUNCT iajs-700 84 14 so so CCONJ iajs-700 84 15 ( ( PUNCT iajs-700 84 16 ax)k ax)k NOUN iajs-700 84 17 = = PUNCT iajs-700 84 18 ok ok ADJ iajs-700 84 19   PROPN iajs-700 84 20 o1 o1 PROPN iajs-700 84 21 . . PUNCT iajs-700 85 1 thus thus ADV iajs-700 85 2 ax ax NOUN iajs-700 85 3 = = SYM iajs-700 85 4 0 0 NUM iajs-700 85 5 , , PUNCT iajs-700 85 6 for for SCONJ iajs-700 85 7 any any DET iajs-700 85 8 x x ADJ iajs-700 85 9   PROPN iajs-700 85 10 n. n. NOUN iajs-700 85 11 ( ( PUNCT iajs-700 85 12   NOUN iajs-700 85 13 ) ) PUNCT iajs-700 85 14 conversely conversely ADV iajs-700 85 15 , , PUNCT iajs-700 85 16 to to PART iajs-700 85 17 prove prove VERB iajs-700 85 18 f f NOUN iajs-700 85 19 - - PUNCT iajs-700 85 20 anna anna PROPN iajs-700 85 21 is be AUX iajs-700 85 22 a a DET iajs-700 85 23 semiprime semiprime NOUN iajs-700 85 24 fuzzy fuzzy ADJ iajs-700 85 25 ideal ideal NOUN iajs-700 85 26 of of ADP iajs-700 85 27 r r NOUN iajs-700 85 28 for for ADP iajs-700 85 29 each each DET iajs-700 85 30 non non ADJ iajs-700 85 31 zero zero NUM iajs-700 85 32 fuzzy fuzzy ADJ iajs-700 85 33 submodule submodule NOUN iajs-700 85 34 a a PRON iajs-700 85 35 of of ADP iajs-700 85 36 x. x. NOUN iajs-700 85 37 let let VERB iajs-700 85 38 2 2 NUM iajs-700 85 39 kr kr PROPN iajs-700 85 40 f f PROPN iajs-700 85 41 - - PUNCT iajs-700 85 42 anna anna NOUN iajs-700 85 43 , , PUNCT iajs-700 85 44 so so ADV iajs-700 85 45 2 2 NUM iajs-700 85 46 kr kr PROPN iajs-700 85 47 xt xt PROPN iajs-700 85 48   PROPN iajs-700 85 49 o1 o1 PROPN iajs-700 85 50 , , PUNCT iajs-700 85 51 for for ADP iajs-700 85 52 all all DET iajs-700 85 53 xt xt X iajs-700 85 54   PROPN iajs-700 85 55 a. a. NOUN iajs-700 86 1 this this PRON iajs-700 86 2 implies imply VERB iajs-700 86 3 ( ( PUNCT iajs-700 86 4 r r NOUN iajs-700 86 5 2 2 NUM iajs-700 86 6 x) x) PROPN iajs-700 86 7 = = SYM iajs-700 86 8 0 0 PROPN iajs-700 86 9 , , PUNCT iajs-700 86 10 where where SCONJ iajs-700 86 11   ADJ iajs-700 86 12 = = SYM iajs-700 86 13 min{k min{k NOUN iajs-700 86 14 , , PUNCT iajs-700 86 15 t t PROPN iajs-700 86 16 } } PUNCT iajs-700 86 17 hence hence ADV iajs-700 86 18 r2x r2x VERB iajs-700 86 19 = = SYM iajs-700 86 20 0 0 NUM iajs-700 86 21 , , PUNCT iajs-700 86 22 x x SYM iajs-700 86 23   NOUN iajs-700 86 24 at at ADP iajs-700 86 25 . . PUNCT iajs-700 87 1 but but CCONJ iajs-700 87 2 at at ADP iajs-700 87 3   PROPN iajs-700 87 4 xt xt PROPN iajs-700 87 5 and and CCONJ iajs-700 87 6 xt xt PROPN iajs-700 87 7 is be AUX iajs-700 87 8 semiprime semiprime NOUN iajs-700 87 9 by by ADP iajs-700 87 10 hypothesis hypothesis NOUN iajs-700 87 11 . . PUNCT iajs-700 88 1 hence hence ADV iajs-700 88 2 rx rx VERB iajs-700 88 3 = = NOUN iajs-700 88 4 0 0 PROPN iajs-700 88 5 . . PUNCT iajs-700 89 1 this this PRON iajs-700 89 2 implies imply VERB iajs-700 89 3 ( ( PUNCT iajs-700 89 4 rx) rx) DET iajs-700 89 5   PROPN iajs-700 89 6 o1 o1 PROPN iajs-700 89 7 . . PUNCT iajs-700 90 1 that that PRON iajs-700 90 2 is be AUX iajs-700 90 3 rkxt rkxt ADJ iajs-700 90 4   PROPN iajs-700 90 5 o1 o1 PROPN iajs-700 90 6 . . PUNCT iajs-700 91 1 therefore therefore ADV iajs-700 91 2 rk rk VERB iajs-700 91 3   NOUN iajs-700 91 4 f f NOUN iajs-700 91 5 - - PUNCT iajs-700 91 6 anna anna NOUN iajs-700 91 7 . . PUNCT iajs-700 92 1 by by ADP iajs-700 92 2 def def PROPN iajs-700 92 3 . . PUNCT iajs-700 93 1 ( ( PUNCT iajs-700 93 2 2.3 2.3 NUM iajs-700 93 3 ) ) PUNCT iajs-700 93 4 we we PRON iajs-700 93 5 get get VERB iajs-700 93 6 the the DET iajs-700 93 7 result result NOUN iajs-700 93 8 . . PUNCT iajs-700 94 1 the the DET iajs-700 94 2 following follow VERB iajs-700 94 3 proposition proposition NOUN iajs-700 94 4 gives give VERB iajs-700 94 5 another another DET iajs-700 94 6 characterization characterization NOUN iajs-700 94 7 of of ADP iajs-700 94 8 semiprime semiprime NOUN iajs-700 94 9 fuzzy fuzzy ADJ iajs-700 94 10 module module NOUN iajs-700 94 11 . . PUNCT iajs-700 95 1 2.6 2.6 NUM iajs-700 95 2 proposition proposition NOUN iajs-700 95 3 : : PUNCT iajs-700 95 4 let let VERB iajs-700 95 5 x x PRON iajs-700 95 6 be be AUX iajs-700 95 7 a a DET iajs-700 95 8 fuzzy fuzzy ADJ iajs-700 95 9 module module NOUN iajs-700 95 10 of of ADP iajs-700 95 11 an an DET iajs-700 95 12 r r NOUN iajs-700 95 13 - - PUNCT iajs-700 95 14 module module NOUN iajs-700 95 15 m. m. NOUN iajs-700 95 16 then then ADV iajs-700 95 17 x x PRON iajs-700 95 18 is be AUX iajs-700 95 19 a a DET iajs-700 95 20 semiprime semiprime NOUN iajs-700 95 21 fuzzy fuzzy ADJ iajs-700 95 22 module module NOUN iajs-700 95 23 if if SCONJ iajs-700 95 24 and and CCONJ iajs-700 95 25 only only ADV iajs-700 95 26 if if SCONJ iajs-700 95 27 o1 o1 NOUN iajs-700 95 28 is be AUX iajs-700 95 29 a a DET iajs-700 95 30 semiprime semiprime NOUN iajs-700 95 31 fuzzy fuzzy ADJ iajs-700 95 32 submodule submodule NOUN iajs-700 95 33 of of ADP iajs-700 95 34 x. x. PROPN iajs-700 95 35 مجلة مجلة PROPN iajs-700 95 36 إبن إبن VERB iajs-700 95 37 الھیثم الھیثم NOUN iajs-700 95 38 للعلوم للعلوم NOUN iajs-700 95 39 الصرفة الصرفة NOUN iajs-700 95 40 و و PRON iajs-700 95 41 التطبیقیة التطبیقیة PROPN iajs-700 95 42 2012 2012 NUM iajs-700 95 43 السنة السنة NOUN iajs-700 96 1 25 25 NUM iajs-700 97 1 المجلد المجلد NOUN iajs-700 97 2 1 1 NUM iajs-700 97 3 العدد العدد PROPN iajs-700 97 4 ibn ibn PROPN iajs-700 97 5 al al PROPN iajs-700 97 6 - - PUNCT iajs-700 97 7 haitham haitham PROPN iajs-700 97 8 journal journal PROPN iajs-700 97 9 for for ADP iajs-700 97 10 pure pure ADJ iajs-700 97 11 and and CCONJ iajs-700 97 12 applied apply VERB iajs-700 97 13 science science NOUN iajs-700 97 14 no no NOUN iajs-700 97 15 . . NOUN iajs-700 97 16 1 1 NUM iajs-700 97 17 vol vol NOUN iajs-700 97 18 . . PUNCT iajs-700 98 1 25 25 NUM iajs-700 98 2 year year NOUN iajs-700 98 3 2012 2012 NUM iajs-700 98 4 proof proof NOUN iajs-700 98 5 : : PUNCT iajs-700 98 6 ( ( PUNCT iajs-700 98 7   NOUN iajs-700 98 8 ) ) PUNCT iajs-700 98 9 let let VERB iajs-700 98 10 2 2 NUM iajs-700 98 11 tr tr NOUN iajs-700 98 12 xk xk PROPN iajs-700 98 13   PROPN iajs-700 98 14 o1 o1 PROPN iajs-700 98 15 , , PUNCT iajs-700 98 16 for for ADP iajs-700 98 17 any any DET iajs-700 98 18 fuzzy fuzzy ADJ iajs-700 98 19 singleton singleton PROPN iajs-700 98 20 rt rt PROPN iajs-700 98 21 of of ADP iajs-700 98 22 r r PROPN iajs-700 98 23 , , PUNCT iajs-700 98 24 xk xk PROPN iajs-700 98 25   PROPN iajs-700 98 26 x x PROPN iajs-700 98 27 , , PUNCT iajs-700 98 28 hence hence ADV iajs-700 98 29 2 2 NUM iajs-700 98 30 tr tr NOUN iajs-700 98 31   NOUN iajs-700 98 32 ( ( PUNCT iajs-700 98 33 o1 o1 NOUN iajs-700 98 34 : : PUNCT iajs-700 98 35 xk xk PROPN iajs-700 98 36 ) ) PUNCT iajs-700 98 37 = = SYM iajs-700 98 38 f f X iajs-700 98 39 - - PUNCT iajs-700 98 40 ann ann PROPN iajs-700 98 41 < < PROPN iajs-700 98 42 xk xk PROPN iajs-700 98 43 > > PROPN iajs-700 98 44 . . PUNCT iajs-700 99 1 but but CCONJ iajs-700 99 2 f f X iajs-700 99 3 - - PUNCT iajs-700 99 4 ann ann PROPN iajs-700 99 5 < < PROPN iajs-700 99 6 xk xk PROPN iajs-700 99 7 > > X iajs-700 99 8 is be AUX iajs-700 99 9 a a DET iajs-700 99 10 semiprime semiprime NOUN iajs-700 99 11 fuzzy fuzzy ADJ iajs-700 99 12 ideal ideal NOUN iajs-700 99 13 . . PUNCT iajs-700 100 1 thus thus ADV iajs-700 100 2 rt rt PROPN iajs-700 100 3   PROPN iajs-700 100 4 ( ( PUNCT iajs-700 100 5 o1 o1 PROPN iajs-700 100 6 : : PUNCT iajs-700 100 7 xk xk PROPN iajs-700 100 8 ) ) PUNCT iajs-700 100 9 , , PUNCT iajs-700 100 10 so so ADV iajs-700 100 11 rtxk rtxk VERB iajs-700 100 12   PROPN iajs-700 100 13 o1 o1 PROPN iajs-700 100 14 . . PUNCT iajs-700 101 1 thus thus ADV iajs-700 101 2 o1 o1 NOUN iajs-700 101 3 is be AUX iajs-700 101 4 a a DET iajs-700 101 5 semiprime semiprime NOUN iajs-700 101 6 fuzzy fuzzy ADJ iajs-700 101 7 submodule submodule NOUN iajs-700 101 8 . . PUNCT iajs-700 102 1 ( ( PUNCT iajs-700 102 2   NOUN iajs-700 102 3 ) ) PUNCT iajs-700 102 4 to to PART iajs-700 102 5 prove prove VERB iajs-700 102 6 x x PUNCT iajs-700 102 7 is be AUX iajs-700 102 8 a a DET iajs-700 102 9 semiprime semiprime NOUN iajs-700 102 10 fuzzy fuzzy ADJ iajs-700 102 11 module module NOUN iajs-700 102 12 . . PUNCT iajs-700 103 1 by by ADP iajs-700 103 2 proposition proposition NOUN iajs-700 103 3 2.5 2.5 NUM iajs-700 103 4 . . PUNCT iajs-700 104 1 it it PRON iajs-700 104 2 is be AUX iajs-700 104 3 enough enough ADJ iajs-700 104 4 to to PART iajs-700 104 5 show show VERB iajs-700 104 6 that that SCONJ iajs-700 104 7 xt xt PROPN iajs-700 104 8 is be AUX iajs-700 104 9 semiprime semiprime NOUN iajs-700 104 10 , , PUNCT iajs-700 104 11   PROPN iajs-700 104 12 t t PROPN iajs-700 104 13   NOUN iajs-700 104 14 [ [ X iajs-700 104 15 0,1 0,1 NUM iajs-700 104 16 ] ] PUNCT iajs-700 104 17 . . PUNCT iajs-700 105 1 ( ( PUNCT iajs-700 105 2 i.e. i.e. X iajs-700 105 3 ) ) PUNCT iajs-700 105 4 to to PART iajs-700 105 5 prove prove VERB iajs-700 105 6 { { PUNCT iajs-700 105 7 0 0 NUM iajs-700 105 8 } } PUNCT iajs-700 105 9 is be AUX iajs-700 105 10 semiprime semiprime NOUN iajs-700 105 11 submodule submodule NOUN iajs-700 105 12 of of ADP iajs-700 105 13 xt xt NUM iajs-700 105 14 by by ADP iajs-700 105 15 [ [ X iajs-700 105 16 9,th.4.1.8 9,th.4.1.8 X iajs-700 105 17 ] ] PUNCT iajs-700 105 18 . . PUNCT iajs-700 106 1 let let VERB iajs-700 106 2 r2 r2 PROPN iajs-700 106 3 = = PUNCT iajs-700 106 4 0 0 NUM iajs-700 106 5 , , PUNCT iajs-700 106 6 to to PART iajs-700 106 7 prove prove VERB iajs-700 106 8 r r NOUN iajs-700 106 9 = = SYM iajs-700 106 10 0 0 NUM iajs-700 106 11 , , PUNCT iajs-700 106 12 hence hence ADV iajs-700 106 13 2 2 NUM iajs-700 106 14 tr tr NOUN iajs-700 106 15 = = NUM iajs-700 106 16 ot ot ADJ iajs-700 106 17   PROPN iajs-700 106 18 o1 o1 PROPN iajs-700 106 19 , , PUNCT iajs-700 106 20 ( ( PUNCT iajs-700 106 21 rt rt NOUN iajs-700 106 22 ) ) PUNCT iajs-700 106 23 2 2 NUM iajs-700 106 24   PROPN iajs-700 106 25 o1 o1 NOUN iajs-700 106 26 , , PUNCT iajs-700 106 27 hence hence ADV iajs-700 106 28 2 2 NUM iajs-700 106 29 tr tr NOUN iajs-700 106 30   PROPN iajs-700 106 31 o1 o1 PROPN iajs-700 106 32 , , PUNCT iajs-700 106 33 which which PRON iajs-700 106 34 implies imply VERB iajs-700 106 35 rt rt PROPN iajs-700 106 36   PROPN iajs-700 106 37 o1 o1 PROPN iajs-700 106 38 , , PUNCT iajs-700 106 39 since since SCONJ iajs-700 106 40 o1 o1 NOUN iajs-700 106 41 is be AUX iajs-700 106 42 semiprime semiprime NOUN iajs-700 106 43 . . PUNCT iajs-700 107 1 thus thus ADV iajs-700 107 2 r r NOUN iajs-700 107 3 = = SYM iajs-700 107 4 0 0 NUM iajs-700 107 5 , , PUNCT iajs-700 107 6 hence hence ADV iajs-700 107 7 xt xt PROPN iajs-700 107 8 is be AUX iajs-700 107 9 a a DET iajs-700 107 10 semiprime semiprime NOUN iajs-700 107 11 module module NOUN iajs-700 107 12 , , PUNCT iajs-700 107 13   PROPN iajs-700 107 14 t t PROPN iajs-700 107 15   NOUN iajs-700 107 16 [ [ X iajs-700 107 17 0,1 0,1 NUM iajs-700 107 18 ] ] PUNCT iajs-700 107 19 . . PUNCT iajs-700 108 1 thus thus ADV iajs-700 108 2 x x X iajs-700 108 3 is be AUX iajs-700 108 4 a a DET iajs-700 108 5 semiprime semiprime NOUN iajs-700 108 6 fuzzy fuzzy ADJ iajs-700 108 7 module module NOUN iajs-700 108 8 . . PUNCT iajs-700 109 1 next next ADV iajs-700 109 2 we we PRON iajs-700 109 3 can can AUX iajs-700 109 4 give give VERB iajs-700 109 5 some some DET iajs-700 109 6 examples example NOUN iajs-700 109 7 of of ADP iajs-700 109 8 semiprime semiprime NOUN iajs-700 109 9 and and CCONJ iajs-700 109 10 not not PART iajs-700 109 11 semiprime semiprime NOUN iajs-700 109 12 fuzzy fuzzy ADJ iajs-700 109 13 module module NOUN iajs-700 109 14 . . PUNCT iajs-700 110 1 2.7 2.7 NUM iajs-700 110 2 examples example NOUN iajs-700 110 3 : : PUNCT iajs-700 110 4 1let 1let NUM iajs-700 110 5 x x NOUN iajs-700 110 6 : : PUNCT iajs-700 110 7 z6 z6 PROPN iajs-700 110 8   PROPN iajs-700 111 1 [ [ X iajs-700 111 2 0,1 0,1 NUM iajs-700 111 3 ] ] PUNCT iajs-700 111 4 defined define VERB iajs-700 111 5 by by ADP iajs-700 111 6 x(a x(a NOUN iajs-700 111 7 ) ) PUNCT iajs-700 111 8 = = SYM iajs-700 111 9 1 1 NUM iajs-700 111 10 , , PUNCT iajs-700 111 11   VERB iajs-700 111 12 a a DET iajs-700 111 13   PROPN iajs-700 111 14 z6 z6 PROPN iajs-700 111 15 . . PUNCT iajs-700 112 1 xt xt PROPN iajs-700 113 1 = = PROPN iajs-700 113 2 z6 z6 PROPN iajs-700 113 3 , , PUNCT iajs-700 113 4 which which PRON iajs-700 113 5 is be AUX iajs-700 113 6 a a DET iajs-700 113 7 semiprime semiprime NOUN iajs-700 113 8 module module NOUN iajs-700 113 9 , , PUNCT iajs-700 113 10   PROPN iajs-700 113 11 t t PROPN iajs-700 113 12   NOUN iajs-700 114 1 [ [ X iajs-700 114 2 0,1 0,1 NUM iajs-700 114 3 ] ] PUNCT iajs-700 114 4 . . PUNCT iajs-700 115 1 thus thus ADV iajs-700 115 2 by by ADP iajs-700 115 3 prop prop NOUN iajs-700 115 4 . . PUNCT iajs-700 116 1 ( ( PUNCT iajs-700 116 2 2.5),x 2.5),x PRON iajs-700 116 3 is be AUX iajs-700 116 4 a a DET iajs-700 116 5 semiprime semiprime NOUN iajs-700 116 6 fuzzy fuzzy ADJ iajs-700 116 7 module module NOUN iajs-700 116 8 . . PUNCT iajs-700 117 1 2let 2let NUM iajs-700 117 2 x x NUM iajs-700 117 3 : : PUNCT iajs-700 117 4 z6 z6 PROPN iajs-700 117 5   PROPN iajs-700 118 1 [ [ X iajs-700 118 2 0,1 0,1 NUM iajs-700 118 3 ] ] PUNCT iajs-700 118 4 defined define VERB iajs-700 118 5 by by ADP iajs-700 118 6 1 1 NUM iajs-700 118 7 if if SCONJ iajs-700 118 8 x x X iajs-700 118 9 { { PUNCT iajs-700 118 10 0 0 NUM iajs-700 118 11 , , PUNCT iajs-700 118 12 3 3 NUM iajs-700 118 13 } } SYM iajs-700 118 14 x(x x(x PROPN iajs-700 118 15 ) ) PUNCT iajs-700 118 16 1 1 NUM iajs-700 118 17 otherwise otherwise ADV iajs-700 118 18 2 2 NUM iajs-700 118 19   NOUN iajs-700 118 20   NOUN iajs-700 118 21   NUM iajs-700 118 22   PROPN iajs-700 118 23   NUM iajs-700 118 24   NOUN iajs-700 118 25 x0 x0 PROPN iajs-700 118 26 = = SYM iajs-700 118 27 z6 z6 PROPN iajs-700 118 28 , , PUNCT iajs-700 118 29 1 1 NUM iajs-700 118 30 2 2 NUM iajs-700 118 31 x x X iajs-700 118 32 = = NOUN iajs-700 118 33 z6 z6 NOUN iajs-700 118 34 which which PRON iajs-700 118 35 is be AUX iajs-700 118 36 semiprime semiprime NOUN iajs-700 118 37 and and CCONJ iajs-700 118 38   NOUN iajs-700 118 39 t t PROPN iajs-700 118 40 > > X iajs-700 118 41 1 1 NUM iajs-700 118 42 2 2 NUM iajs-700 118 43 , , PUNCT iajs-700 118 44 t t NOUN iajs-700 118 45 x x X iajs-700 118 46 = = PUNCT iajs-700 118 47 { { PUNCT iajs-700 118 48 0 0 NUM iajs-700 118 49 , , PUNCT iajs-700 118 50 3 3 NUM iajs-700 118 51 } } PUNCT iajs-700 118 52 is be AUX iajs-700 118 53 a a DET iajs-700 118 54 prime prime ADJ iajs-700 118 55 submodule submodule NOUN iajs-700 118 56 , , PUNCT iajs-700 118 57 hence hence ADV iajs-700 118 58 it it PRON iajs-700 118 59 is be AUX iajs-700 118 60 semiprime semiprime NOUN iajs-700 118 61 . . PUNCT iajs-700 119 1 thus thus ADV iajs-700 119 2 xt xt PROPN iajs-700 119 3 is be AUX iajs-700 119 4 a a DET iajs-700 119 5 semiprime semiprime NOUN iajs-700 119 6 module module NOUN iajs-700 119 7 , , PUNCT iajs-700 119 8   PROPN iajs-700 119 9 t t PROPN iajs-700 119 10   NOUN iajs-700 119 11 [ [ X iajs-700 119 12 0,1 0,1 NUM iajs-700 119 13 ] ] PUNCT iajs-700 119 14 . . PUNCT iajs-700 120 1 thus thus ADV iajs-700 120 2 x x X iajs-700 120 3 is be AUX iajs-700 120 4 a a DET iajs-700 120 5 semiprime semiprime NOUN iajs-700 120 6 fuzzy fuzzy ADJ iajs-700 120 7 module module NOUN iajs-700 120 8 . . PUNCT iajs-700 121 1 3let 3let NUM iajs-700 121 2 x x NOUN iajs-700 121 3 : : PUNCT iajs-700 121 4 z12 z12 NUM iajs-700 121 5   X iajs-700 122 1 [ [ X iajs-700 122 2 0,1 0,1 NUM iajs-700 122 3 ] ] PUNCT iajs-700 122 4 defined define VERB iajs-700 122 5 by by ADP iajs-700 122 6 : : PUNCT iajs-700 122 7 1 1 NUM iajs-700 122 8 if if SCONJ iajs-700 122 9 a a DET iajs-700 122 10 { { PUNCT iajs-700 122 11 0,2 0,2 NUM iajs-700 122 12 , , PUNCT iajs-700 122 13 4 4 NUM iajs-700 122 14 , , PUNCT iajs-700 122 15 6 6 NUM iajs-700 122 16 , , PUNCT iajs-700 122 17 8,10 8,10 NUM iajs-700 122 18 } } PUNCT iajs-700 122 19 x(a x(a NOUN iajs-700 122 20 ) ) PUNCT iajs-700 122 21 0 0 NUM iajs-700 122 22 otherwise otherwise ADV iajs-700 122 23   NOUN iajs-700 122 24   VERB iajs-700 122 25   PRON iajs-700 122 26   NUM iajs-700 122 27   NOUN iajs-700 122 28 it it PRON iajs-700 122 29 is be AUX iajs-700 122 30 clear clear ADJ iajs-700 122 31 that that SCONJ iajs-700 122 32 x x PRON iajs-700 122 33 is be AUX iajs-700 122 34 a a DET iajs-700 122 35 fuzzy fuzzy ADJ iajs-700 122 36 module module NOUN iajs-700 122 37 and and CCONJ iajs-700 123 1 x0 x0 PROPN iajs-700 123 2 = = SYM iajs-700 123 3 z12 z12 PROPN iajs-700 123 4 , , PUNCT iajs-700 123 5 is be AUX iajs-700 123 6 not not PART iajs-700 123 7 semiprime semiprime NOUN iajs-700 123 8 module module NOUN iajs-700 123 9 . . PUNCT iajs-700 124 1 by by ADP iajs-700 124 2 prop.(2.5 prop.(2.5 NOUN iajs-700 124 3 ) ) PUNCT iajs-700 124 4 x x X iajs-700 124 5 is be AUX iajs-700 124 6 not not PART iajs-700 124 7 semiprime semiprime ADJ iajs-700 124 8 fuzzy fuzzy ADJ iajs-700 124 9 module module NOUN iajs-700 124 10 . . PUNCT iajs-700 125 1 2.8 2.8 NUM iajs-700 125 2 lemma lemma PROPN iajs-700 125 3 : : PUNCT iajs-700 125 4 let let VERB iajs-700 125 5 a a PRON iajs-700 125 6 and and CCONJ iajs-700 125 7 b b NOUN iajs-700 125 8 be be AUX iajs-700 125 9 two two NUM iajs-700 125 10 fuzzy fuzzy ADJ iajs-700 125 11 submodules submodule NOUN iajs-700 125 12 of of ADP iajs-700 125 13 fuzzy fuzzy ADJ iajs-700 125 14 module module NOUN iajs-700 125 15 of of ADP iajs-700 125 16 an an DET iajs-700 125 17 r r NOUN iajs-700 125 18 - - PUNCT iajs-700 125 19 module module NOUN iajs-700 125 20 m. m. NOUN iajs-700 125 21 if if SCONJ iajs-700 125 22 for for ADP iajs-700 125 23 each each PRON iajs-700 125 24 xtb xtb NOUN iajs-700 125 25 , , PUNCT iajs-700 125 26 [ [ X iajs-700 125 27 a: > X iajs-700 125 29 ] ] X iajs-700 125 30 is be AUX iajs-700 125 31 a a DET iajs-700 125 32 semiprime semiprime NOUN iajs-700 125 33 fuzzy fuzzy ADJ iajs-700 125 34 ideal ideal NOUN iajs-700 125 35 of of ADP iajs-700 125 36 r r NOUN iajs-700 125 37 , , PUNCT iajs-700 125 38 then then ADV iajs-700 125 39 [ [ X iajs-700 125 40 a a DET iajs-700 125 41 : : PUNCT iajs-700 125 42 b b NOUN iajs-700 125 43 ] ] PUNCT iajs-700 125 44 is be AUX iajs-700 125 45 a a DET iajs-700 125 46 semiprime semiprime NOUN iajs-700 125 47 fuzzy fuzzy ADJ iajs-700 125 48 ideal ideal NOUN iajs-700 125 49 of of ADP iajs-700 125 50 r. r. PROPN iajs-700 125 51 proof proof NOUN iajs-700 125 52 : : PUNCT iajs-700 125 53 let let VERB iajs-700 125 54 2 2 NUM iajs-700 125 55 ka ka NOUN iajs-700 125 56   PROPN iajs-700 126 1 [ [ X iajs-700 126 2 a a X iajs-700 126 3 : : PUNCT iajs-700 126 4 b b NOUN iajs-700 126 5 ] ] X iajs-700 126 6 , , PUNCT iajs-700 126 7 hence hence ADV iajs-700 126 8 2 2 NUM iajs-700 126 9 ka ka PROPN iajs-700 126 10 b b PROPN iajs-700 126 11   PROPN iajs-700 126 12 a. a. NOUN iajs-700 127 1 this this PRON iajs-700 127 2 implies imply VERB iajs-700 127 3 2 2 NUM iajs-700 127 4 ka ka PROPN iajs-700 127 5 xt xt PROPN iajs-700 127 6   PROPN iajs-700 127 7 a a PROPN iajs-700 127 8 , , PUNCT iajs-700 127 9 for for ADP iajs-700 127 10 all all PRON iajs-700 127 11 xt xt X iajs-700 127 12   PROPN iajs-700 127 13 b. b. PROPN iajs-700 128 1 hence hence ADV iajs-700 128 2 2 2 NUM iajs-700 128 3 ka ka X iajs-700 128 4 [a: > X iajs-700 128 6 ] ] X iajs-700 128 7 which which PRON iajs-700 128 8 is be AUX iajs-700 128 9 a a DET iajs-700 128 10 semiprime semiprime NOUN iajs-700 128 11 fuzzy fuzzy ADJ iajs-700 128 12 ideal ideal NOUN iajs-700 128 13 . . PUNCT iajs-700 129 1 thus thus ADV iajs-700 129 2 ak ak PROPN iajs-700 129 3   PROPN iajs-700 129 4 [ [ X iajs-700 129 5 a: > X iajs-700 129 7 ] ] PUNCT iajs-700 129 8 . . PUNCT iajs-700 130 1 so so ADV iajs-700 130 2 akxt akxt ADJ iajs-700 130 3   PROPN iajs-700 130 4 a a PROPN iajs-700 130 5 , , PUNCT iajs-700 130 6 hence hence ADV iajs-700 130 7 akb akb PROPN iajs-700 130 8   PROPN iajs-700 130 9 a. a. NOUN iajs-700 130 10 thus thus ADV iajs-700 130 11 [ [ X iajs-700 130 12 a a DET iajs-700 130 13 : : PUNCT iajs-700 130 14 b b NOUN iajs-700 130 15 ] ] PUNCT iajs-700 130 16 is be AUX iajs-700 130 17 a a DET iajs-700 130 18 semiprime semiprime NOUN iajs-700 130 19 fuzzy fuzzy ADJ iajs-700 130 20 ideal ideal NOUN iajs-700 130 21 . . PUNCT iajs-700 131 1 2.9 2.9 NUM iajs-700 131 2 proposition proposition NOUN iajs-700 131 3 : : PUNCT iajs-700 131 4 let let VERB iajs-700 131 5 x x PRON iajs-700 131 6 be be AUX iajs-700 131 7 a a DET iajs-700 131 8 fuzzy fuzzy ADJ iajs-700 131 9 module module NOUN iajs-700 131 10 of of ADP iajs-700 131 11 an an DET iajs-700 131 12 r r NOUN iajs-700 131 13 - - PUNCT iajs-700 131 14 module module NOUN iajs-700 131 15 m. m. NOUN iajs-700 131 16 then then ADV iajs-700 131 17 the the DET iajs-700 131 18 following follow VERB iajs-700 131 19 are be AUX iajs-700 131 20 equivalent equivalent ADJ iajs-700 131 21 : : PUNCT iajs-700 131 22 مجلة مجلة ADJ iajs-700 131 23 إبن إبن VERB iajs-700 131 24 الھیثم الھیثم NOUN iajs-700 131 25 للعلوم للعلوم NOUN iajs-700 131 26 الصرفة الصرفة NOUN iajs-700 131 27 و و PRON iajs-700 131 28 التطبیقیة التطبیقیة PROPN iajs-700 131 29 2012 2012 NUM iajs-700 131 30 السنة السنة NOUN iajs-700 131 31 25 25 NUM iajs-700 131 32 المجلد المجلد NOUN iajs-700 131 33 1 1 NUM iajs-700 131 34 العدد العدد PROPN iajs-700 131 35 ibn ibn PROPN iajs-700 131 36 al al PROPN iajs-700 131 37 - - PUNCT iajs-700 131 38 haitham haitham PROPN iajs-700 131 39 journal journal PROPN iajs-700 131 40 for for ADP iajs-700 131 41 pure pure ADJ iajs-700 131 42 and and CCONJ iajs-700 131 43 applied apply VERB iajs-700 131 44 science science NOUN iajs-700 131 45 no no NOUN iajs-700 131 46 . . NOUN iajs-700 131 47 1 1 NUM iajs-700 131 48 vol vol NOUN iajs-700 131 49 . . PUNCT iajs-700 132 1 25 25 NUM iajs-700 132 2 year year NOUN iajs-700 132 3 2012 2012 NUM iajs-700 132 4 1x 1x PROPN iajs-700 132 5 is be AUX iajs-700 132 6 semiprime semiprime NOUN iajs-700 132 7 . . PUNCT iajs-700 133 1 2[f 2[f NUM iajs-700 133 2 - - PUNCT iajs-700 133 3 anna anna NOUN iajs-700 133 4 : : PUNCT iajs-700 133 5 b b X iajs-700 133 6 ] ] PUNCT iajs-700 133 7 is be AUX iajs-700 133 8 a a DET iajs-700 133 9 semiprime semiprime NOUN iajs-700 133 10 fuzzy fuzzy ADJ iajs-700 133 11 ideal ideal NOUN iajs-700 133 12 of of ADP iajs-700 133 13 r r NOUN iajs-700 133 14 for for ADP iajs-700 133 15 every every DET iajs-700 133 16 nonzero nonzero ADJ iajs-700 133 17 fuzzy fuzzy ADJ iajs-700 133 18 submodule submodule NOUN iajs-700 133 19 a a PRON iajs-700 133 20 of of ADP iajs-700 133 21 x x X iajs-700 133 22 and and CCONJ iajs-700 133 23 for for ADP iajs-700 133 24 every every DET iajs-700 133 25 non non ADJ iajs-700 133 26 - - ADJ iajs-700 133 27 zero zero ADJ iajs-700 133 28 fuzzy fuzzy ADJ iajs-700 133 29 ideal ideal ADJ iajs-700 133 30 b b PROPN iajs-700 133 31 of of ADP iajs-700 133 32 r r NOUN iajs-700 133 33 such such ADJ iajs-700 133 34 that that SCONJ iajs-700 133 35 f f PROPN iajs-700 133 36 - - PUNCT iajs-700 133 37 anna anna PROPN iajs-700 133 38   PROPN iajs-700 133 39 f f X iajs-700 133 40 - - PUNCT iajs-700 133 41 annb annb ADJ iajs-700 133 42 . . PUNCT iajs-700 134 1 3[f 3[f NUM iajs-700 134 2 - - PUNCT iajs-700 134 3 anna anna NOUN iajs-700 134 4 : : PUNCT iajs-700 134 5 xt xt X iajs-700 134 6 ] ] PUNCT iajs-700 134 7 is be AUX iajs-700 134 8 a a DET iajs-700 134 9 semiprime semiprime NOUN iajs-700 134 10 fuzzy fuzzy ADJ iajs-700 134 11 ideal ideal NOUN iajs-700 134 12 of of ADP iajs-700 134 13 r r NOUN iajs-700 134 14 for for ADP iajs-700 134 15 every every DET iajs-700 134 16 non non ADJ iajs-700 134 17 - - ADJ iajs-700 134 18 zero zero ADJ iajs-700 134 19 fuzzy fuzzy ADJ iajs-700 134 20 submodule submodule NOUN iajs-700 134 21 a a PRON iajs-700 134 22 of of ADP iajs-700 134 23 x x X iajs-700 134 24 and and CCONJ iajs-700 134 25 for for ADP iajs-700 134 26 every every DET iajs-700 134 27 fuzzy fuzzy ADJ iajs-700 134 28 singleton singleton PROPN iajs-700 134 29 xt xt PROPN iajs-700 135 1   PROPN iajs-700 135 2 r r NOUN iajs-700 135 3 such such ADJ iajs-700 135 4 that that PRON iajs-700 135 5 xt xt PROPN iajs-700 135 6   VERB iajs-700 136 1 f f X iajs-700 136 2 - - PUNCT iajs-700 136 3 anna anna NOUN iajs-700 136 4 . . PUNCT iajs-700 137 1 4f 4f NOUN iajs-700 137 2 - - PUNCT iajs-700 137 3 ann(xk ann(xk NOUN iajs-700 137 4 ) ) PUNCT iajs-700 137 5 is be AUX iajs-700 137 6 a a DET iajs-700 137 7 semiprime semiprime NOUN iajs-700 137 8 fuzzy fuzzy ADJ iajs-700 137 9 ideal ideal NOUN iajs-700 137 10 of of ADP iajs-700 137 11 r r NOUN iajs-700 137 12 for for ADP iajs-700 137 13 non non ADJ iajs-700 137 14 - - ADJ iajs-700 137 15 zero zero ADJ iajs-700 137 16 fuzzy fuzzy ADJ iajs-700 137 17 singleton singleton PROPN iajs-700 137 18 xk xk PROPN iajs-700 138 1   PROPN iajs-700 138 2 x. x. PUNCT iajs-700 139 1 proof proof NOUN iajs-700 139 2 : : PUNCT iajs-700 139 3 ( ( PUNCT iajs-700 139 4 1 1 X iajs-700 139 5 ) ) PUNCT iajs-700 139 6   NOUN iajs-700 139 7 ( ( PUNCT iajs-700 139 8 2 2 X iajs-700 139 9 ) ) PUNCT iajs-700 139 10 let let VERB iajs-700 139 11 2 2 NUM iajs-700 139 12 kr kr PROPN iajs-700 139 13   NOUN iajs-700 140 1 [ [ X iajs-700 140 2 f f X iajs-700 140 3 - - PUNCT iajs-700 140 4 anna anna NOUN iajs-700 140 5 : : PUNCT iajs-700 140 6 b b NOUN iajs-700 140 7 ] ] X iajs-700 140 8 , , PUNCT iajs-700 140 9 hence hence ADV iajs-700 140 10 2 2 NUM iajs-700 140 11 kr kr PROPN iajs-700 140 12 b b PROPN iajs-700 140 13   PROPN iajs-700 140 14 f f X iajs-700 140 15 - - NOUN iajs-700 140 16 anna anna NOUN iajs-700 140 17 . . PUNCT iajs-700 141 1 so so ADV iajs-700 141 2 2 2 NUM iajs-700 141 3 kr kr PROPN iajs-700 141 4 bs bs PROPN iajs-700 141 5   PROPN iajs-700 141 6 f f PROPN iajs-700 141 7 - - PUNCT iajs-700 141 8 anna anna NOUN iajs-700 141 9 , , PUNCT iajs-700 141 10 for for ADP iajs-700 141 11 all all DET iajs-700 141 12 bs bs PROPN iajs-700 141 13 b. b. PROPN iajs-700 141 14 hence hence ADV iajs-700 141 15 2 2 NUM iajs-700 141 16 2 2 NUM iajs-700 141 17 k k NOUN iajs-700 141 18 sr sr PROPN iajs-700 141 19 b b PROPN iajs-700 141 20   PROPN iajs-700 141 21 f f X iajs-700 141 22 - - PUNCT iajs-700 141 23 anna anna NOUN iajs-700 141 24 , , PUNCT iajs-700 141 25 so so ADV iajs-700 141 26 2(rb) 2(rb) NUM iajs-700 141 27   PROPN iajs-700 141 28 f f X iajs-700 141 29 - - PUNCT iajs-700 141 30 anna anna NOUN iajs-700 141 31 , , PUNCT iajs-700 141 32 where where SCONJ iajs-700 141 33   ADJ iajs-700 141 34 = = SYM iajs-700 141 35 min{k min{k X iajs-700 141 36 , , PUNCT iajs-700 141 37 s s PART iajs-700 141 38 } } PUNCT iajs-700 141 39 . . PUNCT iajs-700 142 1 thus thus ADV iajs-700 142 2 ( ( PUNCT iajs-700 142 3 rb) rb) PROPN iajs-700 142 4   PROPN iajs-700 142 5 f f PROPN iajs-700 142 6 - - PUNCT iajs-700 142 7 anna anna NOUN iajs-700 142 8 , , PUNCT iajs-700 142 9 since since SCONJ iajs-700 142 10 f f NOUN iajs-700 142 11 - - PUNCT iajs-700 142 12 anna anna PROPN iajs-700 142 13 is be AUX iajs-700 142 14 a a DET iajs-700 142 15 semiprime semiprime NOUN iajs-700 142 16 fuzzy fuzzy ADJ iajs-700 142 17 ideal ideal NOUN iajs-700 142 18 . . PUNCT iajs-700 143 1 so so ADV iajs-700 143 2 rkbs rkb VERB iajs-700 143 3   NOUN iajs-700 143 4 f f PROPN iajs-700 143 5 - - PUNCT iajs-700 143 6 anna anna NOUN iajs-700 143 7 . . PUNCT iajs-700 144 1 hence hence ADV iajs-700 144 2 rk rk VERB iajs-700 144 3   NOUN iajs-700 145 1 [ [ X iajs-700 145 2 f f X iajs-700 145 3 - - PUNCT iajs-700 145 4 anna anna NOUN iajs-700 145 5 : : PUNCT iajs-700 145 6 b b NOUN iajs-700 145 7 ] ] X iajs-700 145 8 . . PUNCT iajs-700 146 1 thus thus ADV iajs-700 146 2 [ [ X iajs-700 146 3 f f X iajs-700 146 4 - - PUNCT iajs-700 146 5 anna anna NOUN iajs-700 146 6 : : PUNCT iajs-700 146 7 b b X iajs-700 146 8 ] ] PUNCT iajs-700 146 9 is be AUX iajs-700 146 10 a a DET iajs-700 146 11 semiprime semiprime NOUN iajs-700 146 12 fuzzy fuzzy ADJ iajs-700 146 13 ideal ideal NOUN iajs-700 146 14 . . PUNCT iajs-700 147 1 ( ( PUNCT iajs-700 147 2 2 2 X iajs-700 147 3 ) ) PUNCT iajs-700 147 4   NOUN iajs-700 147 5 ( ( PUNCT iajs-700 147 6 3 3 X iajs-700 147 7 ) ) PUNCT iajs-700 147 8 it it PRON iajs-700 147 9 is be AUX iajs-700 147 10 followed follow VERB iajs-700 147 11 by by ADP iajs-700 147 12 putting put VERB iajs-700 147 13 < < X iajs-700 147 14 xt xt X iajs-700 147 15 > > X iajs-700 147 16 = = PROPN iajs-700 147 17 b. b. PROPN iajs-700 147 18 ( ( PUNCT iajs-700 147 19 3 3 NUM iajs-700 147 20 ) ) PUNCT iajs-700 147 21   NOUN iajs-700 147 22 ( ( PUNCT iajs-700 147 23 4 4 X iajs-700 147 24 ) ) PUNCT iajs-700 147 25 it it PRON iajs-700 147 26 is be AUX iajs-700 147 27 easy easy ADJ iajs-700 147 28 to to PART iajs-700 147 29 check check VERB iajs-700 147 30 that that PRON iajs-700 147 31 [ [ X iajs-700 147 32 f f X iajs-700 147 33 - - PUNCT iajs-700 147 34 annxt:<11 annxt:<11 NOUN iajs-700 147 35 > > X iajs-700 147 36 ] ] X iajs-700 147 37 = = SYM iajs-700 147 38 f f X iajs-700 147 39 - - PUNCT iajs-700 147 40 ann ann PROPN iajs-700 147 41 < < X iajs-700 147 42 xt xt X iajs-700 147 43 > > X iajs-700 147 44 . . PUNCT iajs-700 148 1 but but CCONJ iajs-700 148 2 [ [ X iajs-700 148 3 f f X iajs-700 148 4 - - PUNCT iajs-700 148 5 ann ann PROPN iajs-700 148 6 < < X iajs-700 148 7 xt>:<11 xt>:<11 PROPN iajs-700 148 8 > > X iajs-700 148 9 ] ] X iajs-700 148 10 is be AUX iajs-700 148 11 a a DET iajs-700 148 12 semiprime semiprime NOUN iajs-700 148 13 fuzzy fuzzy ADJ iajs-700 148 14 ideal ideal NOUN iajs-700 148 15 by by ADP iajs-700 148 16 ( ( PUNCT iajs-700 148 17 3 3 NUM iajs-700 148 18 ) ) PUNCT iajs-700 148 19 . . PUNCT iajs-700 149 1 thus thus ADV iajs-700 149 2 f f X iajs-700 149 3 - - PUNCT iajs-700 149 4 annxt annxt PROPN iajs-700 149 5 is be AUX iajs-700 149 6 a a DET iajs-700 149 7 semiprime semiprime NOUN iajs-700 149 8 fuzzy fuzzy ADJ iajs-700 149 9 ideal ideal NOUN iajs-700 149 10 . . PUNCT iajs-700 150 1 the the DET iajs-700 150 2 following follow VERB iajs-700 150 3 proposition proposition NOUN iajs-700 150 4 shows show VERB iajs-700 150 5 that that SCONJ iajs-700 150 6 the the DET iajs-700 150 7 direct direct ADJ iajs-700 150 8 sum sum NOUN iajs-700 150 9 of of ADP iajs-700 150 10 semiprime semiprime NOUN iajs-700 150 11 fuzzy fuzzy ADJ iajs-700 150 12 modules module NOUN iajs-700 150 13 is be AUX iajs-700 150 14 semiprime semiprime ADJ iajs-700 150 15 fuzzy fuzzy ADJ iajs-700 150 16 module module NOUN iajs-700 150 17 . . PUNCT iajs-700 151 1 2.10 2.10 NUM iajs-700 151 2 proposition proposition NOUN iajs-700 151 3 : : PUNCT iajs-700 151 4 let let VERB iajs-700 151 5 x x PRON iajs-700 151 6 and and CCONJ iajs-700 151 7 y y PROPN iajs-700 151 8 be be AUX iajs-700 151 9 two two NUM iajs-700 151 10 fuzzy fuzzy ADJ iajs-700 151 11 modules module NOUN iajs-700 151 12 of of ADP iajs-700 151 13 m m PROPN iajs-700 151 14 1 1 NUM iajs-700 151 15 and and CCONJ iajs-700 151 16 m2 m2 PROPN iajs-700 151 17 r r PROPN iajs-700 151 18 - - PUNCT iajs-700 151 19 modules module NOUN iajs-700 151 20 respectively respectively ADV iajs-700 151 21 . . PUNCT iajs-700 152 1 then then ADV iajs-700 152 2 x x X iajs-700 152 3 and and CCONJ iajs-700 152 4 y y PROPN iajs-700 152 5 are be AUX iajs-700 152 6 semiprime semiprime NOUN iajs-700 152 7 if if SCONJ iajs-700 152 8 and and CCONJ iajs-700 152 9 only only ADV iajs-700 152 10 if if SCONJ iajs-700 152 11 xy xy PROPN iajs-700 152 12 is be AUX iajs-700 152 13 a a DET iajs-700 152 14 semiprime semiprime NOUN iajs-700 152 15 fuzzy fuzzy ADJ iajs-700 152 16 module module NOUN iajs-700 152 17 . . PUNCT iajs-700 153 1 proof proof NOUN iajs-700 153 2 : : PUNCT iajs-700 153 3 ( ( PUNCT iajs-700 153 4   NOUN iajs-700 153 5 ) ) PUNCT iajs-700 153 6 if if SCONJ iajs-700 153 7 x x PRON iajs-700 153 8 and and CCONJ iajs-700 153 9 y y PROPN iajs-700 153 10 are be AUX iajs-700 153 11 semiprime semiprime NOUN iajs-700 153 12 , , PUNCT iajs-700 153 13 then then ADV iajs-700 153 14 xt xt PROPN iajs-700 153 15 and and CCONJ iajs-700 153 16 yt yt PROPN iajs-700 153 17 are be AUX iajs-700 153 18 semiprime semiprime NOUN iajs-700 153 19 modules module NOUN iajs-700 153 20 by by ADP iajs-700 153 21 proposition proposition NOUN iajs-700 153 22 2.5 2.5 NUM iajs-700 153 23 . . PUNCT iajs-700 154 1 hence hence ADV iajs-700 154 2 xtyt xtyt PUNCT iajs-700 154 3 is be AUX iajs-700 154 4 a a DET iajs-700 154 5 semiprime semiprime NOUN iajs-700 154 6 module module NOUN iajs-700 154 7 by by ADP iajs-700 154 8 [ [ X iajs-700 154 9 9,prop.4.1.11 9,prop.4.1.11 X iajs-700 154 10 ] ] PUNCT iajs-700 154 11 . . PUNCT iajs-700 155 1 but but CCONJ iajs-700 155 2 xtyt xtyt X iajs-700 156 1 = = PUNCT iajs-700 156 2 ( ( PUNCT iajs-700 156 3 xy)t xy)t VERB iajs-700 156 4 by by ADP iajs-700 156 5 remark remark NOUN iajs-700 156 6 1.19 1.19 NUM iajs-700 156 7 . . PUNCT iajs-700 157 1 thus thus ADV iajs-700 157 2 xy xy PROPN iajs-700 157 3 is be AUX iajs-700 157 4 a a DET iajs-700 157 5 semiprime semiprime NOUN iajs-700 157 6 fuzzy fuzzy ADJ iajs-700 157 7 module module NOUN iajs-700 157 8 by by ADP iajs-700 157 9 proposition proposition NOUN iajs-700 157 10 2.5 2.5 NUM iajs-700 157 11 . . PUNCT iajs-700 158 1 ( ( PUNCT iajs-700 158 2   NOUN iajs-700 158 3 ) ) PUNCT iajs-700 158 4 the the DET iajs-700 158 5 proof proof NOUN iajs-700 158 6 is be AUX iajs-700 158 7 similarly similarly ADV iajs-700 158 8 . . PUNCT iajs-700 159 1 now now ADV iajs-700 159 2 we we PRON iajs-700 159 3 turn turn VERB iajs-700 159 4 our our PRON iajs-700 159 5 at at ADP iajs-700 159 6 tention tention NOUN iajs-700 159 7 to to ADP iajs-700 159 8 image image NOUN iajs-700 159 9 and and CCONJ iajs-700 159 10 inverse inverse NOUN iajs-700 159 11 image image NOUN iajs-700 159 12 of of ADP iajs-700 159 13 semiprime semiprime NOUN iajs-700 159 14 fuzzy fuzzy ADJ iajs-700 159 15 module module NOUN iajs-700 159 16 . . PUNCT iajs-700 160 1 we we PRON iajs-700 160 2 have have VERB iajs-700 160 3 the the DET iajs-700 160 4 following follow VERB iajs-700 160 5 : : PUNCT iajs-700 160 6 2.11 2.11 NUM iajs-700 160 7 proposition proposition NOUN iajs-700 160 8 : : PUNCT iajs-700 160 9 let let VERB iajs-700 160 10 x x PRON iajs-700 160 11 and and CCONJ iajs-700 160 12 y y PROPN iajs-700 160 13 be be AUX iajs-700 160 14 two two NUM iajs-700 160 15 fuzzy fuzzy ADJ iajs-700 160 16 modules module NOUN iajs-700 160 17 of of ADP iajs-700 160 18 r r NOUN iajs-700 160 19 - - PUNCT iajs-700 160 20 modules module NOUN iajs-700 160 21 m1 m1 NOUN iajs-700 160 22 and and CCONJ iajs-700 160 23 m2 m2 PROPN iajs-700 160 24 respectively respectively ADV iajs-700 160 25 . . PUNCT iajs-700 161 1 let let VERB iajs-700 161 2 f f X iajs-700 161 3 : : PUNCT iajs-700 161 4 m1 m1 PROPN iajs-700 161 5   PROPN iajs-700 161 6 m2 m2 PROPN iajs-700 161 7 be be VERB iajs-700 161 8 rhomomorphism rhomomorphism NOUN iajs-700 161 9 , , PUNCT iajs-700 161 10 then then ADV iajs-700 161 11 3semiprime 3semiprime NUM iajs-700 161 12 fuzzy fuzzy ADJ iajs-700 161 13 modules module NOUN iajs-700 161 14 and and CCONJ iajs-700 161 15 other other ADJ iajs-700 161 16 related relate VERB iajs-700 161 17 fuzzy fuzzy ADJ iajs-700 161 18 modules module NOUN iajs-700 161 19 in in ADP iajs-700 161 20 this this DET iajs-700 161 21 section section NOUN iajs-700 161 22 we we PRON iajs-700 161 23 study study VERB iajs-700 161 24 the the DET iajs-700 161 25 relationship relationship NOUN iajs-700 161 26 between between ADP iajs-700 161 27 semiprime semiprime NOUN iajs-700 161 28 fuzzy fuzzy ADJ iajs-700 161 29 module module NOUN iajs-700 161 30 and and CCONJ iajs-700 161 31 divisible divisible ADJ iajs-700 161 32 , , PUNCT iajs-700 161 33 uniform uniform NOUN iajs-700 161 34 and and CCONJ iajs-700 161 35 f f NOUN iajs-700 161 36 - - PUNCT iajs-700 161 37 regular regular ADJ iajs-700 161 38 fuzzy fuzzy ADJ iajs-700 161 39 modules module NOUN iajs-700 161 40 . . PUNCT iajs-700 162 1 3.1 3.1 NUM iajs-700 162 2 de de NOUN iajs-700 162 3 finition finition NOUN iajs-700 162 4 : : PUNCT iajs-700 163 1 [ [ X iajs-700 163 2 17 17 NUM iajs-700 163 3 ] ] PUNCT iajs-700 163 4 a a DET iajs-700 163 5 fuzzy fuzzy ADJ iajs-700 163 6 module module NOUN iajs-700 163 7 x x PRON iajs-700 163 8 is be AUX iajs-700 163 9 divisible divisible ADJ iajs-700 163 10 if if SCONJ iajs-700 163 11 rtx rtx PROPN iajs-700 163 12 = = SYM iajs-700 163 13 x x PROPN iajs-700 163 14 , , PUNCT iajs-700 163 15 for for ADP iajs-700 163 16 all all DET iajs-700 163 17 rt rt PROPN iajs-700 163 18   INTJ iajs-700 163 19 ot ot INTJ iajs-700 163 20 ( ( PUNCT iajs-700 163 21 rt rt PROPN iajs-700 163 22 is be AUX iajs-700 163 23 a a DET iajs-700 163 24 fuzzy fuzzy ADJ iajs-700 163 25 singleton singleton NOUN iajs-700 163 26 of of ADP iajs-700 163 27 r r NOUN iajs-700 163 28 ) ) PUNCT iajs-700 163 29 . . PUNCT iajs-700 164 1 3.2 3.2 NUM iajs-700 164 2 definition definition NOUN iajs-700 164 3 : : PUNCT iajs-700 164 4 [ [ X iajs-700 164 5 17 17 NUM iajs-700 164 6 ] ] PUNCT iajs-700 164 7 a a DET iajs-700 164 8 fuzzy fuzzy ADJ iajs-700 164 9 module module NOUN iajs-700 164 10 is be AUX iajs-700 164 11 called call VERB iajs-700 164 12 uniform uniform ADJ iajs-700 164 13 if if SCONJ iajs-700 164 14 ab ab NOUN iajs-700 164 15   NOUN iajs-700 164 16 o1 o1 NOUN iajs-700 164 17 , , PUNCT iajs-700 164 18 for for ADP iajs-700 164 19 any any DET iajs-700 164 20 non non ADJ iajs-700 164 21 trivial trivial ADJ iajs-700 164 22 fuzzy fuzzy ADJ iajs-700 164 23 submodules submodule NOUN iajs-700 164 24 a a PRON iajs-700 164 25 and and CCONJ iajs-700 164 26 b. b. PROPN iajs-700 164 27 3.3 3.3 NUM iajs-700 164 28 definition definition NOUN iajs-700 164 29 : : PUNCT iajs-700 165 1 [ [ X iajs-700 165 2 17 17 NUM iajs-700 165 3 ] ] PUNCT iajs-700 165 4 let let VERB iajs-700 165 5 a a PRON iajs-700 165 6 be be AUX iajs-700 165 7 a a DET iajs-700 165 8 fuzzy fuzzy ADJ iajs-700 165 9 submodule submodule NOUN iajs-700 165 10 of of ADP iajs-700 165 11 fuzzy fuzzy ADJ iajs-700 165 12 module module NOUN iajs-700 165 13 x. x. NOUN iajs-700 165 14 then then ADV iajs-700 165 15 a a PRON iajs-700 165 16 is be AUX iajs-700 165 17 called call VERB iajs-700 165 18 an an DET iajs-700 165 19 essential essential ADJ iajs-700 165 20 fuzzy fuzzy ADJ iajs-700 165 21 submodule submodule NOUN iajs-700 165 22 if if SCONJ iajs-700 165 23 ab ab NOUN iajs-700 165 24   NOUN iajs-700 165 25 o1 o1 NOUN iajs-700 165 26 , , PUNCT iajs-700 165 27 for for SCONJ iajs-700 165 28 any any DET iajs-700 165 29 nontrivial nontrivial ADJ iajs-700 165 30 fuzzy fuzzy ADJ iajs-700 165 31 submodule submodule PROPN iajs-700 165 32 b b PROPN iajs-700 165 33 of of ADP iajs-700 165 34 x. x. NOUN iajs-700 165 35 مجلة مجلة PROPN iajs-700 165 36 إبن إبن VERB iajs-700 165 37 الھیثم الھیثم NOUN iajs-700 165 38 للعلوم للعلوم NOUN iajs-700 165 39 الصرفة الصرفة NOUN iajs-700 165 40 و و PRON iajs-700 165 41 التطبیقیة التطبیقیة PROPN iajs-700 165 42 2012 2012 NUM iajs-700 165 43 السنة السنة NOUN iajs-700 165 44 25 25 NUM iajs-700 165 45 المجلد المجلد NOUN iajs-700 165 46 1 1 NUM iajs-700 165 47 العدد العدد PROPN iajs-700 165 48 ibn ibn PROPN iajs-700 165 49 al al PROPN iajs-700 165 50 - - PUNCT iajs-700 165 51 haitham haitham PROPN iajs-700 165 52 journal journal PROPN iajs-700 165 53 for for ADP iajs-700 165 54 pure pure ADJ iajs-700 165 55 and and CCONJ iajs-700 165 56 applied apply VERB iajs-700 165 57 science science NOUN iajs-700 165 58 no no NOUN iajs-700 165 59 . . NOUN iajs-700 165 60 1 1 NUM iajs-700 165 61 vol vol NOUN iajs-700 165 62 . . PUNCT iajs-700 166 1 25 25 NUM iajs-700 166 2 year year NOUN iajs-700 166 3 2012 2012 NUM iajs-700 166 4 3.4 3.4 NUM iajs-700 166 5 proposition proposition NOUN iajs-700 166 6 : : PUNCT iajs-700 166 7 let let VERB iajs-700 166 8 x x PRON iajs-700 166 9 be be AUX iajs-700 166 10 a a DET iajs-700 166 11 uniform uniform ADJ iajs-700 166 12 fuzzy fuzzy ADJ iajs-700 166 13 module module NOUN iajs-700 166 14 . . PUNCT iajs-700 167 1 then then ADV iajs-700 167 2 x x PRON iajs-700 167 3 is be AUX iajs-700 167 4 a a DET iajs-700 167 5 prime prime ADJ iajs-700 167 6 fuzzy fuzzy ADJ iajs-700 167 7 module module NOUN iajs-700 167 8 if if SCONJ iajs-700 167 9 and and CCONJ iajs-700 167 10 only only ADV iajs-700 167 11 if if SCONJ iajs-700 167 12 x x PRON iajs-700 167 13 is be AUX iajs-700 167 14 semiprime semiprime ADJ iajs-700 167 15 fuzzy fuzzy ADJ iajs-700 167 16 module module NOUN iajs-700 167 17 . . PUNCT iajs-700 168 1 proof proof NOUN iajs-700 168 2 : : PUNCT iajs-700 168 3 ( ( PUNCT iajs-700 168 4   NOUN iajs-700 168 5 ) ) PUNCT iajs-700 168 6 it it PRON iajs-700 168 7 is be AUX iajs-700 168 8 easy easy ADJ iajs-700 168 9 by by ADP iajs-700 168 10 prop.2.2 prop.2.2 PROPN iajs-700 168 11 . . PUNCT iajs-700 168 12 ( ( PUNCT iajs-700 168 13   NOUN iajs-700 168 14 ) ) PUNCT iajs-700 168 15 to to PART iajs-700 168 16 prove prove VERB iajs-700 168 17 f f PROPN iajs-700 168 18 - - PUNCT iajs-700 168 19 annx annx PROPN iajs-700 168 20 = = PUNCT iajs-700 169 1 f f X iajs-700 169 2 - - PUNCT iajs-700 169 3 anna anna NOUN iajs-700 169 4 , , PUNCT iajs-700 169 5 for for ADP iajs-700 169 6 any any DET iajs-700 169 7 non non ADJ iajs-700 169 8 trivial trivial ADJ iajs-700 169 9 fuzzy fuzzy ADJ iajs-700 169 10 submodule submodule NOUN iajs-700 169 11 a a PRON iajs-700 169 12 of of ADP iajs-700 169 13 x x X iajs-700 170 1 [ [ X iajs-700 170 2 17,def.3.1.1 17,def.3.1.1 NUM iajs-700 170 3 ] ] PUNCT iajs-700 170 4 . . PUNCT iajs-700 171 1 it it PRON iajs-700 171 2 is be AUX iajs-700 171 3 clear clear ADJ iajs-700 171 4 that that SCONJ iajs-700 171 5 f f X iajs-700 171 6 - - PUNCT iajs-700 171 7 annx annx PROPN iajs-700 171 8   PROPN iajs-700 171 9 f f X iajs-700 171 10 - - PUNCT iajs-700 171 11 anna anna NOUN iajs-700 171 12 . . PUNCT iajs-700 172 1 to to PART iajs-700 172 2 prove prove VERB iajs-700 172 3 f f NOUN iajs-700 172 4 - - PUNCT iajs-700 172 5 anna anna NOUN iajs-700 172 6   PROPN iajs-700 172 7 f f PROPN iajs-700 172 8 - - PUNCT iajs-700 172 9 annx annx PROPN iajs-700 172 10 . . PUNCT iajs-700 173 1 let let VERB iajs-700 173 2 rt rt PROPN iajs-700 173 3   PROPN iajs-700 173 4 f f NOUN iajs-700 173 5 - - PUNCT iajs-700 173 6 anna anna PROPN iajs-700 173 7 and and CCONJ iajs-700 173 8 rt rt PROPN iajs-700 173 9   CCONJ iajs-700 173 10 f f PROPN iajs-700 173 11 - - PUNCT iajs-700 173 12 annx annx PROPN iajs-700 173 13 . . PUNCT iajs-700 174 1 thus thus ADV iajs-700 174 2 there there PRON iajs-700 174 3 exists exist VERB iajs-700 174 4 xk xk PROPN iajs-700 174 5   PROPN iajs-700 174 6 x x PROPN iajs-700 174 7 , , PUNCT iajs-700 174 8 xk xk PROPN iajs-700 174 9   NOUN iajs-700 174 10 0k 0k NOUN iajs-700 174 11 such such ADJ iajs-700 174 12 that that PRON iajs-700 174 13 rtxk rtxk VERB iajs-700 174 14 ⊈ ⊈ PROPN iajs-700 174 15 o1 o1 NOUN iajs-700 174 16 . . PUNCT iajs-700 175 1 since since SCONJ iajs-700 175 2 x x PROPN iajs-700 175 3 is be AUX iajs-700 175 4 uniform uniform ADJ iajs-700 175 5 , , PUNCT iajs-700 175 6 a > X iajs-700 175 8   NOUN iajs-700 175 9 o1 o1 NOUN iajs-700 175 10 , , PUNCT iajs-700 175 11 then then ADV iajs-700 175 12 there there PRON iajs-700 175 13 exists exist VERB iajs-700 175 14 y y PROPN iajs-700 175 15 s s PROPN iajs-700 175 16   PROPN iajs-700 175 17 a a PRON iajs-700 175 18 and and CCONJ iajs-700 175 19 ys ys NOUN iajs-700 176 1 < < AUX iajs-700 176 2 rtxk rtxk VERB iajs-700 176 3 > > X iajs-700 176 4 such such ADJ iajs-700 176 5 that that DET iajs-700 176 6 ys ys PROPN iajs-700 176 7   PROPN iajs-700 176 8 o1 o1 PROPN iajs-700 176 9 . . PUNCT iajs-700 177 1 thus thus ADV iajs-700 177 2 ys ys VERB iajs-700 177 3 = = NOUN iajs-700 177 4 aℓrtxk aℓrtxk PROPN iajs-700 177 5 , , PUNCT iajs-700 177 6 aℓ aℓ PROPN iajs-700 177 7 is be AUX iajs-700 177 8 a a DET iajs-700 177 9 fuzzy fuzzy ADJ iajs-700 177 10 singleton singleton NOUN iajs-700 177 11 of of ADP iajs-700 177 12 r. r. PROPN iajs-700 177 13 o1 o1 PROPN iajs-700 177 14 = = SYM iajs-700 177 15 rtys rtys NOUN iajs-700 178 1 = = SYM iajs-700 178 2 aℓ aℓ PROPN iajs-700 178 3 2 2 NUM iajs-700 178 4 tr tr VERB iajs-700 178 5 xk xk PROPN iajs-700 178 6 , , PUNCT iajs-700 178 7 it it PRON iajs-700 178 8 follows follow VERB iajs-700 178 9 2 2 NUM iajs-700 178 10 tr tr NOUN iajs-700 178 11   NOUN iajs-700 178 12 f f PROPN iajs-700 178 13 - - PUNCT iajs-700 178 14 ann ann PROPN iajs-700 178 15 < < X iajs-700 178 16 aℓxk aℓxk NOUN iajs-700 178 17 > > X iajs-700 178 18 since since SCONJ iajs-700 178 19 aℓxk aℓxk PROPN iajs-700 178 20   NOUN iajs-700 178 21 o1 o1 PROPN iajs-700 178 22 , , PUNCT iajs-700 178 23 so so CCONJ iajs-700 178 24 f f PROPN iajs-700 178 25 - - PUNCT iajs-700 178 26 ann ann PROPN iajs-700 178 27 < < X iajs-700 178 28 aℓxk aℓxk NOUN iajs-700 178 29 > > X iajs-700 178 30 is be AUX iajs-700 178 31 a a DET iajs-700 178 32 semiprime semiprime NOUN iajs-700 178 33 fuzzy fuzzy ADJ iajs-700 178 34 ideal ideal NOUN iajs-700 178 35 of of ADP iajs-700 178 36 r. r. PROPN iajs-700 178 37 therefore therefore ADV iajs-700 178 38 rt rt PROPN iajs-700 179 1   PROPN iajs-700 179 2 f f PROPN iajs-700 179 3 - - PUNCT iajs-700 179 4 ann ann PROPN iajs-700 179 5 < < X iajs-700 179 6 aℓxk aℓxk NOUN iajs-700 179 7 > > PUNCT iajs-700 179 8 . . PUNCT iajs-700 180 1 this this PRON iajs-700 180 2 implies imply VERB iajs-700 180 3 that that PRON iajs-700 180 4 o1 o1 NOUN iajs-700 180 5 = = SYM iajs-700 180 6 rtaℓxk rtaℓxk NOUN iajs-700 180 7 = = SYM iajs-700 180 8 y y PROPN iajs-700 180 9 s. s. PROPN iajs-700 180 10 thus thus ADV iajs-700 180 11 y y PROPN iajs-700 180 12 s s PROPN iajs-700 180 13 = = NOUN iajs-700 180 14 o1 o1 NOUN iajs-700 180 15 which which PRON iajs-700 180 16 is be AUX iajs-700 180 17 a a DET iajs-700 180 18 contradiction contradiction NOUN iajs-700 180 19 . . PUNCT iajs-700 181 1 3.5 3.5 NUM iajs-700 181 2 proposition proposition NOUN iajs-700 181 3 : : PUNCT iajs-700 181 4 if if SCONJ iajs-700 181 5 x x PRON iajs-700 181 6 is be AUX iajs-700 181 7 a a DET iajs-700 181 8 uniform uniform ADJ iajs-700 181 9 fuzzy fuzzy ADJ iajs-700 181 10 module module NOUN iajs-700 181 11 , , PUNCT iajs-700 181 12 then then ADV iajs-700 181 13 xt xt PROPN iajs-700 181 14 is be AUX iajs-700 181 15 a a DET iajs-700 181 16 uniform uniform ADJ iajs-700 181 17 module module NOUN iajs-700 181 18 , , PUNCT iajs-700 181 19   PROPN iajs-700 181 20 t t PROPN iajs-700 181 21   PROPN iajs-700 181 22 ( ( PUNCT iajs-700 181 23 0,1 0,1 NOUN iajs-700 181 24 ] ] PUNCT iajs-700 181 25 . . PUNCT iajs-700 182 1 proof proof NOUN iajs-700 182 2 : : PUNCT iajs-700 182 3 let let VERB iajs-700 182 4 n n PRON iajs-700 182 5 and and CCONJ iajs-700 182 6 w w PROPN iajs-700 182 7 be be AUX iajs-700 182 8 submodules submodule NOUN iajs-700 182 9 of of ADP iajs-700 182 10 xt xt NOUN iajs-700 182 11 such such ADJ iajs-700 182 12 that that SCONJ iajs-700 182 13 n n PROPN iajs-700 182 14   NOUN iajs-700 182 15 o o PROPN iajs-700 182 16 , , PUNCT iajs-700 182 17 w w PROPN iajs-700 182 18   PROPN iajs-700 182 19 o. o. INTJ iajs-700 182 20 define define VERB iajs-700 182 21 a a DET iajs-700 182 22 : : PUNCT iajs-700 182 23 m m PROPN iajs-700 182 24   X iajs-700 183 1 [ [ X iajs-700 183 2 0,1 0,1 NUM iajs-700 183 3 ] ] PUNCT iajs-700 183 4 , , PUNCT iajs-700 183 5 b b X iajs-700 183 6 : : PUNCT iajs-700 183 7 m m PROPN iajs-700 183 8   NOUN iajs-700 184 1 [ [ X iajs-700 184 2 0,1 0,1 NUM iajs-700 184 3 ] ] PUNCT iajs-700 184 4 by by ADP iajs-700 184 5 t t PROPN iajs-700 184 6 if if SCONJ iajs-700 184 7 x x PROPN iajs-700 184 8 n n CCONJ iajs-700 184 9 , , PUNCT iajs-700 184 10 t t PROPN iajs-700 184 11 0 0 SYM iajs-700 184 12 a(x a(x PROPN iajs-700 184 13 ) ) PUNCT iajs-700 184 14 0 0 PUNCT iajs-700 185 1 otherwise otherwise ADV iajs-700 185 2   NOUN iajs-700 185 3   PUNCT iajs-700 185 4   NUM iajs-700 185 5   NOUN iajs-700 185 6   INTJ iajs-700 185 7 , , PUNCT iajs-700 185 8 t t X iajs-700 185 9 if if SCONJ iajs-700 185 10 x x X iajs-700 185 11 w w PROPN iajs-700 185 12 , , PUNCT iajs-700 185 13 t t NOUN iajs-700 185 14 0 0 NUM iajs-700 185 15 b(x b(x NOUN iajs-700 185 16 ) ) PUNCT iajs-700 185 17 0 0 PUNCT iajs-700 186 1 otherwise otherwise ADV iajs-700 186 2   NOUN iajs-700 186 3   PUNCT iajs-700 186 4   NUM iajs-700 186 5   NOUN iajs-700 186 6   INTJ iajs-700 186 7 this this PRON iajs-700 186 8 implies imply VERB iajs-700 186 9 a a PRON iajs-700 186 10 and and CCONJ iajs-700 186 11 b b NOUN iajs-700 186 12 are be AUX iajs-700 186 13 fuzzy fuzzy ADJ iajs-700 186 14 submodules submodule NOUN iajs-700 186 15 of of ADP iajs-700 186 16 x x X iajs-700 186 17 and and CCONJ iajs-700 186 18 at at ADP iajs-700 186 19 = = PUNCT iajs-700 186 20 n n CCONJ iajs-700 186 21 , , PUNCT iajs-700 186 22 bt bt NOUN iajs-700 186 23 = = SYM iajs-700 186 24 w w PROPN iajs-700 186 25 , , PUNCT iajs-700 186 26 for for ADP iajs-700 186 27 all all DET iajs-700 186 28 t t NOUN iajs-700 186 29   NOUN iajs-700 186 30 ( ( PUNCT iajs-700 186 31 0,1 0,1 NOUN iajs-700 186 32 ] ] PUNCT iajs-700 186 33 . . PUNCT iajs-700 187 1 since since SCONJ iajs-700 187 2 x x PROPN iajs-700 187 3 is be AUX iajs-700 187 4 uniform uniform ADJ iajs-700 187 5 , , PUNCT iajs-700 187 6 then then ADV iajs-700 187 7 ab ab PROPN iajs-700 187 8   NOUN iajs-700 187 9 o1 o1 PROPN iajs-700 187 10 . . PUNCT iajs-700 188 1 but but CCONJ iajs-700 188 2 t t PROPN iajs-700 188 3 if if SCONJ iajs-700 188 4 x x PROPN iajs-700 188 5 n n X iajs-700 188 6 w w PROPN iajs-700 188 7 ( ( PUNCT iajs-700 188 8 a a DET iajs-700 188 9 b)(x b)(x PROPN iajs-700 188 10 ) ) PUNCT iajs-700 188 11 0 0 PUNCT iajs-700 189 1 if if SCONJ iajs-700 189 2 x x PROPN iajs-700 189 3 n n X iajs-700 189 4 w w PROPN iajs-700 189 5   PROPN iajs-700 189 6   NOUN iajs-700 189 7   PUNCT iajs-700 189 8   NOUN iajs-700 189 9   NOUN iajs-700 189 10   NOUN iajs-700 189 11   NOUN iajs-700 189 12 . . PUNCT iajs-700 190 1 on on ADP iajs-700 190 2 the the DET iajs-700 190 3 other other ADJ iajs-700 190 4 hand hand NOUN iajs-700 190 5 , , PUNCT iajs-700 190 6 ( ( PUNCT iajs-700 190 7 ab)t ab)t PROPN iajs-700 190 8 = = PUNCT iajs-700 190 9 at at ADP iajs-700 190 10   NOUN iajs-700 190 11 bt bt NOUN iajs-700 190 12 = = SYM iajs-700 190 13 n n PROPN iajs-700 190 14   PUNCT iajs-700 190 15 w. w. NOUN iajs-700 190 16 hence hence ADV iajs-700 190 17 n n PROPN iajs-700 190 18   PUNCT iajs-700 190 19 w w PROPN iajs-700 190 20   NOUN iajs-700 190 21 { { PUNCT iajs-700 190 22 0 0 NUM iajs-700 190 23 } } PUNCT iajs-700 190 24 . . PUNCT iajs-700 191 1 thus thus ADV iajs-700 191 2 xt xt PROPN iajs-700 191 3 is be AUX iajs-700 191 4 a a DET iajs-700 191 5 uniform uniform ADJ iajs-700 191 6 module module NOUN iajs-700 191 7 , , PUNCT iajs-700 191 8   PROPN iajs-700 191 9 t t PROPN iajs-700 191 10   PROPN iajs-700 191 11 ( ( PUNCT iajs-700 191 12 0,1 0,1 NOUN iajs-700 191 13 ] ] PUNCT iajs-700 191 14 . . PUNCT iajs-700 192 1 recall recall VERB iajs-700 192 2 that that SCONJ iajs-700 192 3 an an DET iajs-700 192 4 r r NOUN iajs-700 192 5 - - PUNCT iajs-700 192 6 submodule submodule NOUN iajs-700 192 7 n n NOUN iajs-700 192 8 of of ADP iajs-700 192 9 module module NOUN iajs-700 192 10 m m VERB iajs-700 192 11 is be AUX iajs-700 192 12 called call VERB iajs-700 192 13 quasi quasi ADJ iajs-700 192 14 - - ADJ iajs-700 192 15 invertible invertible ADJ iajs-700 192 16 if if SCONJ iajs-700 192 17 hom hom NOUN iajs-700 192 18 ( ( PUNCT iajs-700 192 19 m m VERB iajs-700 192 20 n n PRON iajs-700 192 21 , , PUNCT iajs-700 192 22 m m NOUN iajs-700 192 23 ) ) PUNCT iajs-700 193 1 = = SYM iajs-700 193 2 0 0 X iajs-700 193 3 . . PUNCT iajs-700 194 1 and and CCONJ iajs-700 194 2 an an DET iajs-700 194 3 r r NOUN iajs-700 194 4 - - PUNCT iajs-700 194 5 module module NOUN iajs-700 194 6 m m NOUN iajs-700 194 7 is be AUX iajs-700 194 8 called call VERB iajs-700 194 9 quasi quasi ADJ iajs-700 194 10 - - NOUN iajs-700 194 11 dedekind dedekind ADJ iajs-700 194 12 if if SCONJ iajs-700 194 13 every every DET iajs-700 194 14 non non ADJ iajs-700 194 15 - - ADJ iajs-700 194 16 zero zero ADJ iajs-700 194 17 r r NOUN iajs-700 194 18 - - PUNCT iajs-700 194 19 submodule submodule NOUN iajs-700 194 20 of of ADP iajs-700 194 21 m m PROPN iajs-700 194 22 is be AUX iajs-700 194 23 quasiinvertible(18 quasiinvertible(18 NOUN iajs-700 194 24 ) ) PUNCT iajs-700 194 25 . . PUNCT iajs-700 195 1 3.6 3.6 NUM iajs-700 195 2 proposition proposition NOUN iajs-700 195 3 : : PUNCT iajs-700 195 4 let let VERB iajs-700 195 5 x x PRON iajs-700 195 6 be be AUX iajs-700 195 7 a a DET iajs-700 195 8 uniform uniform ADJ iajs-700 195 9 and and CCONJ iajs-700 195 10 semiprime semiprime NOUN iajs-700 195 11 fuzzy fuzzy ADJ iajs-700 195 12 module module NOUN iajs-700 195 13 . . PUNCT iajs-700 196 1 then then ADV iajs-700 196 2 xt xt PROPN iajs-700 196 3 is be AUX iajs-700 196 4 a a DET iajs-700 196 5 quasi quasi ADJ iajs-700 196 6 - - ADJ iajs-700 196 7 dedekind dedekind ADJ iajs-700 196 8 module module NOUN iajs-700 196 9 , , PUNCT iajs-700 196 10   PROPN iajs-700 196 11 t t PROPN iajs-700 196 12   PROPN iajs-700 196 13 ( ( PUNCT iajs-700 196 14 0,1 0,1 NOUN iajs-700 196 15 ] ] PUNCT iajs-700 196 16 . . PUNCT iajs-700 197 1 proof proof NOUN iajs-700 197 2 : : PUNCT iajs-700 197 3 x x X iajs-700 197 4 is be AUX iajs-700 197 5 uniform uniform ADJ iajs-700 197 6 , , PUNCT iajs-700 197 7 implies imply VERB iajs-700 197 8 xt xt PROPN iajs-700 197 9 is be AUX iajs-700 197 10 uniform uniform ADJ iajs-700 197 11   PROPN iajs-700 197 12 t t PROPN iajs-700 197 13   PROPN iajs-700 197 14 ( ( PUNCT iajs-700 197 15 0,1 0,1 NUM iajs-700 197 16 ] ] PUNCT iajs-700 197 17 and and CCONJ iajs-700 197 18 x x X iajs-700 197 19 is be AUX iajs-700 197 20 semiprime semiprime NOUN iajs-700 197 21 , , PUNCT iajs-700 197 22 then then ADV iajs-700 197 23 xt xt PROPN iajs-700 197 24 is be AUX iajs-700 197 25 semiprime semiprime NOUN iajs-700 197 26 by by ADP iajs-700 197 27 prop prop NOUN iajs-700 197 28 . . PUNCT iajs-700 198 1 2.5 2.5 NUM iajs-700 198 2 . . PUNCT iajs-700 199 1 thus thus ADV iajs-700 199 2 xt xt PROPN iajs-700 199 3 is be AUX iajs-700 199 4 a a DET iajs-700 199 5 quasi quasi ADJ iajs-700 199 6 - - ADJ iajs-700 199 7 dedekind dedekind ADJ iajs-700 199 8 module module NOUN iajs-700 199 9   PROPN iajs-700 199 10 t t PROPN iajs-700 199 11   PROPN iajs-700 199 12 ( ( PUNCT iajs-700 199 13 0,1 0,1 NUM iajs-700 199 14 ] ] PUNCT iajs-700 199 15 by by ADP iajs-700 199 16 [ [ X iajs-700 199 17 9,prop.2.4 9,prop.2.4 NUM iajs-700 199 18 ] ] PUNCT iajs-700 199 19 . . PUNCT iajs-700 200 1 مجلة مجلة PROPN iajs-700 200 2 إبن إبن VERB iajs-700 200 3 الھیثم الھیثم NOUN iajs-700 200 4 للعلوم للعلوم NOUN iajs-700 200 5 الصرفة الصرفة NOUN iajs-700 200 6 و و PRON iajs-700 200 7 التطبیقیة التطبیقیة PROPN iajs-700 200 8 2012 2012 NUM iajs-700 200 9 السنة السنة NOUN iajs-700 201 1 25 25 NUM iajs-700 201 2 المجلد المجلد NOUN iajs-700 201 3 1 1 NUM iajs-700 201 4 العدد العدد PROPN iajs-700 201 5 ibn ibn PROPN iajs-700 201 6 al al PROPN iajs-700 201 7 - - PUNCT iajs-700 201 8 haitham haitham PROPN iajs-700 201 9 journal journal PROPN iajs-700 201 10 for for ADP iajs-700 201 11 pure pure ADJ iajs-700 201 12 and and CCONJ iajs-700 201 13 applied apply VERB iajs-700 201 14 science science NOUN iajs-700 201 15 no no NOUN iajs-700 201 16 . . NOUN iajs-700 201 17 1 1 NUM iajs-700 201 18 vol vol NOUN iajs-700 201 19 . . PUNCT iajs-700 202 1 25 25 NUM iajs-700 202 2 year year NOUN iajs-700 202 3 2012 2012 NUM iajs-700 202 4 3.7 3.7 NUM iajs-700 202 5 proposition proposition NOUN iajs-700 202 6 : : PUNCT iajs-700 202 7 let let VERB iajs-700 202 8 x x PRON iajs-700 202 9 be be AUX iajs-700 202 10 a a DET iajs-700 202 11 fuzzy fuzzy ADJ iajs-700 202 12 module module NOUN iajs-700 202 13 of of ADP iajs-700 202 14 an an DET iajs-700 202 15 r r NOUN iajs-700 202 16 - - PUNCT iajs-700 202 17 module module NOUN iajs-700 202 18 m m NOUN iajs-700 202 19 such such ADJ iajs-700 202 20 that that SCONJ iajs-700 202 21 every every DET iajs-700 202 22 fuzzy fuzzy ADJ iajs-700 202 23 submodule submodule NOUN iajs-700 202 24 of of ADP iajs-700 202 25 x x PUNCT iajs-700 202 26 is be AUX iajs-700 202 27 divisible divisible ADJ iajs-700 202 28 , , PUNCT iajs-700 202 29 then then ADV iajs-700 202 30 x x PUNCT iajs-700 202 31 is be AUX iajs-700 202 32 semiprime semiprime NOUN iajs-700 202 33 . . PUNCT iajs-700 203 1 proof proof NOUN iajs-700 203 2 : : PUNCT iajs-700 203 3 let let VERB iajs-700 203 4 o1 o1 PROPN iajs-700 203 5   PROPN iajs-700 203 6 xt xt PROPN iajs-700 203 7   PROPN iajs-700 203 8 x. x. NOUN iajs-700 204 1 it it PRON iajs-700 204 2 is be AUX iajs-700 204 3 enough enough ADJ iajs-700 204 4 to to PART iajs-700 204 5 show show VERB iajs-700 204 6 that that SCONJ iajs-700 204 7 f f PROPN iajs-700 204 8 - - PUNCT iajs-700 204 9 ann ann PROPN iajs-700 204 10 < < X iajs-700 204 11 xt xt X iajs-700 204 12 > > X iajs-700 204 13 is be AUX iajs-700 204 14 a a DET iajs-700 204 15 semiprime semiprime NOUN iajs-700 204 16 fuzzy fuzzy ADJ iajs-700 204 17 ideal ideal NOUN iajs-700 204 18 by by ADP iajs-700 204 19 prop.(2.9)(4 prop.(2.9)(4 NOUN iajs-700 204 20 ) ) PUNCT iajs-700 204 21 . . PUNCT iajs-700 205 1 let let VERB iajs-700 205 2 2 2 NUM iajs-700 205 3 kr kr PROPN iajs-700 205 4   PROPN iajs-700 205 5 f f PROPN iajs-700 205 6 - - PUNCT iajs-700 205 7 ann ann PROPN iajs-700 205 8 < < X iajs-700 205 9 xt xt X iajs-700 205 10 > > NOUN iajs-700 205 11 , , PUNCT iajs-700 205 12 hence hence ADV iajs-700 205 13 2 2 NUM iajs-700 205 14 kr kr PROPN iajs-700 205 15 xt xt PROPN iajs-700 205 16 = = SYM iajs-700 205 17 o1 o1 PROPN iajs-700 205 18 . . PUNCT iajs-700 206 1 but but CCONJ iajs-700 206 2 < < X iajs-700 206 3 xt xt X iajs-700 206 4 > > X iajs-700 206 5 is be AUX iajs-700 206 6 a a DET iajs-700 206 7 divisible divisible ADJ iajs-700 206 8 fuzzy fuzzy ADJ iajs-700 206 9 submodule submodule NOUN iajs-700 206 10 . . PUNCT iajs-700 207 1 then then ADV iajs-700 207 2 < < X iajs-700 207 3 xt xt X iajs-700 207 4 > > X iajs-700 207 5 = = PUNCT iajs-700 207 6 rk rk X iajs-700 207 7 < < X iajs-700 207 8 xt xt X iajs-700 207 9 > > PRON iajs-700 207 10 ; ; PUNCT iajs-700 207 11 rk rk PROPN iajs-700 207 12 is be AUX iajs-700 207 13 a a DET iajs-700 207 14 fuzzy fuzzy ADJ iajs-700 207 15 singleton singleton NOUN iajs-700 207 16 of of ADP iajs-700 207 17 r. r. PROPN iajs-700 207 18 hence hence ADV iajs-700 207 19 xt xt PUNCT iajs-700 208 1 = = PUNCT iajs-700 208 2 rkcℓxt rkcℓxt PROPN iajs-700 208 3 , , PUNCT iajs-700 208 4 cℓ cℓ ADP iajs-700 208 5   PROPN iajs-700 208 6 x. x. PUNCT iajs-700 209 1 so so ADV iajs-700 209 2 rkxt rkxt NOUN iajs-700 209 3 = = NOUN iajs-700 209 4 rkrkcℓxt rkrkcℓxt NOUN iajs-700 209 5 = = SYM iajs-700 209 6 2 2 NUM iajs-700 209 7 kr kr PROPN iajs-700 209 8 cℓxt cℓxt NOUN iajs-700 209 9 , , PUNCT iajs-700 209 10 but but CCONJ iajs-700 209 11 2 2 NUM iajs-700 209 12 kr kr PROPN iajs-700 209 13 cℓxt cℓxt NOUN iajs-700 209 14 = = PUNCT iajs-700 209 15 cℓ cℓ ADP iajs-700 209 16 2 2 NUM iajs-700 209 17 kr kr PROPN iajs-700 209 18 xt xt PROPN iajs-700 209 19 = = NOUN iajs-700 209 20 cℓo1 cℓo1 PROPN iajs-700 209 21 = = SYM iajs-700 209 22 o1 o1 PROPN iajs-700 209 23 . . PUNCT iajs-700 210 1 thus thus ADV iajs-700 210 2 rkxt rkxt VERB iajs-700 210 3 = = SYM iajs-700 210 4 o1 o1 NOUN iajs-700 210 5 . . PUNCT iajs-700 211 1 so so ADV iajs-700 211 2 rk rk VERB iajs-700 211 3   PROPN iajs-700 211 4 f f PROPN iajs-700 211 5 - - PUNCT iajs-700 211 6 ann ann PROPN iajs-700 211 7 < < X iajs-700 211 8 xt xt X iajs-700 211 9 > > X iajs-700 211 10 . . PUNCT iajs-700 212 1 thus thus ADV iajs-700 212 2 f f X iajs-700 212 3 - - PUNCT iajs-700 212 4 ann ann PROPN iajs-700 212 5 < < X iajs-700 212 6 xt xt X iajs-700 212 7 > > X iajs-700 212 8 is be AUX iajs-700 212 9 a a DET iajs-700 212 10 semiprime semiprime NOUN iajs-700 212 11 fuzzy fuzzy ADJ iajs-700 212 12 ideal ideal NOUN iajs-700 212 13 . . PUNCT iajs-700 213 1 thus thus ADV iajs-700 213 2 x x X iajs-700 213 3 is be AUX iajs-700 213 4 semiprime semiprime ADJ iajs-700 213 5 fuzzy fuzzy ADJ iajs-700 213 6 module module NOUN iajs-700 213 7 . . PUNCT iajs-700 214 1 3.8 3.8 NUM iajs-700 214 2 corollary corollary NOUN iajs-700 214 3 : : PUNCT iajs-700 214 4 let let VERB iajs-700 214 5 x x PRON iajs-700 214 6 be be AUX iajs-700 214 7 a a DET iajs-700 214 8 fuzzy fuzzy ADJ iajs-700 214 9 module module NOUN iajs-700 214 10 and and CCONJ iajs-700 214 11 every every DET iajs-700 214 12 fuzzy fuzzy ADJ iajs-700 214 13 submodule submodule NOUN iajs-700 214 14 of of ADP iajs-700 214 15 x x PUNCT iajs-700 214 16 is be AUX iajs-700 214 17 divisible divisible ADJ iajs-700 214 18 . . PUNCT iajs-700 215 1 then then ADV iajs-700 215 2 x x X iajs-700 215 3 is be AUX iajs-700 215 4 a a DET iajs-700 215 5 prime prime ADJ iajs-700 215 6 fuzzy fuzzy ADJ iajs-700 215 7 module module NOUN iajs-700 215 8 . . PUNCT iajs-700 216 1 proof proof NOUN iajs-700 216 2 : : PUNCT iajs-700 216 3 by by ADP iajs-700 216 4 prop.3.8 prop.3.8 PROPN iajs-700 216 5 , , PUNCT iajs-700 216 6 x x X iajs-700 216 7 is be AUX iajs-700 216 8 semiprime semiprime NOUN iajs-700 216 9 . . PUNCT iajs-700 217 1 but but CCONJ iajs-700 217 2 x x PRON iajs-700 217 3 is be AUX iajs-700 217 4 divisible divisible ADJ iajs-700 217 5 , , PUNCT iajs-700 217 6 then then ADV iajs-700 217 7 x x PUNCT iajs-700 217 8 is be AUX iajs-700 217 9 prime prime ADJ iajs-700 217 10 by by ADP iajs-700 217 11 [ [ X iajs-700 217 12 17,prop.3.1.13 17,prop.3.1.13 NUM iajs-700 217 13 ] ] X iajs-700 217 14 . . PUNCT iajs-700 218 1 3.9 3.9 NUM iajs-700 218 2 proposition proposition NOUN iajs-700 218 3 : : PUNCT iajs-700 218 4 let let VERB iajs-700 218 5 x x PRON iajs-700 218 6 be be AUX iajs-700 218 7 an an DET iajs-700 218 8 f f NOUN iajs-700 218 9 - - PUNCT iajs-700 218 10 regular regular ADJ iajs-700 218 11 fuzzy fuzzy ADJ iajs-700 218 12 module module NOUN iajs-700 218 13 of of ADP iajs-700 218 14 an an DET iajs-700 218 15 r r NOUN iajs-700 218 16 - - PUNCT iajs-700 218 17 module module NOUN iajs-700 218 18 m m NOUN iajs-700 218 19 , , PUNCT iajs-700 218 20 where where SCONJ iajs-700 218 21 r r NOUN iajs-700 218 22 is be AUX iajs-700 218 23 a a DET iajs-700 218 24 principle principle ADJ iajs-700 218 25 ideal ideal ADJ iajs-700 218 26 domain domain NOUN iajs-700 218 27 . . PUNCT iajs-700 219 1 then then ADV iajs-700 219 2 x x X iajs-700 219 3 is be AUX iajs-700 219 4 a a DET iajs-700 219 5 semiprime semiprime NOUN iajs-700 219 6 fuzzy fuzzy ADJ iajs-700 219 7 module module NOUN iajs-700 219 8 . . PUNCT iajs-700 220 1 proof proof NOUN iajs-700 220 2 : : PUNCT iajs-700 220 3 since since SCONJ iajs-700 220 4 x x PRON iajs-700 220 5 is be AUX iajs-700 220 6 f f NOUN iajs-700 220 7 - - PUNCT iajs-700 220 8 regular regular ADJ iajs-700 220 9 , , PUNCT iajs-700 220 10 then then ADV iajs-700 220 11 xt xt PROPN iajs-700 220 12 is be AUX iajs-700 220 13 f f ADJ iajs-700 220 14 - - PUNCT iajs-700 220 15 regular regular ADJ iajs-700 220 16   PROPN iajs-700 220 17 t t PROPN iajs-700 220 18   PROPN iajs-700 220 19 ( ( PUNCT iajs-700 220 20 0,1 0,1 NUM iajs-700 220 21 ] ] PUNCT iajs-700 220 22 by by ADP iajs-700 220 23 [ [ X iajs-700 220 24 11 11 NUM iajs-700 220 25 ] ] PUNCT iajs-700 220 26 . . PUNCT iajs-700 221 1 hence hence ADV iajs-700 221 2 xt xt PROPN iajs-700 221 3 is be AUX iajs-700 221 4 a a DET iajs-700 221 5 semiprime semiprime NOUN iajs-700 221 6 module module NOUN iajs-700 221 7   PROPN iajs-700 221 8 t t PROPN iajs-700 221 9   PROPN iajs-700 221 10 ( ( PUNCT iajs-700 221 11 0,1 0,1 NUM iajs-700 221 12 ] ] PUNCT iajs-700 221 13 by by ADP iajs-700 221 14 [ [ X iajs-700 221 15 9,prop.4.2.6 9,prop.4.2.6 X iajs-700 221 16 ] ] PUNCT iajs-700 221 17 . . PUNCT iajs-700 222 1 thus thus ADV iajs-700 222 2 x x X iajs-700 222 3 is be AUX iajs-700 222 4 a a DET iajs-700 222 5 semiprime semiprime NOUN iajs-700 222 6 fuzzy fuzzy ADJ iajs-700 222 7 module module NOUN iajs-700 222 8 . . PUNCT iajs-700 223 1 references reference NOUN iajs-700 223 2 1 1 NUM iajs-700 223 3 . . PUNCT iajs-700 224 1 zadeh zadeh PROPN iajs-700 224 2 , , PUNCT iajs-700 224 3 l.a l.a PROPN iajs-700 224 4 . . PROPN iajs-700 224 5 , , PUNCT iajs-700 224 6 ( ( PUNCT iajs-700 224 7 1965 1965 NUM iajs-700 224 8 ) ) PUNCT iajs-700 224 9 , , PUNCT iajs-700 224 10 fuzzy fuzzy ADJ iajs-700 224 11 sets set NOUN iajs-700 224 12 , , PUNCT iajs-700 224 13 information information NOUN iajs-700 224 14 and and CCONJ iajs-700 224 15 control control NOUN iajs-700 224 16 , , PUNCT iajs-700 224 17 8 8 NUM iajs-700 224 18 : : SYM iajs-700 224 19 338 338 NUM iajs-700 224 20 - - SYM iajs-700 224 21 353 353 NUM iajs-700 224 22 . . NOUN iajs-700 224 23 2 2 NUM iajs-700 224 24 . . X iajs-700 224 25 negoita negoita PROPN iajs-700 224 26 , , PUNCT iajs-700 224 27 c.v c.v PROPN iajs-700 224 28 . . PROPN iajs-700 224 29 and and CCONJ iajs-700 224 30 ralescu ralescu NOUN iajs-700 224 31 , , PUNCT iajs-700 224 32 d.a d.a PROPN iajs-700 224 33 . . PROPN iajs-700 224 34 , , PUNCT iajs-700 224 35 ( ( PUNCT iajs-700 224 36 1975 1975 NUM iajs-700 224 37 ) ) PUNCT iajs-700 224 38 , , PUNCT iajs-700 224 39 applications application NOUN iajs-700 224 40 of of ADP iajs-700 224 41 fuzzy fuzzy ADJ iajs-700 224 42 sets set NOUN iajs-700 224 43 and and CCONJ iajs-700 224 44 system system NOUN iajs-700 224 45 analysis analysis NOUN iajs-700 224 46 ( ( PUNCT iajs-700 224 47 birkhous birkhous ADJ iajs-700 224 48 , , PUNCT iajs-700 224 49 basel basel PROPN iajs-700 224 50 ) ) PUNCT iajs-700 224 51 . . PUNCT iajs-700 225 1 3 3 X iajs-700 225 2 . . X iajs-700 225 3 maschinchi maschinchi PROPN iajs-700 225 4 , , PUNCT iajs-700 225 5 m. m. NOUN iajs-700 225 6 and and CCONJ iajs-700 225 7 zahedi zahedi PROPN iajs-700 225 8 , , PUNCT iajs-700 225 9 m.m m.m PROPN iajs-700 225 10 . . PROPN iajs-700 225 11 , , PUNCT iajs-700 225 12 ( ( PUNCT iajs-700 225 13 1992 1992 NUM iajs-700 225 14 ) ) PUNCT iajs-700 225 15 , , PUNCT iajs-700 225 16 on on ADP iajs-700 225 17 l l ADJ iajs-700 225 18 - - ADJ iajs-700 225 19 fuzzy fuzzy ADJ iajs-700 225 20 primary primary ADJ iajs-700 225 21 submodules submodule NOUN iajs-700 225 22 , , PUNCT iajs-700 225 23 fuzzy fuzzy ADJ iajs-700 225 24 sets set NOUN iajs-700 225 25 and and CCONJ iajs-700 225 26 systems system NOUN iajs-700 225 27 , , PUNCT iajs-700 225 28 49:231 49:231 PROPN iajs-700 225 29 - - SYM iajs-700 225 30 236 236 NUM iajs-700 225 31 . . NOUN iajs-700 225 32 4 4 NUM iajs-700 225 33 . . X iajs-700 226 1 liu liu PROPN iajs-700 226 2 , , PUNCT iajs-700 226 3 w.j w.j PROPN iajs-700 226 4 . . PROPN iajs-700 226 5 ( ( PUNCT iajs-700 226 6 1982 1982 NUM iajs-700 226 7 ) ) PUNCT iajs-700 226 8 , , PUNCT iajs-700 226 9 fuzzy fuzzy ADJ iajs-700 226 10 invariant invariant ADJ iajs-700 226 11 subgroups subgroup NOUN iajs-700 226 12 and and CCONJ iajs-700 226 13 fuzzy fuzzy ADJ iajs-700 226 14 ideals ideal NOUN iajs-700 226 15 , , PUNCT iajs-700 226 16 fuzzy fuzzy ADJ iajs-700 226 17 sets set NOUN iajs-700 226 18 and and CCONJ iajs-700 226 19 systems system NOUN iajs-700 226 20 , , PUNCT iajs-700 226 21 8:133 8:133 NUM iajs-700 226 22 - - SYM iajs-700 226 23 139 139 NUM iajs-700 226 24 . . PUNCT iajs-700 226 25 5 5 NUM iajs-700 226 26 . . PUNCT iajs-700 226 27 dauns daun NOUN iajs-700 226 28 , , PUNCT iajs-700 226 29 ( ( PUNCT iajs-700 226 30 1980 1980 NUM iajs-700 226 31 ) ) PUNCT iajs-700 226 32 , , PUNCT iajs-700 226 33 prime prime ADJ iajs-700 226 34 modules module NOUN iajs-700 226 35 and and CCONJ iajs-700 226 36 one one NUM iajs-700 226 37 - - PUNCT iajs-700 226 38 sided sided ADJ iajs-700 226 39 ideals ideal NOUN iajs-700 226 40 in in ADP iajs-700 226 41 " " PUNCT iajs-700 226 42 ring ring NOUN iajs-700 226 43 theory theory NOUN iajs-700 226 44 and and CCONJ iajs-700 226 45 algebra algebra NOUN iajs-700 226 46 iii iii PROPN iajs-700 226 47 " " PUNCT iajs-700 226 48 ( ( PUNCT iajs-700 226 49 proceeding proceeding NOUN iajs-700 226 50 of of ADP iajs-700 226 51 the the DET iajs-700 226 52 trird trird PROPN iajs-700 226 53 oklahama oklahama PROPN iajs-700 226 54 conference conference NOUN iajs-700 226 55 ) ) PUNCT iajs-700 226 56 , , PUNCT iajs-700 226 57 b.r b.r PROPN iajs-700 226 58 . . PROPN iajs-700 226 59 mcdonald mcdonald PROPN iajs-700 226 60 , , PUNCT iajs-700 226 61 newyork newyork PROPN iajs-700 226 62 , , PUNCT iajs-700 226 63 .301 .301 NUM iajs-700 226 64 - - SYM iajs-700 226 65 314 314 NUM iajs-700 226 66 . . PUNCT iajs-700 227 1 6 6 NUM iajs-700 227 2 . . NOUN iajs-700 227 3 athab athab PROPN iajs-700 227 4 , , PUNCT iajs-700 227 5 e.a e.a PROPN iajs-700 227 6 . . PROPN iajs-700 227 7 ( ( PUNCT iajs-700 227 8 1996 1996 NUM iajs-700 227 9 ) ) PUNCT iajs-700 227 10 , , PUNCT iajs-700 227 11 prime prime ADJ iajs-700 227 12 submodule submodule NOUN iajs-700 227 13 and and CCONJ iajs-700 227 14 semiprime semiprime NOUN iajs-700 227 15 submodules submodule NOUN iajs-700 227 16 , , PUNCT iajs-700 227 17 m.sc m.sc PROPN iajs-700 227 18 thesis thesis NOUN iajs-700 227 19 , , PUNCT iajs-700 227 20 college college NOUN iajs-700 227 21 of of ADP iajs-700 227 22 science science NOUN iajs-700 227 23 , , PUNCT iajs-700 227 24 univ univ PROPN iajs-700 227 25 . . PROPN iajs-700 227 26 of of ADP iajs-700 227 27 baghdad baghdad PROPN iajs-700 227 28 . . PUNCT iajs-700 228 1 7 7 X iajs-700 228 2 . . X iajs-700 228 3 hadi hadi PROPN iajs-700 228 4 , , PUNCT iajs-700 228 5 i.m.a i.m.a NOUN iajs-700 228 6 . . PUNCT iajs-700 228 7 ( ( PUNCT iajs-700 228 8 2000 2000 NUM iajs-700 228 9 ) ) PUNCT iajs-700 228 10 , , PUNCT iajs-700 228 11 semiprime semiprime NOUN iajs-700 228 12 fuzzy fuzzy ADJ iajs-700 228 13 ideals ideal NOUN iajs-700 228 14 of of ADP iajs-700 228 15 a a DET iajs-700 228 16 ring ring NOUN iajs-700 228 17 , , PUNCT iajs-700 228 18 journal journal NOUN iajs-700 228 19 published publish VERB iajs-700 228 20 by by ADP iajs-700 228 21 iraqi iraqi ADJ iajs-700 228 22 society society NOUN iajs-700 228 23 of of ADP iajs-700 228 24 phy phy NOUN iajs-700 228 25 . . PUNCT iajs-700 228 26 and and CCONJ iajs-700 228 27 m m PROPN iajs-700 228 28 ath ath PROPN iajs-700 228 29 . . PROPN iajs-700 228 30 , , PUNCT iajs-700 228 31 15(2 15(2 NUM iajs-700 228 32 ) ) PUNCT iajs-700 228 33 . . PUNCT iajs-700 229 1 8 8 X iajs-700 229 2 . . X iajs-700 230 1 hadi hadi PROPN iajs-700 230 2 , , PUNCT iajs-700 230 3 i.m.a i.m.a NOUN iajs-700 230 4 . . PROPN iajs-700 230 5 , , PUNCT iajs-700 230 6 ( ( PUNCT iajs-700 230 7 2004 2004 NUM iajs-700 230 8 ) ) PUNCT iajs-700 230 9 , , PUNCT iajs-700 230 10 semiprime semiprime NOUN iajs-700 230 11 fuzzy fuzzy ADJ iajs-700 230 12 submodules submodule NOUN iajs-700 230 13 of of ADP iajs-700 230 14 fuzzy fuzzy ADJ iajs-700 230 15 modules module NOUN iajs-700 230 16 , , PUNCT iajs-700 230 17 ibn ibn PROPN iajs-700 230 18 - - PUNCT iajs-700 230 19 al al PROPN iajs-700 230 20 - - PUNCT iajs-700 230 21 haitham haitham PROPN iajs-700 230 22 j. j. PROPN iajs-700 230 23 for for ADP iajs-700 230 24 pure pure ADJ iajs-700 230 25 and and CCONJ iajs-700 230 26 appl appl NOUN iajs-700 230 27 . . PUNCT iajs-700 231 1 sci sci PROPN iajs-700 231 2 . . PROPN iajs-700 231 3 , , PUNCT iajs-700 231 4 17(3 17(3 NUM iajs-700 231 5 ): ): PUNCT iajs-700 231 6 112 112 NUM iajs-700 231 7 – – PUNCT iajs-700 231 8 123 123 NUM iajs-700 231 9 . . PUNCT iajs-700 232 1 9 9 NUM iajs-700 232 2 . . X iajs-700 232 3 al al PROPN iajs-700 232 4 - - PUNCT iajs-700 232 5 sharid sharid PROPN iajs-700 232 6 , , PUNCT iajs-700 232 7 f.a f.a PROPN iajs-700 232 8 . . PROPN iajs-700 232 9 ( ( PUNCT iajs-700 232 10 2008 2008 NUM iajs-700 232 11 ) ) PUNCT iajs-700 232 12 , , PUNCT iajs-700 232 13 s s X iajs-700 232 14 - - PUNCT iajs-700 232 15 compactly compactly ADV iajs-700 232 16 packed pack VERB iajs-700 232 17 submodules submodule NOUN iajs-700 232 18 and and CCONJ iajs-700 232 19 semi semi ADJ iajs-700 232 20 - - ADJ iajs-700 232 21 prime prime ADJ iajs-700 232 22 modules module NOUN iajs-700 232 23 , , PUNCT iajs-700 232 24 m.sc m.sc PROPN iajs-700 232 25 . . PUNCT iajs-700 233 1 thesis thesis NOUN iajs-700 233 2 , , PUNCT iajs-700 233 3 tikrit tikrit NOUN iajs-700 233 4 university university NOUN iajs-700 233 5 . . PUNCT iajs-700 234 1 10 10 NUM iajs-700 234 2 . . X iajs-700 235 1 zahedi zahedi PROPN iajs-700 235 2 , , PUNCT iajs-700 235 3 m.m m.m PROPN iajs-700 235 4 . . PROPN iajs-700 235 5 ( ( PUNCT iajs-700 235 6 1992 1992 NUM iajs-700 235 7 ) ) PUNCT iajs-700 235 8 , , PUNCT iajs-700 235 9 on on ADP iajs-700 235 10 l l ADJ iajs-700 235 11 - - ADJ iajs-700 235 12 fuzzy fuzzy ADJ iajs-700 235 13 residual residual ADJ iajs-700 235 14 quotient quotient NOUN iajs-700 235 15 modules module NOUN iajs-700 235 16 and and CCONJ iajs-700 235 17 p.primary p.primary ADJ iajs-700 235 18 submodules submodule NOUN iajs-700 235 19 , , PUNCT iajs-700 235 20 fuzzy fuzzy ADJ iajs-700 235 21 sets set NOUN iajs-700 235 22 and and CCONJ iajs-700 235 23 systems system NOUN iajs-700 235 24 , , PUNCT iajs-700 235 25 51:33 51:33 NUM iajs-700 235 26 - - SYM iajs-700 235 27 344 344 NUM iajs-700 235 28 . . PUNCT iajs-700 235 29 11 11 NUM iajs-700 235 30 . . PUNCT iajs-700 236 1 hamil hamil PROPN iajs-700 236 2 , , PUNCT iajs-700 236 3 m.a m.a PROPN iajs-700 236 4 . . PROPN iajs-700 236 5 ( ( PUNCT iajs-700 236 6 2002 2002 NUM iajs-700 236 7 ) ) PUNCT iajs-700 236 8 , , PUNCT iajs-700 236 9 f f X iajs-700 236 10 - - PUNCT iajs-700 236 11 regular regular ADJ iajs-700 236 12 fuzzy fuzzy ADJ iajs-700 236 13 modules module NOUN iajs-700 236 14 , , PUNCT iajs-700 236 15 m m PROPN iajs-700 236 16 .sc .sc NOUN iajs-700 236 17 . . PUNCT iajs-700 237 1 thesis thesis NOUN iajs-700 237 2 , , PUNCT iajs-700 237 3 univ univ PROPN iajs-700 237 4 . . PROPN iajs-700 237 5 of of ADP iajs-700 237 6 baghdad baghdad PROPN iajs-700 237 7 . . PUNCT iajs-700 238 1 12 12 NUM iajs-700 238 2 . . PUNCT iajs-700 239 1 zahedi zahedi PROPN iajs-700 239 2 , , PUNCT iajs-700 239 3 m.m m.m PROPN iajs-700 239 4 . . PROPN iajs-700 239 5 ( ( PUNCT iajs-700 239 6 1991 1991 NUM iajs-700 239 7 ) ) PUNCT iajs-700 239 8 , , PUNCT iajs-700 239 9 characterization characterization NOUN iajs-700 239 10 of of ADP iajs-700 239 11 l l ADJ iajs-700 239 12 - - ADJ iajs-700 239 13 fuzzy fuzzy ADJ iajs-700 239 14 prime prime ADJ iajs-700 239 15 ideals ideal NOUN iajs-700 239 16 , , PUNCT iajs-700 239 17 fuzzy fuzzy ADJ iajs-700 239 18 sets set NOUN iajs-700 239 19 and and CCONJ iajs-700 239 20 fuzzy fuzzy ADJ iajs-700 239 21 system system NOUN iajs-700 239 22 , , PUNCT iajs-700 239 23 44:147 44:147 PROPN iajs-700 239 24 - - PUNCT iajs-700 239 25 160 160 PROPN iajs-700 239 26 . . PUNCT iajs-700 240 1 مجلة مجلة NOUN iajs-700 240 2 إبن إبن VERB iajs-700 240 3 الھیثم الھیثم NOUN iajs-700 240 4 للعلوم للعلوم NOUN iajs-700 240 5 الصرفة الصرفة NOUN iajs-700 240 6 و و PRON iajs-700 240 7 التطبیقیة التطبیقیة PROPN iajs-700 240 8 2012 2012 NUM iajs-700 240 9 السنة السنة NOUN iajs-700 241 1 25 25 NUM iajs-700 241 2 المجلد المجلد NOUN iajs-700 241 3 1 1 NUM iajs-700 241 4 العدد العدد PROPN iajs-700 241 5 ibn ibn PROPN iajs-700 241 6 al al PROPN iajs-700 241 7 - - PUNCT iajs-700 241 8 haitham haitham PROPN iajs-700 241 9 journal journal PROPN iajs-700 241 10 for for ADP iajs-700 241 11 pure pure ADJ iajs-700 241 12 and and CCONJ iajs-700 241 13 applied apply VERB iajs-700 241 14 science science NOUN iajs-700 241 15 no no NOUN iajs-700 241 16 . . NOUN iajs-700 241 17 1 1 NUM iajs-700 241 18 vol vol NOUN iajs-700 241 19 . . PUNCT iajs-700 242 1 25 25 NUM iajs-700 242 2 year year NOUN iajs-700 242 3 2012 2012 NUM iajs-700 242 4 13 13 NUM iajs-700 242 5 . . PUNCT iajs-700 243 1 martinex martinex PROPN iajs-700 243 2 , , PUNCT iajs-700 243 3 l. l. PROPN iajs-700 243 4 ( ( PUNCT iajs-700 243 5 1996 1996 NUM iajs-700 243 6 ) ) PUNCT iajs-700 243 7 , , PUNCT iajs-700 243 8 fuzzy fuzzy ADJ iajs-700 243 9 modules module NOUN iajs-700 243 10 over over ADP iajs-700 243 11 fuzzy fuzzy ADJ iajs-700 243 12 rings ring NOUN iajs-700 243 13 in in ADP iajs-700 243 14 connection connection NOUN iajs-700 243 15 with with ADP iajs-700 243 16 fuzzy fuzzy ADJ iajs-700 243 17 ideal ideal NOUN iajs-700 243 18 of of ADP iajs-700 243 19 ring ring NOUN iajs-700 243 20 , , PUNCT iajs-700 243 21 j. j. PROPN iajs-700 243 22 fuzzy fuzzy PROPN iajs-700 243 23 math math PROPN iajs-700 243 24 . . PUNCT iajs-700 243 25 , , PUNCT iajs-700 243 26 4 4 NUM iajs-700 243 27 : : SYM iajs-700 243 28 843 843 NUM iajs-700 243 29 - - NOUN iajs-700 243 30 857 857 NUM iajs-700 243 31 . . PUNCT iajs-700 244 1 14 14 NUM iajs-700 244 2 . . PUNCT iajs-700 245 1 mukhegee mukhegee NOUN iajs-700 245 2 , , PUNCT iajs-700 245 3 t.k t.k PROPN iajs-700 245 4 . . PROPN iajs-700 245 5 ; ; PUNCT iajs-700 245 6 sen sen PROPN iajs-700 245 7 , , PUNCT iajs-700 245 8 m m PROPN iajs-700 245 9 , , PUNCT iajs-700 245 10 k k NOUN iajs-700 245 11 , , PUNCT iajs-700 245 12 and and CCONJ iajs-700 245 13 roy roy PROPN iajs-700 245 14 , , PUNCT iajs-700 245 15 d. d. PROPN iajs-700 245 16 , , PUNCT iajs-700 245 17 ( ( PUNCT iajs-700 245 18 1996 1996 NUM iajs-700 245 19 ) ) PUNCT iajs-700 245 20 , , PUNCT iajs-700 245 21 on on ADP iajs-700 245 22 submodules submodule NOUN iajs-700 245 23 and and CCONJ iajs-700 245 24 their their PRON iajs-700 245 25 radicals radical NOUN iajs-700 245 26 , , PUNCT iajs-700 245 27 j. j. PROPN iajs-700 245 28 fuzzy fuzzy PROPN iajs-700 245 29 math math PROPN iajs-700 245 30 . . PUNCT iajs-700 245 31 , , PUNCT iajs-700 245 32 4 4 NUM iajs-700 245 33 , , PUNCT iajs-700 245 34 pp.549 pp.549 PROPN iajs-700 245 35 - - ADJ iajs-700 245 36 558 558 NUM iajs-700 245 37 . . PUNCT iajs-700 245 38 15 15 NUM iajs-700 245 39 . . PUNCT iajs-700 246 1 kumar kumar PROPN iajs-700 246 2 , , PUNCT iajs-700 246 3 r. r. PROPN iajs-700 246 4 , , PUNCT iajs-700 246 5 ( ( PUNCT iajs-700 246 6 1992 1992 NUM iajs-700 246 7 ) ) PUNCT iajs-700 246 8 , , PUNCT iajs-700 246 9 fuzzy fuzzy ADJ iajs-700 246 10 cosets coset NOUN iajs-700 246 11 and and CCONJ iajs-700 246 12 some some DET iajs-700 246 13 fuzzy fuzzy ADJ iajs-700 246 14 radicals radical NOUN iajs-700 246 15 , , PUNCT iajs-700 246 16 fuzzy fuzzy ADJ iajs-700 246 17 sets set NOUN iajs-700 246 18 and and CCONJ iajs-700 246 19 systems system NOUN iajs-700 246 20 , , PUNCT iajs-700 246 21 46 46 NUM iajs-700 246 22 : : SYM iajs-700 246 23 261 261 NUM iajs-700 246 24 - - SYM iajs-700 246 25 265 265 NUM iajs-700 246 26 . . PUNCT iajs-700 247 1 16 16 NUM iajs-700 247 2 . . PUNCT iajs-700 248 1 majumdar majumdar PROPN iajs-700 248 2 , , PUNCT iajs-700 248 3 s. s. PROPN iajs-700 248 4 ( ( PUNCT iajs-700 248 5 1990 1990 NUM iajs-700 248 6 ) ) PUNCT iajs-700 248 7 , , PUNCT iajs-700 248 8 theory theory NOUN iajs-700 248 9 of of ADP iajs-700 248 10 fuzzy fuzzy ADJ iajs-700 248 11 modules module NOUN iajs-700 248 12 , , PUNCT iajs-700 248 13 eull eull NOUN iajs-700 248 14 . . PUNCT iajs-700 248 15 col col PROPN iajs-700 248 16 . . PROPN iajs-700 248 17 math math PROPN iajs-700 248 18 . . PUNCT iajs-700 249 1 sce sce PROPN iajs-700 249 2 . . PROPN iajs-700 249 3 , , PUNCT iajs-700 249 4 82:395 82:395 PROPN iajs-700 249 5 - - SYM iajs-700 249 6 399 399 NUM iajs-700 249 7 . . PUNCT iajs-700 250 1 17 17 NUM iajs-700 250 2 . . PUNCT iajs-700 251 1 rabi rabi NOUN iajs-700 251 2 , , PUNCT iajs-700 251 3 h.j h.j PROPN iajs-700 251 4 . . PROPN iajs-700 251 5 ( ( PUNCT iajs-700 251 6 2001 2001 NUM iajs-700 251 7 ) ) PUNCT iajs-700 251 8 , , PUNCT iajs-700 251 9 prime prime ADJ iajs-700 251 10 fuzzy fuzzy ADJ iajs-700 251 11 submodules submodule NOUN iajs-700 251 12 and and CCONJ iajs-700 251 13 prime prime ADJ iajs-700 251 14 fuzzy fuzzy ADJ iajs-700 251 15 modules module NOUN iajs-700 251 16 , , PUNCT iajs-700 251 17 m.sc.thesis m.sc.thesis NOUN iajs-700 251 18 , , PUNCT iajs-700 251 19 univ univ PROPN iajs-700 251 20 . . PROPN iajs-700 251 21 of of ADP iajs-700 251 22 baghdad baghdad PROPN iajs-700 251 23 . . PUNCT iajs-700 252 1 18 18 NUM iajs-700 252 2 . . X iajs-700 252 3 ali ali PROPN iajs-700 252 4 , , PUNCT iajs-700 252 5 s.mijbass s.mijbass PROPN iajs-700 252 6 , , PUNCT iajs-700 252 7 ( ( PUNCT iajs-700 252 8 1997 1997 NUM iajs-700 252 9 ) ) PUNCT iajs-700 252 10 , , PUNCT iajs-700 252 11 quasi quasi ADJ iajs-700 252 12 - - ADJ iajs-700 252 13 dedekind dedekind ADJ iajs-700 252 14 modules module NOUN iajs-700 252 15 , , PUNCT iajs-700 252 16 ph.d ph.d PROPN iajs-700 252 17 . . PUNCT iajs-700 253 1 thesis thesis PROPN iajs-700 253 2 , , PUNCT iajs-700 253 3 college college NOUN iajs-700 253 4 of of ADP iajs-700 253 5 science science NOUN iajs-700 253 6 , , PUNCT iajs-700 253 7 univ univ PROPN iajs-700 253 8 . . PROPN iajs-700 253 9 of of ADP iajs-700 253 10 baghdad baghdad PROPN iajs-700 253 11 . . PUNCT iajs-700 254 1 مجلة مجلة NOUN iajs-700 254 2 إبن إبن VERB iajs-700 254 3 الھیثم الھیثم NOUN iajs-700 254 4 للعلوم للعلوم NOUN iajs-700 254 5 الصرفة الصرفة NOUN iajs-700 254 6 و و PRON iajs-700 254 7 التطبیقیة التطبیقیة PROPN iajs-700 254 8 2012 2012 NUM iajs-700 254 9 السنة السنة NOUN iajs-700 255 1 25 25 NUM iajs-700 255 2 المجلد المجلد NOUN iajs-700 255 3 1 1 NUM iajs-700 255 4 العدد العدد PROPN iajs-700 255 5 ibn ibn PROPN iajs-700 255 6 al al PROPN iajs-700 255 7 - - PUNCT iajs-700 255 8 haitham haitham PROPN iajs-700 255 9 journal journal PROPN iajs-700 255 10 for for ADP iajs-700 255 11 pure pure ADJ iajs-700 255 12 and and CCONJ iajs-700 255 13 applied apply VERB iajs-700 255 14 science science NOUN iajs-700 255 15 no no NOUN iajs-700 255 16 . . NOUN iajs-700 255 17 1 1 NUM iajs-700 255 18 vol vol NOUN iajs-700 255 19 . . PUNCT iajs-700 256 1 25 25 NUM iajs-700 256 2 year year NOUN iajs-700 256 3 2012 2012 NUM iajs-700 256 4 ة ة PRON iajs-700 256 5 الضبابیةالمقاسات الضبابیةالمقاسات NOUN iajs-700 256 6 شبه شبه VERB iajs-700 256 7 األولی األولی ADJ iajs-700 256 8 میسون میسون NOUN iajs-700 256 9 عبد عبد VERB iajs-700 256 10 هامل هامل NOUN iajs-700 256 11 جامعة جامعة NOUN iajs-700 256 12 بغداد بغداد PROPN iajs-700 256 13 ، ، PROPN iajs-700 256 14 ابن ابن PROPN iajs-700 256 15 الهیثم الهیثم PROPN iajs-700 256 16 -كلیة -كلیة PROPN iajs-700 256 17 التربیة التربیة NOUN iajs-700 256 18 ، ، NOUN iajs-700 256 19 قسم قسم PROPN iajs-700 256 20 الریاضیات الریاضیات VERB iajs-700 256 21 2011 2011 NUM iajs-700 256 22 ایلول ایلول NOUN iajs-700 256 23 20 20 NUM iajs-700 256 24 : : PUNCT iajs-700 256 25 ، ، PROPN iajs-700 257 1 قبل قبل PROPN iajs-700 257 2 البحث البحث VERB iajs-700 257 3 في في ADP iajs-700 257 4 2011 2011 NUM iajs-700 257 5 حزیران حزیران NOUN iajs-700 257 6 19 19 NUM iajs-700 257 7 : : PUNCT iajs-700 257 8 استلم استلم PROPN iajs-700 257 9 البحث البحث VERB iajs-700 257 10 في في ADP iajs-700 257 11 الخالصة الخالصة PROPN iajs-700 257 12 م م PROPN iajs-700 257 13 اعطیـت اعطیـت PROPN iajs-700 257 14 العدیـد العدیـد PROPN iajs-700 257 15 مــن مــن PROPN iajs-700 257 16 ثــ.ةالولیـ ثــ.ةالولیـ PROPN iajs-700 257 17 للمقاســات للمقاســات PROPN iajs-700 257 18 شـبه شـبه PROPN iajs-700 257 19 ا"اعمامـا ا"اعمامـا NOUN iajs-700 257 20 ة ة DET iajs-700 257 21 الـضبابیةفـي الـضبابیةفـي PROPN iajs-700 257 22 هـذا هـذا NOUN iajs-700 257 23 البحـث البحـث VERB iajs-700 257 24 قــدم قــدم PROPN iajs-700 258 1 مفهـوم مفهـوم PROPN iajs-700 258 2 المقاسـات المقاسـات PROPN iajs-700 258 3 شــبه شــبه PROPN iajs-700 258 4 االولیـ االولیـ PROPN iajs-700 258 5 .التشخیصات .التشخیصات ADP iajs-700 258 6 والخواص والخواص NOUN iajs-700 258 7 لهذا لهذا VERB iajs-700 258 8 المفهوم المفهوم NOUN iajs-700 258 9 . . PUNCT iajs-700 259 1 المقاس المقاس PROPN iajs-700 259 2 األولي األولي PROPN iajs-700 259 3 الضبابي الضبابي PROPN iajs-700 259 4 ، ، PROPN iajs-700 259 5 المقاس المقاس PROPN iajs-700 259 6 شبه شبه PROPN iajs-700 259 7 األولي األولي PROPN iajs-700 259 8 ، ، PROPN iajs-700 259 9 المقاس المقاس PROPN iajs-700 259 10 شبه شبه PROPN iajs-700 259 11 األولي األولي NOUN iajs-700 259 12 الضبابي الضبابي PROPN iajs-700 259 13 : : PUNCT iajs-700 259 14 الكلمات الكلمات VERB iajs-700 259 15 المفتاحیة المفتاحیة ADV