id sid tid token lemma pos cana-3154 1 1 communications communication NOUN cana-3154 1 2 on on ADP cana-3154 1 3 applied apply VERB cana-3154 1 4 nonlinear nonlinear ADJ cana-3154 1 5 analysis analysis NOUN cana-3154 1 6 issn issn NOUN cana-3154 1 7 : : PUNCT cana-3154 1 8 1074 1074 NUM cana-3154 1 9 - - PUNCT cana-3154 1 10 133x 133x NUM cana-3154 1 11 vol vol NOUN cana-3154 1 12 32 32 NUM cana-3154 1 13 no no NOUN cana-3154 1 14 . . PUNCT cana-3154 2 1 5s 5s NUM cana-3154 2 2 ( ( PUNCT cana-3154 2 3 2025 2025 NUM cana-3154 2 4 ) ) PUNCT cana-3154 2 5 481 481 NUM cana-3154 2 6 https://internationalpubls.com https://internationalpubls.com X cana-3154 2 7 principal principal ADJ cana-3154 2 8 intersection intersection NOUN cana-3154 2 9 graph graph NOUN cana-3154 2 10 of of ADP cana-3154 2 11 commutative commutative ADJ cana-3154 2 12 rings ring NOUN cana-3154 2 13 lamine lamine PROPN cana-3154 2 14 ngom1 ngom1 PROPN cana-3154 2 15 , , PUNCT cana-3154 2 16 mankagna mankagna PROPN cana-3154 2 17 albert albert PROPN cana-3154 2 18 diompy diompy VERB cana-3154 2 19 2 2 NUM cana-3154 2 20 1,2department 1,2department NUM cana-3154 2 21 of of ADP cana-3154 2 22 mathematics mathematic NOUN cana-3154 2 23 and and CCONJ cana-3154 2 24 computer computer NOUN cana-3154 2 25 science science NOUN cana-3154 2 26 , , PUNCT cana-3154 2 27 university university NOUN cana-3154 2 28 of of ADP cana-3154 2 29 cheikh cheikh PROPN cana-3154 2 30 anta anta PROPN cana-3154 2 31 diop diop PROPN cana-3154 2 32 , , PUNCT cana-3154 2 33 dakar dakar NOUN cana-3154 2 34 , , PUNCT cana-3154 2 35 senegal senegal ADJ cana-3154 2 36 email email NOUN cana-3154 3 1 i i PROPN cana-3154 3 2 d d PROPN cana-3154 3 3 : : PUNCT cana-3154 3 4 laminengo89@hotmail.fr1 laminengo89@hotmail.fr1 PROPN cana-3154 3 5 corresponding corresponding ADJ cana-3154 3 6 author author NOUN cana-3154 3 7 : : PUNCT cana-3154 3 8 albertdiompy@yahoo.fr albertdiompy@yahoo.fr PROPN cana-3154 3 9 article article PROPN cana-3154 3 10 history history NOUN cana-3154 3 11 : : PUNCT cana-3154 3 12 received receive VERB cana-3154 3 13 : : PUNCT cana-3154 3 14 13 13 NUM cana-3154 3 15 - - SYM cana-3154 3 16 10 10 NUM cana-3154 3 17 - - PUNCT cana-3154 3 18 2024 2024 NUM cana-3154 3 19 revised revise VERB cana-3154 3 20 : : PUNCT cana-3154 3 21 28 28 NUM cana-3154 3 22 - - SYM cana-3154 3 23 11 11 NUM cana-3154 3 24 - - PUNCT cana-3154 3 25 2024 2024 NUM cana-3154 3 26 accepted accept VERB cana-3154 3 27 : : PUNCT cana-3154 3 28 09 09 NUM cana-3154 3 29 - - SYM cana-3154 3 30 12 12 NUM cana-3154 3 31 - - PUNCT cana-3154 3 32 2024 2024 NUM cana-3154 3 33 abstract abstract NOUN cana-3154 3 34 : : PUNCT cana-3154 3 35 let let VERB cana-3154 3 36 𝑅 𝑅 PROPN cana-3154 3 37 be be AUX cana-3154 3 38 a a DET cana-3154 3 39 commutative commutative ADJ cana-3154 3 40 ring ring NOUN cana-3154 3 41 . . PUNCT cana-3154 4 1 the the DET cana-3154 4 2 principal principal ADJ cana-3154 4 3 intersection intersection NOUN cana-3154 4 4 graph graph NOUN cana-3154 4 5 of of ADP cana-3154 4 6 a a DET cana-3154 4 7 commutative commutative ADJ cana-3154 4 8 ring ring PROPN cana-3154 4 9 𝑅 𝑅 PROPN cana-3154 4 10 , , PUNCT cana-3154 4 11 noted note VERB cana-3154 4 12 𝐺𝑐(𝑅 𝐺𝑐(𝑅 VERB cana-3154 4 13 ) ) PUNCT cana-3154 4 14 , , PUNCT cana-3154 4 15 consist consist VERB cana-3154 4 16 of of ADP cana-3154 4 17 all all DET cana-3154 4 18 proper proper ADJ cana-3154 4 19 ideals ideal NOUN cana-3154 4 20 of of ADP cana-3154 4 21 𝑅 𝑅 NOUN cana-3154 4 22 as as ADP cana-3154 4 23 vertices vertex NOUN cana-3154 4 24 . . PUNCT cana-3154 5 1 two two NUM cana-3154 5 2 distinct distinct ADJ cana-3154 5 3 vertices vertex NOUN cana-3154 5 4 𝐼 𝐼 PROPN cana-3154 5 5 and and CCONJ cana-3154 5 6 𝐽 𝐽 PROPN cana-3154 5 7 are be AUX cana-3154 5 8 adjacent adjacent ADJ cana-3154 5 9 if if SCONJ cana-3154 5 10 𝐼 𝐼 PROPN cana-3154 5 11 ∩ ∩ NOUN cana-3154 5 12 𝐽 𝐽 NOUN cana-3154 5 13 β‰  β‰  PROPN cana-3154 5 14 0 0 NUM cana-3154 5 15 and and CCONJ cana-3154 5 16 either either CCONJ cana-3154 5 17 𝐼 𝐼 PROPN cana-3154 5 18 or or CCONJ cana-3154 5 19 𝐽 𝐽 PROPN cana-3154 5 20 is be AUX cana-3154 5 21 a a DET cana-3154 5 22 principal principal ADJ cana-3154 5 23 ( ( PUNCT cana-3154 5 24 cyclic cyclic ADJ cana-3154 5 25 ) ) PUNCT cana-3154 5 26 ideal ideal NOUN cana-3154 5 27 . . PUNCT cana-3154 6 1 in in ADP cana-3154 6 2 this this DET cana-3154 6 3 paper paper NOUN cana-3154 6 4 , , PUNCT cana-3154 6 5 we we PRON cana-3154 6 6 investigate investigate VERB cana-3154 6 7 some some DET cana-3154 6 8 properties property NOUN cana-3154 6 9 from from ADP cana-3154 6 10 graph graph NOUN cana-3154 6 11 theory theory NOUN cana-3154 6 12 of of ADP cana-3154 6 13 𝐺𝑐(𝑅 𝐺𝑐(𝑅 DET cana-3154 6 14 ) ) PUNCT cana-3154 6 15 and and CCONJ cana-3154 6 16 its its PRON cana-3154 6 17 algebraic algebraic ADJ cana-3154 6 18 properties property NOUN cana-3154 6 19 where where SCONJ cana-3154 6 20 𝑅 𝑅 PROPN cana-3154 6 21 is be AUX cana-3154 6 22 a a DET cana-3154 6 23 ring ring NOUN cana-3154 6 24 . . PUNCT cana-3154 7 1 keywords keyword NOUN cana-3154 7 2 : : PUNCT cana-3154 7 3 principal principal ADJ cana-3154 7 4 intersection intersection NOUN cana-3154 7 5 graph graph NOUN cana-3154 7 6 , , PUNCT cana-3154 7 7 principal principal ADJ cana-3154 7 8 ideal ideal ADJ cana-3154 7 9 domain domain NOUN cana-3154 7 10 , , PUNCT cana-3154 7 11 ore ore NOUN cana-3154 7 12 domain domain NOUN cana-3154 7 13 , , PUNCT cana-3154 7 14 bezout bezout NOUN cana-3154 7 15 domain domain NOUN cana-3154 7 16 , , PUNCT cana-3154 7 17 connected connected ADJ cana-3154 7 18 graph graph NOUN cana-3154 7 19 , , PUNCT cana-3154 7 20 complete complete ADJ cana-3154 7 21 graph graph NOUN cana-3154 7 22 , , PUNCT cana-3154 7 23 hamiltonian hamiltonian ADJ cana-3154 7 24 graph graph NOUN cana-3154 7 25 . . PUNCT cana-3154 8 1 ams am NOUN cana-3154 8 2 subject subject ADJ cana-3154 8 3 classification classification NOUN cana-3154 8 4 : : PUNCT cana-3154 8 5 13a70 13a70 NUM cana-3154 8 6 , , PUNCT cana-3154 8 7 05c25 05c25 NUM cana-3154 8 8 , , PUNCT cana-3154 8 9 05c45 05c45 NOUN cana-3154 8 10 . . PUNCT cana-3154 9 1 1 1 X cana-3154 9 2 . . X cana-3154 9 3 introduction introduction NOUN cana-3154 9 4 the the DET cana-3154 9 5 intersection intersection NOUN cana-3154 9 6 graph graph NOUN cana-3154 9 7 of of ADP cana-3154 9 8 ideals ideal NOUN cana-3154 9 9 of of ADP cana-3154 9 10 a a DET cana-3154 9 11 ring ring NOUN cana-3154 9 12 r r NOUN cana-3154 9 13 is be AUX cana-3154 9 14 the the DET cana-3154 9 15 graph graph NOUN cana-3154 9 16 having have VERB cana-3154 9 17 the the DET cana-3154 9 18 set set NOUN cana-3154 9 19 of of ADP cana-3154 9 20 all all DET cana-3154 9 21 ideals ideal NOUN cana-3154 9 22 as as ADP cana-3154 9 23 its its PRON cana-3154 9 24 set set NOUN cana-3154 9 25 of of ADP cana-3154 9 26 vertices vertex NOUN cana-3154 9 27 . . PUNCT cana-3154 10 1 two two NUM cana-3154 10 2 distinct distinct ADJ cana-3154 10 3 vertices vertex NOUN cana-3154 10 4 i i PRON cana-3154 10 5 and and CCONJ cana-3154 10 6 j j PROPN cana-3154 10 7 are be AUX cana-3154 10 8 adjacent adjacent ADJ cana-3154 10 9 if if SCONJ cana-3154 10 10 and and CCONJ cana-3154 10 11 only only ADV cana-3154 10 12 if if SCONJ cana-3154 10 13 their their PRON cana-3154 10 14 intersection intersection NOUN cana-3154 10 15 is be AUX cana-3154 10 16 non non ADJ cana-3154 10 17 - - ADJ cana-3154 10 18 zero zero NUM cana-3154 10 19 idea idea NOUN cana-3154 10 20 and and CCONJ cana-3154 10 21 either either CCONJ cana-3154 10 22 i i PRON cana-3154 10 23 or or CCONJ cana-3154 10 24 j j PROPN cana-3154 10 25 is be AUX cana-3154 10 26 a a DET cana-3154 10 27 principal principal ADJ cana-3154 10 28 ( ( PUNCT cana-3154 10 29 cyclic cyclic ADJ cana-3154 10 30 ) ) PUNCT cana-3154 10 31 ideal ideal NOUN cana-3154 10 32 . . PUNCT cana-3154 11 1 intersection intersection NOUN cana-3154 11 2 graph graph NOUN cana-3154 11 3 were be AUX cana-3154 11 4 introduced introduce VERB cana-3154 11 5 by by ADP cana-3154 11 6 bosak bosak NOUN cana-3154 11 7 in in ADP cana-3154 11 8 1964 1964 NUM cana-3154 11 9 [ [ X cana-3154 11 10 6 6 NUM cana-3154 11 11 ] ] PUNCT cana-3154 11 12 . . PUNCT cana-3154 12 1 since since SCONJ cana-3154 12 2 , , PUNCT cana-3154 12 3 particular particular ADJ cana-3154 12 4 intersection intersection NOUN cana-3154 12 5 graph graph NOUN cana-3154 12 6 like like ADP cana-3154 12 7 small small ADJ cana-3154 12 8 intersection intersection NOUN cana-3154 12 9 graph graph NOUN cana-3154 12 10 , , PUNCT cana-3154 12 11 prime prime ADJ cana-3154 12 12 intersection intersection NOUN cana-3154 12 13 graph graph NOUN cana-3154 12 14 , , PUNCT cana-3154 12 15 semisimple semisimple ADJ cana-3154 12 16 intersection intersection NOUN cana-3154 12 17 graph graph NOUN cana-3154 12 18 are be AUX cana-3154 12 19 studied study VERB cana-3154 12 20 respectively respectively ADV cana-3154 12 21 in in ADP cana-3154 12 22 [ [ X cana-3154 12 23 3 3 NUM cana-3154 12 24 , , PUNCT cana-3154 12 25 1 1 NUM cana-3154 12 26 , , PUNCT cana-3154 12 27 11 11 NUM cana-3154 12 28 , , PUNCT cana-3154 12 29 5 5 NUM cana-3154 12 30 , , PUNCT cana-3154 12 31 7 7 NUM cana-3154 12 32 ] ] PUNCT cana-3154 12 33 . . PUNCT cana-3154 13 1 recently recently ADV cana-3154 13 2 , , PUNCT cana-3154 13 3 several several ADJ cana-3154 13 4 properties property NOUN cana-3154 13 5 of of ADP cana-3154 13 6 these these DET cana-3154 13 7 kinds kind NOUN cana-3154 13 8 of of ADP cana-3154 13 9 graphs graph NOUN cana-3154 13 10 were be AUX cana-3154 13 11 investigated investigate VERB cana-3154 13 12 by by ADP cana-3154 13 13 many many ADJ cana-3154 13 14 authors author NOUN cana-3154 13 15 as as ADP cana-3154 13 16 ansari ansari ADJ cana-3154 13 17 - - PUNCT cana-3154 13 18 toroghy toroghy ADJ cana-3154 13 19 , , PUNCT cana-3154 13 20 nikmehr nikmehr NOUN cana-3154 13 21 soleymanzadeh soleymanzadeh NOUN cana-3154 13 22 and and CCONJ cana-3154 13 23 alwan alwan PROPN cana-3154 13 24 in in ADP cana-3154 13 25 2016 2016 NUM cana-3154 13 26 ; ; PUNCT cana-3154 13 27 2017 2017 NUM cana-3154 13 28 and and CCONJ cana-3154 13 29 2023 2023 NUM cana-3154 13 30 respectively respectively ADV cana-3154 13 31 . . PUNCT cana-3154 14 1 in in ADP cana-3154 14 2 this this DET cana-3154 14 3 paper paper NOUN cana-3154 14 4 , , PUNCT cana-3154 14 5 r r NOUN cana-3154 14 6 is be AUX cana-3154 14 7 a a DET cana-3154 14 8 commutative commutative ADJ cana-3154 14 9 ring ring NOUN cana-3154 14 10 with with ADP cana-3154 14 11 identity identity NOUN cana-3154 14 12 ( ( PUNCT cana-3154 14 13 or or CCONJ cana-3154 14 14 eventually eventually ADV cana-3154 14 15 a a DET cana-3154 14 16 domain domain NOUN cana-3154 14 17 ) ) PUNCT cana-3154 14 18 . . PUNCT cana-3154 15 1 here here ADV cana-3154 15 2 , , PUNCT cana-3154 15 3 we we PRON cana-3154 15 4 introduce introduce VERB cana-3154 15 5 a a DET cana-3154 15 6 particular particular ADJ cana-3154 15 7 intersection intersection NOUN cana-3154 15 8 graph graph NOUN cana-3154 15 9 gc(r gc(r NOUN cana-3154 15 10 ) ) PUNCT cana-3154 15 11 named name VERB cana-3154 15 12 principal principal ADJ cana-3154 15 13 intersection intersection NOUN cana-3154 15 14 graph graph NOUN cana-3154 15 15 , , PUNCT cana-3154 15 16 whose whose DET cana-3154 15 17 set set NOUN cana-3154 15 18 of of ADP cana-3154 15 19 vertices vertex NOUN cana-3154 15 20 is be AUX cana-3154 15 21 the the DET cana-3154 15 22 proper proper ADJ cana-3154 15 23 ideals ideal NOUN cana-3154 15 24 of of ADP cana-3154 15 25 r. r. NOUN cana-3154 15 26 we we PRON cana-3154 15 27 will will AUX cana-3154 15 28 study study VERB cana-3154 15 29 the the DET cana-3154 15 30 algebraic algebraic ADJ cana-3154 15 31 properties property NOUN cana-3154 15 32 of of ADP cana-3154 15 33 gc(r gc(r NOUN cana-3154 15 34 ) ) PUNCT cana-3154 15 35 and and CCONJ cana-3154 15 36 also also ADV cana-3154 15 37 its its PRON cana-3154 15 38 properties property NOUN cana-3154 15 39 when when SCONJ cana-3154 15 40 seen see VERB cana-3154 15 41 as as ADP cana-3154 15 42 a a DET cana-3154 15 43 graph graph NOUN cana-3154 15 44 . . PUNCT cana-3154 16 1 this this DET cana-3154 16 2 paper paper NOUN cana-3154 16 3 is be AUX cana-3154 16 4 organized organize VERB cana-3154 16 5 as as ADP cana-3154 16 6 follow follow NOUN cana-3154 16 7 : : PUNCT cana-3154 16 8 in in ADP cana-3154 16 9 the the DET cana-3154 16 10 first first ADJ cana-3154 16 11 section section NOUN cana-3154 16 12 , , PUNCT cana-3154 16 13 we we PRON cana-3154 16 14 recall recall VERB cana-3154 16 15 some some DET cana-3154 16 16 properties property NOUN cana-3154 16 17 of of ADP cana-3154 16 18 rings ring NOUN cana-3154 16 19 and and CCONJ cana-3154 16 20 graph graph NOUN cana-3154 16 21 theory theory NOUN cana-3154 16 22 . . PUNCT cana-3154 17 1 in in ADP cana-3154 17 2 the the DET cana-3154 17 3 second second ADJ cana-3154 17 4 section section NOUN cana-3154 17 5 , , PUNCT cana-3154 17 6 we we PRON cana-3154 17 7 study study VERB cana-3154 17 8 connectedness connectedness NOUN cana-3154 17 9 , , PUNCT cana-3154 17 10 completeness completeness NOUN cana-3154 17 11 , , PUNCT cana-3154 17 12 k k NOUN cana-3154 17 13 - - PUNCT cana-3154 17 14 partite partite ADJ cana-3154 17 15 and and CCONJ cana-3154 17 16 hamiltonian hamiltonian ADJ cana-3154 17 17 properties property NOUN cana-3154 17 18 of of ADP cana-3154 17 19 this this DET cana-3154 17 20 intersection intersection NOUN cana-3154 17 21 graph graph NOUN cana-3154 17 22 . . PUNCT cana-3154 18 1 we we PRON cana-3154 18 2 gave give VERB cana-3154 18 3 a a DET cana-3154 18 4 characterization characterization NOUN cana-3154 18 5 of of ADP cana-3154 18 6 the the DET cana-3154 18 7 connectedness connectedness NOUN cana-3154 18 8 , , PUNCT cana-3154 18 9 completeness completeness NOUN cana-3154 18 10 and and CCONJ cana-3154 18 11 hamiltonian hamiltonian ADJ cana-3154 18 12 properties property NOUN cana-3154 18 13 of of ADP cana-3154 18 14 gc(r gc(r NOUN cana-3154 18 15 ) ) PUNCT cana-3154 18 16 as as ADP cana-3154 18 17 a a DET cana-3154 18 18 principal principal ADJ cana-3154 18 19 ideal ideal ADJ cana-3154 18 20 domain domain NOUN cana-3154 18 21 , , PUNCT cana-3154 18 22 an an DET cana-3154 18 23 ore ore NOUN cana-3154 18 24 domain domain NOUN cana-3154 18 25 and and CCONJ cana-3154 18 26 a a DET cana-3154 18 27 bezout bezout NOUN cana-3154 18 28 domain domain NOUN cana-3154 18 29 . . PUNCT cana-3154 19 1 2 2 X cana-3154 19 2 . . X cana-3154 19 3 definitions definition NOUN cana-3154 19 4 and and CCONJ cana-3154 19 5 preliminary preliminary ADJ cana-3154 19 6 results result NOUN cana-3154 19 7 definition definition NOUN cana-3154 19 8 2.1 2.1 NUM cana-3154 19 9 . . PUNCT cana-3154 20 1 : : PUNCT cana-3154 20 2 definitions definition NOUN cana-3154 20 3 from from ADP cana-3154 20 4 ring ring NOUN cana-3154 20 5 theory theory NOUN cana-3154 20 6 β€’ β€’ ADP cana-3154 20 7 an an DET cana-3154 20 8 ideal ideal NOUN cana-3154 20 9 i i PRON cana-3154 20 10 of of ADP cana-3154 20 11 commutative commutative ADJ cana-3154 20 12 ring ring NOUN cana-3154 20 13 r r NOUN cana-3154 20 14 is be AUX cana-3154 20 15 principal principal NOUN cana-3154 20 16 written write VERB cana-3154 20 17 as as ADP cana-3154 20 18 i i PRON cana-3154 20 19 = = SYM cana-3154 20 20 ar ar PROPN cana-3154 20 21 for for ADP cana-3154 20 22 some some PRON cana-3154 20 23 a a DET cana-3154 20 24 ∈ ∈ NOUN cana-3154 20 25 r r NOUN cana-3154 20 26 , , PUNCT cana-3154 20 27 if if SCONJ cana-3154 20 28 it it PRON cana-3154 20 29 is be AUX cana-3154 20 30 generated generate VERB cana-3154 20 31 by by ADP cana-3154 20 32 one one NUM cana-3154 20 33 element element NOUN cana-3154 20 34 . . PUNCT cana-3154 21 1 β€’ β€’ NUM cana-3154 21 2 a a DET cana-3154 21 3 ring ring NOUN cana-3154 21 4 r r NOUN cana-3154 21 5 is be AUX cana-3154 21 6 principal principal ADJ cana-3154 21 7 if if SCONJ cana-3154 21 8 every every DET cana-3154 21 9 proper proper ADJ cana-3154 21 10 ideal ideal NOUN cana-3154 21 11 is be AUX cana-3154 21 12 a a DET cana-3154 21 13 principal principal ADJ cana-3154 21 14 ideal ideal NOUN cana-3154 21 15 . . PUNCT cana-3154 22 1 communications communication NOUN cana-3154 22 2 on on ADP cana-3154 22 3 applied apply VERB cana-3154 22 4 nonlinear nonlinear ADJ cana-3154 22 5 analysis analysis NOUN cana-3154 22 6 issn issn NOUN cana-3154 22 7 : : PUNCT cana-3154 22 8 1074 1074 NUM cana-3154 22 9 - - PUNCT cana-3154 22 10 133x 133x NUM cana-3154 22 11 vol vol NOUN cana-3154 22 12 32 32 NUM cana-3154 22 13 no no NOUN cana-3154 22 14 . . PUNCT cana-3154 23 1 5s 5s NUM cana-3154 23 2 ( ( PUNCT cana-3154 23 3 2025 2025 NUM cana-3154 23 4 ) ) PUNCT cana-3154 23 5 482 482 NUM cana-3154 24 1 https://internationalpubls.com https://internationalpubls.com X cana-3154 24 2 β€’ β€’ ADP cana-3154 24 3 a a DET cana-3154 24 4 ring ring NOUN cana-3154 24 5 r r NOUN cana-3154 24 6 is be AUX cana-3154 24 7 an an DET cana-3154 24 8 ore ore NOUN cana-3154 24 9 ring ring NOUN cana-3154 24 10 if if SCONJ cana-3154 24 11 it it PRON cana-3154 24 12 satisfies satisfy VERB cana-3154 24 13 the the DET cana-3154 24 14 ore ore NOUN cana-3154 24 15 condition condition NOUN cana-3154 24 16 . . PUNCT cana-3154 25 1 that that PRON cana-3154 25 2 is be AUX cana-3154 25 3 : : PUNCT cana-3154 25 4 for for ADP cana-3154 25 5 all all DET cana-3154 25 6 elements element NOUN cana-3154 25 7 a a PRON cana-3154 25 8 and and CCONJ cana-3154 25 9 b b NOUN cana-3154 25 10 in in ADP cana-3154 25 11 r r NOUN cana-3154 25 12 , , PUNCT cana-3154 25 13 ar ar NOUN cana-3154 25 14 ∩ ∩ NOUN cana-3154 25 15 br br PROPN cana-3154 25 16 β‰  β‰  PROPN cana-3154 25 17 { { PUNCT cana-3154 25 18 0 0 NUM cana-3154 25 19 } } PUNCT cana-3154 25 20 . . PUNCT cana-3154 26 1 β€’ β€’ NUM cana-3154 26 2 a a DET cana-3154 26 3 ring ring NOUN cana-3154 26 4 r r NOUN cana-3154 26 5 is be AUX cana-3154 26 6 a a DET cana-3154 26 7 domain domain NOUN cana-3154 26 8 if if SCONJ cana-3154 26 9 it it PRON cana-3154 26 10 has have VERB cana-3154 26 11 no no DET cana-3154 26 12 zero zero NUM cana-3154 26 13 - - PUNCT cana-3154 26 14 divisor divisor NOUN cana-3154 26 15 . . NOUN cana-3154 27 1 β€’ β€’ NUM cana-3154 27 2 a a DET cana-3154 27 3 ring ring NOUN cana-3154 27 4 r r NOUN cana-3154 27 5 is be AUX cana-3154 27 6 an an DET cana-3154 27 7 ore ore NOUN cana-3154 27 8 ring ring NOUN cana-3154 27 9 if if SCONJ cana-3154 27 10 for for ADP cana-3154 27 11 all all DET cana-3154 27 12 elements element NOUN cana-3154 27 13 a a PRON cana-3154 27 14 and and CCONJ cana-3154 27 15 b b NOUN cana-3154 27 16 in in ADP cana-3154 27 17 r r NOUN cana-3154 27 18 , , PUNCT cana-3154 27 19 ar ar NOUN cana-3154 27 20 ∩ ∩ NOUN cana-3154 27 21 br br PROPN cana-3154 27 22 β‰  β‰  PROPN cana-3154 27 23 { { PUNCT cana-3154 27 24 0 0 NUM cana-3154 27 25 } } PUNCT cana-3154 27 26 . . PUNCT cana-3154 28 1 β€’ β€’ NUM cana-3154 28 2 a a DET cana-3154 28 3 principal principal ADJ cana-3154 28 4 ideal ideal ADJ cana-3154 28 5 domain domain NOUN cana-3154 28 6 is be AUX cana-3154 28 7 a a DET cana-3154 28 8 domain domain NOUN cana-3154 28 9 such such ADJ cana-3154 28 10 that that SCONJ cana-3154 28 11 every every DET cana-3154 28 12 ideal ideal NOUN cana-3154 28 13 of of ADP cana-3154 28 14 r r NOUN cana-3154 28 15 is be AUX cana-3154 28 16 a a DET cana-3154 28 17 principal principal ADJ cana-3154 28 18 ideal ideal NOUN cana-3154 28 19 . . PUNCT cana-3154 29 1 β€’ β€’ ADP cana-3154 29 2 bezout bezout NOUN cana-3154 29 3 ring ring NOUN cana-3154 29 4 is be AUX cana-3154 29 5 a a DET cana-3154 29 6 domain domain NOUN cana-3154 29 7 in in ADP cana-3154 29 8 which which PRON cana-3154 29 9 for for ADP cana-3154 29 10 any any PRON cana-3154 29 11 to to ADP cana-3154 29 12 elements element NOUN cana-3154 29 13 a a DET cana-3154 29 14 , , PUNCT cana-3154 29 15 b b X cana-3154 29 16 ∈ ∈ PROPN cana-3154 29 17 r r NOUN cana-3154 29 18 , , PUNCT cana-3154 29 19 there there PRON cana-3154 29 20 is be VERB cana-3154 29 21 n n PRON cana-3154 29 22 β‰₯ β‰₯ NOUN cana-3154 29 23 0 0 NUM cana-3154 29 24 such such ADJ cana-3154 29 25 that that PRON cana-3154 29 26 ran run VERB cana-3154 29 27 + + CCONJ cana-3154 29 28 rb rb NOUN cana-3154 29 29 n n PRON cana-3154 29 30 is be AUX cana-3154 29 31 a a DET cana-3154 29 32 principal principal ADJ cana-3154 29 33 ideal ideal NOUN cana-3154 29 34 . . PUNCT cana-3154 30 1 definition definition NOUN cana-3154 30 2 2.2 2.2 NUM cana-3154 30 3 . . PUNCT cana-3154 31 1 : : PUNCT cana-3154 31 2 definitions definition NOUN cana-3154 31 3 from from ADP cana-3154 31 4 graph graph NOUN cana-3154 31 5 theory theory NOUN cana-3154 31 6 β€’ β€’ ADP cana-3154 31 7 let let VERB cana-3154 31 8 i i PRON cana-3154 31 9 and and CCONJ cana-3154 31 10 j j PROPN cana-3154 31 11 two two NUM cana-3154 31 12 distinct distinct ADJ cana-3154 31 13 vertices vertex NOUN cana-3154 31 14 , , PUNCT cana-3154 31 15 iβˆ’ iβˆ’ PROPN cana-3154 31 16 j j PROPN cana-3154 31 17 means mean VERB cana-3154 31 18 that that SCONJ cana-3154 31 19 i i PRON cana-3154 31 20 and and CCONJ cana-3154 31 21 j j PROPN cana-3154 31 22 are be AUX cana-3154 31 23 adjacent adjacent ADJ cana-3154 31 24 . . PUNCT cana-3154 32 1 β€’ β€’ NUM cana-3154 32 2 the the DET cana-3154 32 3 degree degree NOUN cana-3154 32 4 of of ADP cana-3154 32 5 a a DET cana-3154 32 6 vertex vertex NOUN cana-3154 32 7 i i PRON cana-3154 32 8 of of ADP cana-3154 32 9 graph graph NOUN cana-3154 32 10 gc(r gc(r NOUN cana-3154 32 11 ) ) PUNCT cana-3154 32 12 ) ) PUNCT cana-3154 32 13 which which PRON cana-3154 32 14 denoted denote VERB cana-3154 32 15 by by ADP cana-3154 32 16 deg(i deg(i PROPN cana-3154 32 17 ) ) PUNCT cana-3154 32 18 is be AUX cana-3154 32 19 the the DET cana-3154 32 20 number number NOUN cana-3154 32 21 of of ADP cana-3154 32 22 edges edge NOUN cana-3154 32 23 incident incident NOUN cana-3154 32 24 on on ADP cana-3154 32 25 i. i. PROPN cana-3154 32 26 β€’ β€’ ADP cana-3154 32 27 if if SCONJ cana-3154 32 28 |v(gc(r)))| |v(gc(r)))| PROPN cana-3154 32 29 > > X cana-3154 32 30 2 2 NUM cana-3154 32 31 , , PUNCT cana-3154 32 32 a a DET cana-3154 32 33 path path NOUN cana-3154 32 34 from from ADP cana-3154 32 35 i i PRON cana-3154 32 36 to to ADP cana-3154 32 37 j j PROPN cana-3154 32 38 is be AUX cana-3154 32 39 a a DET cana-3154 32 40 sequence sequence NOUN cana-3154 32 41 of of ADP cana-3154 32 42 adjacent adjacent ADJ cana-3154 32 43 vertices vertex NOUN cana-3154 32 44 iβˆ’ iβˆ’ PROPN cana-3154 32 45 i1 i1 PROPN cana-3154 32 46 βˆ’ βˆ’ PROPN cana-3154 32 47 i2 i2 PROPN cana-3154 32 48 βˆ’β‹―βˆ’ βˆ’β‹―βˆ’ ADV cana-3154 32 49 in in ADP cana-3154 32 50 βˆ’ βˆ’ PROPN cana-3154 32 51 j j PROPN cana-3154 32 52 , , PUNCT cana-3154 32 53 where where SCONJ cana-3154 32 54 ii ii PROPN cana-3154 32 55 ∈ ∈ PROPN cana-3154 32 56 v(gc(r v(gc(r PROPN cana-3154 32 57 ) ) PUNCT cana-3154 32 58 . . PUNCT cana-3154 33 1 β€’ β€’ NOUN cana-3154 33 2 the the DET cana-3154 33 3 length length NOUN cana-3154 33 4 a a DET cana-3154 33 5 path path NOUN cana-3154 33 6 graph graph NOUN cana-3154 33 7 of of ADP cana-3154 33 8 a a DET cana-3154 33 9 graph graph NOUN cana-3154 33 10 is be AUX cana-3154 33 11 the the DET cana-3154 33 12 number number NOUN cana-3154 33 13 of of ADP cana-3154 33 14 edges edge NOUN cana-3154 33 15 in in ADP cana-3154 33 16 this this DET cana-3154 33 17 path path NOUN cana-3154 33 18 . . PUNCT cana-3154 34 1 β€’ β€’ ADP cana-3154 34 2 a a DET cana-3154 34 3 path path NOUN cana-3154 34 4 using use VERB cana-3154 34 5 k k PROPN cana-3154 34 6 distinct distinct ADJ cana-3154 34 7 vertices vertex NOUN cana-3154 34 8 has have VERB cana-3154 34 9 length length ADJ cana-3154 34 10 kβˆ’ kβˆ’ PROPN cana-3154 34 11 1 1 NUM cana-3154 34 12 . . NOUN cana-3154 34 13 β€’ β€’ NUM cana-3154 34 14 the the DET cana-3154 34 15 distance distance NOUN cana-3154 34 16 between between ADP cana-3154 34 17 two two NUM cana-3154 34 18 distinct distinct ADJ cana-3154 34 19 vertices vertex NOUN cana-3154 34 20 i i PRON cana-3154 34 21 and and CCONJ cana-3154 34 22 j j PROPN cana-3154 34 23 is be AUX cana-3154 34 24 denoted denote VERB cana-3154 34 25 by by ADP cana-3154 34 26 d(i d(i PROPN cana-3154 34 27 ; ; PUNCT cana-3154 34 28 j j PROPN cana-3154 34 29 ) ) PUNCT cana-3154 34 30 is be AUX cana-3154 34 31 the the DET cana-3154 34 32 length length NOUN cana-3154 34 33 of of ADP cana-3154 34 34 the the DET cana-3154 34 35 shortest short ADJ cana-3154 34 36 path path NOUN cana-3154 34 37 connecting connect VERB cana-3154 34 38 i i PRON cana-3154 34 39 and and CCONJ cana-3154 34 40 j. j. PROPN cana-3154 34 41 β€’ β€’ INTJ cana-3154 34 42 if if SCONJ cana-3154 34 43 there there PRON cana-3154 34 44 is be VERB cana-3154 34 45 not not PART cana-3154 34 46 a a DET cana-3154 34 47 path path NOUN cana-3154 34 48 between between ADP cana-3154 34 49 i i PROPN cana-3154 34 50 and and CCONJ cana-3154 34 51 j j PROPN cana-3154 34 52 , , PUNCT cana-3154 34 53 d(i d(i PROPN cana-3154 34 54 ; ; PUNCT cana-3154 34 55 j j X cana-3154 34 56 ) ) PUNCT cana-3154 35 1 = = PUNCT cana-3154 35 2 0 0 NUM cana-3154 35 3 . . NOUN cana-3154 35 4 β€’ β€’ NOUN cana-3154 35 5 the the DET cana-3154 35 6 number number NOUN cana-3154 35 7 of of ADP cana-3154 35 8 vertices vertex NOUN cana-3154 35 9 of of ADP cana-3154 35 10 gc(r gc(r NOUN cana-3154 35 11 ) ) PUNCT cana-3154 35 12 is be AUX cana-3154 35 13 the the DET cana-3154 35 14 order order NOUN cana-3154 35 15 of of ADP cana-3154 35 16 the the DET cana-3154 35 17 graph graph NOUN cana-3154 35 18 . . PUNCT cana-3154 36 1 β€’ β€’ NUM cana-3154 36 2 the the DET cana-3154 36 3 diameter diameter NOUN cana-3154 36 4 of of ADP cana-3154 36 5 a a DET cana-3154 36 6 graph graph NOUN cana-3154 36 7 gc(r gc(r NOUN cana-3154 36 8 ) ) PUNCT cana-3154 36 9 ) ) PUNCT cana-3154 36 10 is be AUX cana-3154 36 11 diam(gc(r diam(gc(r NOUN cana-3154 36 12 ) ) PUNCT cana-3154 36 13 ) ) PUNCT cana-3154 36 14 ) ) PUNCT cana-3154 37 1 = = PUNCT cana-3154 37 2 sup{d(i sup{d(i PROPN cana-3154 37 3 ; ; PUNCT cana-3154 37 4 j)/i j)/i PROPN cana-3154 37 5 ; ; PUNCT cana-3154 37 6 j j PROPN cana-3154 37 7 ∈ ∈ PROPN cana-3154 37 8 v(gc(r v(gc(r PROPN cana-3154 37 9 ) ) PUNCT cana-3154 37 10 ) ) PUNCT cana-3154 37 11 ) ) PUNCT cana-3154 37 12 } } PUNCT cana-3154 37 13 . . PUNCT cana-3154 38 1 β€’ β€’ NUM cana-3154 38 2 a a DET cana-3154 38 3 graph graph NOUN cana-3154 38 4 gc(r gc(r NOUN cana-3154 38 5 ) ) PUNCT cana-3154 38 6 ) ) PUNCT cana-3154 38 7 is be AUX cana-3154 38 8 connected connect VERB cana-3154 38 9 , , PUNCT cana-3154 38 10 if if SCONJ cana-3154 38 11 for for ADP cana-3154 38 12 any any DET cana-3154 38 13 vertices vertex NOUN cana-3154 38 14 i i PRON cana-3154 38 15 and and CCONJ cana-3154 38 16 j j PROPN cana-3154 38 17 of of ADP cana-3154 38 18 gc(r gc(r PROPN cana-3154 38 19 ) ) PUNCT cana-3154 38 20 ) ) PUNCT cana-3154 39 1 there there PRON cana-3154 39 2 is be VERB cana-3154 39 3 a a DET cana-3154 39 4 path path NOUN cana-3154 39 5 between between ADP cana-3154 39 6 i i PRON cana-3154 39 7 and and CCONJ cana-3154 39 8 j. j. PROPN cana-3154 39 9 if if SCONJ cana-3154 39 10 not not PART cana-3154 39 11 , , PUNCT cana-3154 39 12 gc(r gc(r NOUN cana-3154 39 13 ) ) PUNCT cana-3154 39 14 ) ) PUNCT cana-3154 39 15 is be AUX cana-3154 39 16 disconnected disconnect VERB cana-3154 39 17 . . PUNCT cana-3154 40 1 β€’ β€’ NUM cana-3154 40 2 a a DET cana-3154 40 3 closed closed ADJ cana-3154 40 4 path path NOUN cana-3154 41 1 i i PRON cana-3154 41 2 βˆ’ βˆ’ PROPN cana-3154 41 3 i1 i1 PROPN cana-3154 41 4 βˆ’ βˆ’ PROPN cana-3154 41 5 i2 i2 PROPN cana-3154 41 6 βˆ’β‹―βˆ’ βˆ’β‹―βˆ’ ADV cana-3154 41 7 in in ADP cana-3154 41 8 βˆ’ βˆ’ PROPN cana-3154 41 9 i i PRON cana-3154 41 10 is be AUX cana-3154 41 11 a a DET cana-3154 41 12 cycle cycle NOUN cana-3154 41 13 . . PUNCT cana-3154 42 1 β€’ β€’ NUM cana-3154 42 2 the the DET cana-3154 42 3 girth girth NOUN cana-3154 42 4 of of ADP cana-3154 42 5 gc(r gc(r NOUN cana-3154 42 6 ) ) PUNCT cana-3154 42 7 ) ) PUNCT cana-3154 42 8 is be AUX cana-3154 42 9 the the DET cana-3154 42 10 length length NOUN cana-3154 42 11 of of ADP cana-3154 42 12 the the DET cana-3154 42 13 shortest short ADJ cana-3154 42 14 cycle cycle NOUN cana-3154 42 15 in in ADP cana-3154 42 16 gc(r gc(r PROPN cana-3154 42 17 ) ) PUNCT cana-3154 42 18 ) ) PUNCT cana-3154 42 19 . . PUNCT cana-3154 43 1 β€’ β€’ NOUN cana-3154 43 2 a a DET cana-3154 43 3 hamiltonian hamiltonian ADJ cana-3154 43 4 cycle cycle NOUN cana-3154 43 5 is be AUX cana-3154 43 6 a a DET cana-3154 43 7 cycle cycle NOUN cana-3154 43 8 that that PRON cana-3154 43 9 contains contain VERB cana-3154 43 10 every every DET cana-3154 43 11 vertex vertex NOUN cana-3154 43 12 of of ADP cana-3154 43 13 the the DET cana-3154 43 14 graph graph NOUN cana-3154 43 15 . . PUNCT cana-3154 44 1 β€’ β€’ NUM cana-3154 44 2 a a DET cana-3154 44 3 hamiltonian hamiltonian ADJ cana-3154 44 4 graph graph NOUN cana-3154 44 5 is be AUX cana-3154 44 6 graph graph NOUN cana-3154 44 7 containing contain VERB cana-3154 44 8 a a DET cana-3154 44 9 hamiltonian hamiltonian ADJ cana-3154 44 10 cycle cycle NOUN cana-3154 44 11 . . PUNCT cana-3154 45 1 β€’ β€’ NUM cana-3154 45 2 a a DET cana-3154 45 3 graph graph NOUN cana-3154 45 4 with with ADP cana-3154 45 5 no no DET cana-3154 45 6 loop loop NOUN cana-3154 45 7 or or CCONJ cana-3154 45 8 multiple multiple ADJ cana-3154 45 9 edges edge NOUN cana-3154 45 10 is be AUX cana-3154 45 11 a a DET cana-3154 45 12 simple simple ADJ cana-3154 45 13 graph graph NOUN cana-3154 45 14 . . PUNCT cana-3154 46 1 proposition proposition NOUN cana-3154 46 2 2.3 2.3 NUM cana-3154 46 3 . . PUNCT cana-3154 47 1 let let VERB cana-3154 47 2 r r PRON cana-3154 47 3 be be AUX cana-3154 47 4 a a DET cana-3154 47 5 ring ring NOUN cana-3154 47 6 . . PUNCT cana-3154 48 1 gc(r gc(r PROPN cana-3154 48 2 ) ) PUNCT cana-3154 48 3 is be AUX cana-3154 48 4 an an DET cana-3154 48 5 empty empty ADJ cana-3154 48 6 graph graph NOUN cana-3154 48 7 if if SCONJ cana-3154 49 1 and and CCONJ cana-3154 49 2 only only ADV cana-3154 49 3 if if SCONJ cana-3154 49 4 r r NOUN cana-3154 49 5 is be AUX cana-3154 49 6 a a DET cana-3154 49 7 field field NOUN cana-3154 49 8 . . PUNCT cana-3154 50 1 proof proof NOUN cana-3154 50 2 : : PUNCT cana-3154 50 3 β€’ β€’ NUM cana-3154 50 4 β‡’ β‡’ NOUN cana-3154 50 5 ) ) PUNCT cana-3154 50 6 it it PRON cana-3154 50 7 is be AUX cana-3154 50 8 obvious obvious ADJ cana-3154 50 9 that that SCONJ cana-3154 50 10 if if SCONJ cana-3154 50 11 gc(r gc(r NOUN cana-3154 50 12 ) ) PUNCT cana-3154 50 13 is be AUX cana-3154 50 14 empty empty ADJ cana-3154 50 15 graph graph NOUN cana-3154 50 16 then then ADV cana-3154 50 17 gc(r gc(r VERB cana-3154 50 18 ) ) PUNCT cana-3154 50 19 has have VERB cana-3154 50 20 no no DET cana-3154 50 21 vertices vertex NOUN cana-3154 50 22 . . PUNCT cana-3154 51 1 since since SCONJ cana-3154 51 2 vertices vertex NOUN cana-3154 51 3 of of ADP cana-3154 51 4 gc(r gc(r NOUN cana-3154 51 5 ) ) PUNCT cana-3154 51 6 are be AUX cana-3154 51 7 the the DET cana-3154 51 8 proper proper ADJ cana-3154 51 9 ideals ideal NOUN cana-3154 51 10 of of ADP cana-3154 51 11 r r NOUN cana-3154 51 12 , , PUNCT cana-3154 51 13 then then ADV cana-3154 51 14 r r NOUN cana-3154 51 15 has have VERB cana-3154 51 16 proper proper ADJ cana-3154 51 17 ideal ideal NOUN cana-3154 51 18 , , PUNCT cana-3154 51 19 that that ADV cana-3154 51 20 is is ADV cana-3154 51 21 r r NOUN cana-3154 51 22 is be AUX cana-3154 51 23 a a DET cana-3154 51 24 field field NOUN cana-3154 51 25 . . PUNCT cana-3154 52 1 β€’ β€’ NUM cana-3154 52 2 ⇐ ⇐ NOUN cana-3154 52 3 ) ) PUNCT cana-3154 52 4 conversely conversely ADV cana-3154 52 5 , , PUNCT cana-3154 52 6 if if SCONJ cana-3154 52 7 r r NOUN cana-3154 52 8 is be AUX cana-3154 52 9 a a DET cana-3154 52 10 field field NOUN cana-3154 52 11 , , PUNCT cana-3154 52 12 r r NOUN cana-3154 52 13 has have VERB cana-3154 52 14 no no DET cana-3154 52 15 proper proper ADJ cana-3154 52 16 ideal ideal NOUN cana-3154 52 17 . . PUNCT cana-3154 53 1 β–« β–« DET cana-3154 53 2 lemma lemma PROPN cana-3154 53 3 2.4 2.4 NUM cana-3154 53 4 . . PUNCT cana-3154 54 1 if if SCONJ cana-3154 54 2 the the DET cana-3154 54 3 graph graph NOUN cana-3154 54 4 gc(r gc(r VERB cana-3154 54 5 ) ) PUNCT cana-3154 54 6 is be AUX cana-3154 54 7 a a DET cana-3154 54 8 null null ADJ cana-3154 54 9 graph graph NOUN cana-3154 54 10 , , PUNCT cana-3154 54 11 then then ADV cana-3154 54 12 for for ADP cana-3154 54 13 all all PRON cana-3154 54 14 ( ( PUNCT cana-3154 54 15 a a PRON cana-3154 54 16 ; ; PUNCT cana-3154 54 17 b b X cana-3154 54 18 ) ) PUNCT cana-3154 54 19 ∈ ∈ PROPN cana-3154 54 20 r r NOUN cana-3154 54 21 βˆ– βˆ– X cana-3154 54 22 { { PUNCT cana-3154 54 23 1 1 NUM cana-3154 54 24 } } PUNCT cana-3154 54 25 Γ— Γ— NOUN cana-3154 54 26 r r NOUN cana-3154 54 27 βˆ– βˆ– X cana-3154 54 28 { { PUNCT cana-3154 54 29 1 1 NUM cana-3154 54 30 } } PUNCT cana-3154 54 31 , , PUNCT cana-3154 54 32 ab ab PROPN cana-3154 54 33 = = PUNCT cana-3154 54 34 0 0 X cana-3154 54 35 . . PUNCT cana-3154 55 1 proof proof NOUN cana-3154 55 2 : : PUNCT cana-3154 55 3 assume assume VERB cana-3154 55 4 that that SCONJ cana-3154 55 5 gc(r gc(r NOUN cana-3154 55 6 ) ) PUNCT cana-3154 55 7 is be AUX cana-3154 55 8 a a DET cana-3154 55 9 null null ADJ cana-3154 55 10 graph graph NOUN cana-3154 55 11 . . PUNCT cana-3154 56 1 let let VERB cana-3154 56 2 a a DET cana-3154 56 3 β‰  β‰  PROPN cana-3154 56 4 1 1 NUM cana-3154 56 5 and and CCONJ cana-3154 56 6 b b PROPN cana-3154 56 7 β‰  β‰  PROPN cana-3154 56 8 1 1 NUM cana-3154 56 9 be be VERB cana-3154 56 10 two two NUM cana-3154 56 11 elements element NOUN cana-3154 56 12 of of ADP cana-3154 56 13 r. r. PROPN cana-3154 56 14 since since SCONJ cana-3154 56 15 gc(r gc(r PROPN cana-3154 56 16 ) ) PUNCT cana-3154 56 17 is be AUX cana-3154 56 18 a a DET cana-3154 56 19 null null ADJ cana-3154 56 20 graph graph NOUN cana-3154 56 21 , , PUNCT cana-3154 56 22 then then ADV cana-3154 56 23 ar ar PROPN cana-3154 56 24 ∩ ∩ PROPN cana-3154 56 25 br br NOUN cana-3154 56 26 = = PUNCT cana-3154 56 27 { { PUNCT cana-3154 56 28 0 0 NUM cana-3154 56 29 } } PUNCT cana-3154 56 30 . . PUNCT cana-3154 57 1 therefore therefore ADV cana-3154 57 2 , , PUNCT cana-3154 57 3 we we PRON cana-3154 57 4 have have VERB cana-3154 57 5 that that PRON cana-3154 57 6 , , PUNCT cana-3154 57 7 ab ab PROPN cana-3154 57 8 ∈ ∈ PROPN cana-3154 57 9 ar ar PROPN cana-3154 57 10 ∩ ∩ PROPN cana-3154 57 11 br br PROPN cana-3154 57 12 . . PUNCT cana-3154 58 1 thus thus ADV cana-3154 58 2 , , PUNCT cana-3154 58 3 ab ab PROPN cana-3154 58 4 = = SYM cana-3154 58 5 0 0 NUM cana-3154 58 6 communications communication NOUN cana-3154 58 7 on on ADP cana-3154 58 8 applied apply VERB cana-3154 58 9 nonlinear nonlinear ADJ cana-3154 58 10 analysis analysis NOUN cana-3154 58 11 issn issn NOUN cana-3154 58 12 : : PUNCT cana-3154 58 13 1074 1074 NUM cana-3154 58 14 - - PUNCT cana-3154 58 15 133x 133x NUM cana-3154 58 16 vol vol NOUN cana-3154 58 17 32 32 NUM cana-3154 58 18 no no NOUN cana-3154 58 19 . . PUNCT cana-3154 59 1 5s 5s NUM cana-3154 59 2 ( ( PUNCT cana-3154 59 3 2025 2025 NUM cana-3154 59 4 ) ) PUNCT cana-3154 59 5 483 483 NUM cana-3154 59 6 https://internationalpubls.com https://internationalpubls.com X cana-3154 59 7 proposition proposition NOUN cana-3154 59 8 2.5 2.5 NUM cana-3154 59 9 . . PUNCT cana-3154 60 1 let let VERB cana-3154 60 2 r r PRON cana-3154 60 3 be be AUX cana-3154 60 4 a a DET cana-3154 60 5 commutative commutative ADJ cana-3154 60 6 nonzero nonzero NOUN cana-3154 60 7 ring ring NOUN cana-3154 60 8 . . PUNCT cana-3154 61 1 the the DET cana-3154 61 2 graph graph NOUN cana-3154 61 3 gc(r gc(r NOUN cana-3154 61 4 ) ) PUNCT cana-3154 61 5 is be AUX cana-3154 61 6 a a DET cana-3154 61 7 null null ADJ cana-3154 61 8 graph graph NOUN cana-3154 61 9 if if SCONJ cana-3154 62 1 and and CCONJ cana-3154 62 2 only only ADV cana-3154 62 3 if if SCONJ cana-3154 62 4 r r NOUN cana-3154 62 5 = = SYM cana-3154 62 6 { { PUNCT cana-3154 62 7 0 0 NUM cana-3154 62 8 ; ; PUNCT cana-3154 62 9 1 1 NUM cana-3154 62 10 } } PUNCT cana-3154 62 11 . . PUNCT cana-3154 63 1 proof proof NOUN cana-3154 63 2 : : PUNCT cana-3154 63 3 β€’ β€’ NOUN cana-3154 63 4 ⇐ ⇐ NOUN cana-3154 63 5 ) ) PUNCT cana-3154 63 6 it it PRON cana-3154 63 7 is be AUX cana-3154 63 8 clear clear ADJ cana-3154 63 9 that that SCONJ cana-3154 63 10 if if SCONJ cana-3154 63 11 r r NOUN cana-3154 63 12 = = SYM cana-3154 63 13 { { PUNCT cana-3154 63 14 0 0 NUM cana-3154 63 15 ; ; PUNCT cana-3154 63 16 1 1 NUM cana-3154 63 17 } } PUNCT cana-3154 63 18 , , PUNCT cana-3154 63 19 then then ADV cana-3154 63 20 gc(r gc(r VERB cana-3154 63 21 ) ) PUNCT cana-3154 63 22 is be AUX cana-3154 63 23 a a DET cana-3154 63 24 null null ADJ cana-3154 63 25 graph graph NOUN cana-3154 63 26 . . PUNCT cana-3154 63 27 β€’ β€’ NUM cana-3154 63 28 β‡’ β‡’ NOUN cana-3154 63 29 ) ) PUNCT cana-3154 63 30 let let VERB cana-3154 63 31 1 1 NUM cana-3154 63 32 β‰  β‰  PROPN cana-3154 63 33 x x SYM cana-3154 63 34 ∈ ∈ NOUN cana-3154 63 35 r r NOUN cana-3154 63 36 and and CCONJ cana-3154 63 37 gc(r gc(r NOUN cana-3154 63 38 ) ) PUNCT cana-3154 63 39 a a DET cana-3154 63 40 null null ADJ cana-3154 63 41 graph graph NOUN cana-3154 63 42 . . PUNCT cana-3154 64 1 take take VERB cana-3154 64 2 y y PROPN cana-3154 64 3 ∈ ∈ NOUN cana-3154 64 4 r r NOUN cana-3154 64 5 such such ADJ cana-3154 64 6 that that SCONJ cana-3154 64 7 y y PROPN cana-3154 64 8 β‰  β‰  PROPN cana-3154 64 9 1 1 NUM cana-3154 64 10 and and CCONJ cana-3154 64 11 1βˆ’ 1βˆ’ NUM cana-3154 64 12 y y PROPN cana-3154 64 13 β‰  β‰  PROPN cana-3154 64 14 1 1 NUM cana-3154 64 15 , , PUNCT cana-3154 64 16 then then ADV cana-3154 64 17 xy xy PROPN cana-3154 64 18 = = NOUN cana-3154 64 19 0 0 X cana-3154 64 20 . . PUNCT cana-3154 65 1 by by ADP cana-3154 65 2 lemma lemma PROPN cana-3154 65 3 2.4 2.4 NUM cana-3154 65 4 , , PUNCT cana-3154 65 5 x x PUNCT cana-3154 65 6 = = PUNCT cana-3154 65 7 x x SYM cana-3154 65 8 βˆ’ βˆ’ NOUN cana-3154 65 9 0 0 NUM cana-3154 66 1 = = SYM cana-3154 66 2 xβˆ’ xβˆ’ PROPN cana-3154 66 3 xy xy PROPN cana-3154 67 1 = = PUNCT cana-3154 67 2 x(1βˆ’ x(1βˆ’ PROPN cana-3154 67 3 y y X cana-3154 67 4 ) ) PUNCT cana-3154 68 1 = = NOUN cana-3154 68 2 0 0 X cana-3154 68 3 . . PUNCT cana-3154 69 1 since since SCONJ cana-3154 69 2 r r NOUN cana-3154 69 3 is be AUX cana-3154 69 4 commutative commutative ADJ cana-3154 69 5 nonzero nonzero PROPN cana-3154 69 6 ring ring NOUN cana-3154 69 7 , , PUNCT cana-3154 69 8 then then ADV cana-3154 69 9 r r NOUN cana-3154 69 10 = = SYM cana-3154 69 11 { { PUNCT cana-3154 69 12 0 0 NUM cana-3154 69 13 ; ; PUNCT cana-3154 69 14 1 1 NUM cana-3154 69 15 } } PUNCT cana-3154 69 16 . . PUNCT cana-3154 70 1 example example NOUN cana-3154 70 2 2.6 2.6 NUM cana-3154 70 3 . . PUNCT cana-3154 71 1 if if SCONJ cana-3154 71 2 p p NOUN cana-3154 71 3 is be AUX cana-3154 71 4 prime prime ADJ cana-3154 71 5 integer integer NOUN cana-3154 71 6 , , PUNCT cana-3154 71 7 the the DET cana-3154 71 8 graph graph NOUN cana-3154 71 9 gc(β„€p gc(β„€p NOUN cana-3154 71 10 ) ) PUNCT cana-3154 71 11 is be AUX cana-3154 71 12 null null ADJ cana-3154 71 13 graph graph NOUN cana-3154 71 14 . . PUNCT cana-3154 72 1 lemma lemma PROPN cana-3154 72 2 2.7 2.7 NUM cana-3154 72 3 . . PUNCT cana-3154 73 1 if if SCONJ cana-3154 73 2 r r NOUN cana-3154 73 3 is be AUX cana-3154 73 4 a a DET cana-3154 73 5 domain domain NOUN cana-3154 73 6 , , PUNCT cana-3154 73 7 then then ADV cana-3154 73 8 gc(r gc(r VERB cana-3154 73 9 ) ) PUNCT cana-3154 73 10 is be AUX cana-3154 73 11 a a DET cana-3154 73 12 connected connect VERB cana-3154 73 13 . . PUNCT cana-3154 74 1 proof proof NOUN cana-3154 74 2 : : PUNCT cana-3154 74 3 let let VERB cana-3154 74 4 i i PRON cana-3154 74 5 and and CCONJ cana-3154 74 6 j j PROPN cana-3154 74 7 to to ADP cana-3154 74 8 vertices vertex NOUN cana-3154 74 9 of of ADP cana-3154 74 10 gc(r gc(r NOUN cana-3154 74 11 ) ) PUNCT cana-3154 74 12 . . PUNCT cana-3154 75 1 since since SCONJ cana-3154 75 2 i i PRON cana-3154 75 3 and and CCONJ cana-3154 75 4 j j PROPN cana-3154 75 5 are be AUX cana-3154 75 6 proper proper ADJ cana-3154 75 7 ideals ideal NOUN cana-3154 75 8 of of ADP cana-3154 75 9 r r NOUN cana-3154 75 10 , , PUNCT cana-3154 75 11 there there PRON cana-3154 75 12 exists exist VERB cana-3154 75 13 a a DET cana-3154 75 14 β‰  β‰  PROPN cana-3154 75 15 0 0 NUM cana-3154 75 16 and and CCONJ cana-3154 75 17 b b NOUN cana-3154 75 18 β‰  β‰  PROPN cana-3154 75 19 0 0 NUM cana-3154 75 20 in in ADP cana-3154 75 21 i i PRON cana-3154 75 22 and and CCONJ cana-3154 75 23 j j PROPN cana-3154 75 24 , , PUNCT cana-3154 75 25 respectively respectively ADV cana-3154 75 26 , , PUNCT cana-3154 75 27 such such ADJ cana-3154 75 28 that that SCONJ cana-3154 75 29 ab ab PROPN cana-3154 75 30 β‰  β‰  PROPN cana-3154 75 31 0 0 X cana-3154 75 32 . . PUNCT cana-3154 76 1 so so ADV cana-3154 76 2 , , PUNCT cana-3154 76 3 we we PRON cana-3154 76 4 have have VERB cana-3154 76 5 ab ab PROPN cana-3154 76 6 ∈ ∈ PROPN cana-3154 77 1 i i PRON cana-3154 77 2 ∩ ∩ PROPN cana-3154 77 3 j j PROPN cana-3154 77 4 implies imply VERB cana-3154 77 5 that that SCONJ cana-3154 77 6 i i PRON cana-3154 77 7 ∩ ∩ VERB cana-3154 77 8 j j PROPN cana-3154 77 9 β‰  β‰  PROPN cana-3154 77 10 0 0 NUM cana-3154 77 11 . . PUNCT cana-3154 78 1 if if SCONJ cana-3154 78 2 one one NUM cana-3154 78 3 of of ADP cana-3154 78 4 the the DET cana-3154 78 5 ideals ideal NOUN cana-3154 78 6 i i PRON cana-3154 78 7 or or CCONJ cana-3154 78 8 j j PROPN cana-3154 78 9 is be AUX cana-3154 78 10 principal principal ADJ cana-3154 78 11 , , PUNCT cana-3154 78 12 then then ADV cana-3154 78 13 i i PRON cana-3154 78 14 and and CCONJ cana-3154 78 15 j j PROPN cana-3154 78 16 are be AUX cana-3154 78 17 adjacent adjacent ADJ cana-3154 78 18 . . PUNCT cana-3154 79 1 moreover moreover ADV cana-3154 79 2 , , PUNCT cana-3154 79 3 i i PRON cana-3154 79 4 and and CCONJ cana-3154 79 5 j j PROPN cana-3154 79 6 are be AUX cana-3154 79 7 not not PART cana-3154 79 8 principal principal ADJ cana-3154 79 9 ideals ideal NOUN cana-3154 79 10 . . PUNCT cana-3154 80 1 since since SCONJ cana-3154 80 2 i i PRON cana-3154 80 3 ∩ ∩ NOUN cana-3154 80 4 j j PROPN cana-3154 80 5 β‰  β‰  PROPN cana-3154 80 6 0 0 NUM cana-3154 80 7 , , PUNCT cana-3154 80 8 let let VERB cana-3154 80 9 0 0 NUM cana-3154 80 10 β‰  β‰  PROPN cana-3154 80 11 c c NOUN cana-3154 80 12 ∈ ∈ PROPN cana-3154 80 13 i i PRON cana-3154 80 14 ∩ ∩ ADJ cana-3154 80 15 j j PROPN cana-3154 80 16 and and CCONJ cana-3154 80 17 put put VERB cana-3154 80 18 k k PROPN cana-3154 80 19 = = SYM cana-3154 80 20 cr cr PROPN cana-3154 80 21 . . PUNCT cana-3154 81 1 then then ADV cana-3154 81 2 i i PRON cana-3154 81 3 ∩ ∩ VERB cana-3154 81 4 k k PROPN cana-3154 81 5 β‰  β‰  PROPN cana-3154 81 6 0 0 NUM cana-3154 81 7 and and CCONJ cana-3154 81 8 k k PROPN cana-3154 81 9 ∩ ∩ PROPN cana-3154 81 10 j j PROPN cana-3154 81 11 β‰  β‰  PROPN cana-3154 81 12 0 0 NUM cana-3154 81 13 . . PUNCT cana-3154 82 1 thus thus ADV cana-3154 82 2 i i PRON cana-3154 82 3 βˆ’ βˆ’ PROPN cana-3154 82 4 kβˆ’ kβˆ’ PROPN cana-3154 82 5 j j PROPN cana-3154 82 6 is be AUX cana-3154 82 7 a a DET cana-3154 82 8 path path NOUN cana-3154 82 9 between between ADP cana-3154 82 10 i i PRON cana-3154 82 11 and and CCONJ cana-3154 82 12 j. j. PROPN cana-3154 82 13 3 3 PROPN cana-3154 82 14 . . PUNCT cana-3154 83 1 connectedness connectedness NOUN cana-3154 83 2 , , PUNCT cana-3154 83 3 completeness completeness NOUN cana-3154 83 4 , , PUNCT cana-3154 83 5 hamiltonian hamiltonian ADJ cana-3154 83 6 graph graph NOUN cana-3154 83 7 lemma lemma PROPN cana-3154 83 8 3.1 3.1 NUM cana-3154 83 9 . . PUNCT cana-3154 84 1 if if SCONJ cana-3154 84 2 r r NOUN cana-3154 84 3 is be AUX cana-3154 84 4 a a DET cana-3154 84 5 domain domain NOUN cana-3154 84 6 , , PUNCT cana-3154 84 7 every every DET cana-3154 84 8 connected connected ADJ cana-3154 84 9 graph graph NOUN cana-3154 84 10 gc(r gc(r NOUN cana-3154 84 11 ) ) PUNCT cana-3154 84 12 is be AUX cana-3154 84 13 complete complete ADJ cana-3154 84 14 . . PUNCT cana-3154 85 1 theorem theorem ADJ cana-3154 85 2 3.2 3.2 NUM cana-3154 85 3 . . PUNCT cana-3154 86 1 let let VERB cana-3154 86 2 r r PRON cana-3154 86 3 be be AUX cana-3154 86 4 a a DET cana-3154 86 5 domain domain NOUN cana-3154 86 6 . . PUNCT cana-3154 87 1 the the DET cana-3154 87 2 followings following NOUN cana-3154 87 3 statements statement NOUN cana-3154 87 4 are be AUX cana-3154 87 5 equivalents equivalent NOUN cana-3154 87 6 : : PUNCT cana-3154 87 7 1 1 X cana-3154 87 8 . . NUM cana-3154 87 9 gc(r gc(r NOUN cana-3154 87 10 ) ) PUNCT cana-3154 87 11 is be AUX cana-3154 87 12 a a DET cana-3154 87 13 connected connected ADJ cana-3154 87 14 graph graph NOUN cana-3154 87 15 ; ; PUNCT cana-3154 87 16 2 2 X cana-3154 87 17 . . NUM cana-3154 87 18 gc(r gc(r NOUN cana-3154 87 19 ) ) PUNCT cana-3154 87 20 is be AUX cana-3154 87 21 a a DET cana-3154 87 22 complete complete ADJ cana-3154 87 23 graph graph NOUN cana-3154 87 24 ; ; PUNCT cana-3154 87 25 3 3 X cana-3154 87 26 . . X cana-3154 88 1 r r NOUN cana-3154 88 2 is be AUX cana-3154 88 3 an an DET cana-3154 88 4 ore ore NOUN cana-3154 88 5 domain domain NOUN cana-3154 88 6 . . PUNCT cana-3154 89 1 proof proof NOUN cana-3154 89 2 : : PUNCT cana-3154 89 3 1 1 X cana-3154 89 4 . . PUNCT cana-3154 89 5 β‡’(2 β‡’(2 NOUN cana-3154 89 6 ) ) PUNCT cana-3154 89 7 follows follow VERB cana-3154 89 8 from from ADP cana-3154 89 9 lemma lemma PROPN cana-3154 89 10 3.1 3.1 NUM cana-3154 89 11 2 2 NUM cana-3154 89 12 . . PUNCT cana-3154 89 13 β‡’(3 β‡’(3 NOUN cana-3154 89 14 ) ) PUNCT cana-3154 89 15 let let VERB cana-3154 89 16 a a PRON cana-3154 89 17 and and CCONJ cana-3154 89 18 b b NOUN cana-3154 89 19 to to ADP cana-3154 89 20 non non ADJ cana-3154 89 21 - - ADJ cana-3154 89 22 zero zero NUM cana-3154 89 23 elements element NOUN cana-3154 89 24 in in ADP cana-3154 89 25 r. r. PROPN cana-3154 89 26 put put VERB cana-3154 89 27 on on ADP cana-3154 89 28 i i PROPN cana-3154 89 29 = = SYM cana-3154 89 30 ar ar PROPN cana-3154 89 31 and and CCONJ cana-3154 89 32 j j PROPN cana-3154 90 1 = = SYM cana-3154 90 2 br br PROPN cana-3154 90 3 . . PUNCT cana-3154 91 1 since since SCONJ cana-3154 91 2 gp(r gp(r NOUN cana-3154 91 3 is be AUX cana-3154 91 4 connected connect VERB cana-3154 91 5 , , PUNCT cana-3154 91 6 i i PRON cana-3154 91 7 and and CCONJ cana-3154 91 8 j j PROPN cana-3154 91 9 principal principal ADJ cana-3154 91 10 ideals ideal NOUN cana-3154 91 11 , , PUNCT cana-3154 91 12 then then ADV cana-3154 91 13 i i PRON cana-3154 91 14 ∩ ∩ VERB cana-3154 91 15 j j PROPN cana-3154 91 16 β‰  β‰  PROPN cana-3154 91 17 0 0 NUM cana-3154 91 18 . . PUNCT cana-3154 92 1 hence hence ADV cana-3154 92 2 ar ar NOUN cana-3154 92 3 ∩ ∩ NOUN cana-3154 92 4 br br NOUN cana-3154 92 5 β‰  β‰  PROPN cana-3154 92 6 0 0 NUM cana-3154 92 7 and and CCONJ cana-3154 92 8 ab ab PROPN cana-3154 92 9 β‰  β‰  PROPN cana-3154 92 10 0 0 NUM cana-3154 92 11 . . PUNCT cana-3154 93 1 that that PRON cana-3154 93 2 is is ADV cana-3154 93 3 r r NOUN cana-3154 93 4 is be AUX cana-3154 93 5 an an DET cana-3154 93 6 ore ore NOUN cana-3154 93 7 domain domain NOUN cana-3154 93 8 . . PUNCT cana-3154 94 1 3 3 X cana-3154 94 2 . . X cana-3154 94 3 β‡’ β‡’ NOUN cana-3154 94 4 ( ( PUNCT cana-3154 94 5 1 1 X cana-3154 94 6 ) ) PUNCT cana-3154 94 7 follows follow VERB cana-3154 94 8 from from ADP cana-3154 94 9 lemma lemma PROPN cana-3154 94 10 2.7 2.7 NUM cana-3154 94 11 proposition proposition NOUN cana-3154 94 12 3.3 3.3 NUM cana-3154 94 13 . . PUNCT cana-3154 95 1 let let VERB cana-3154 95 2 r r NOUN cana-3154 95 3 be be AUX cana-3154 95 4 domain domain NOUN cana-3154 95 5 . . PUNCT cana-3154 96 1 if if SCONJ cana-3154 96 2 gc(r gc(r NOUN cana-3154 96 3 ) ) PUNCT cana-3154 96 4 is be AUX cana-3154 96 5 a a DET cana-3154 96 6 connected connected ADJ cana-3154 96 7 graph graph NOUN cana-3154 96 8 , , PUNCT cana-3154 96 9 then then ADV cana-3154 96 10 diam(gc(r diam(gc(r PROPN cana-3154 96 11 ) ) PUNCT cana-3154 96 12 ) ) PUNCT cana-3154 96 13 ≀ ≀ NUM cana-3154 96 14 2 2 NUM cana-3154 96 15 . . X cana-3154 97 1 proof proof NOUN cana-3154 97 2 : : PUNCT cana-3154 97 3 let let VERB cana-3154 97 4 i i PRON cana-3154 97 5 and and CCONJ cana-3154 97 6 j j PROPN cana-3154 97 7 be be VERB cana-3154 97 8 to to ADP cana-3154 97 9 vertices vertex NOUN cana-3154 97 10 of of ADP cana-3154 97 11 gc(r gc(r NOUN cana-3154 97 12 ) ) PUNCT cana-3154 97 13 . . PUNCT cana-3154 98 1 β€’ β€’ INTJ cana-3154 98 2 if if SCONJ cana-3154 98 3 i i PRON cana-3154 98 4 ∩ ∩ VERB cana-3154 98 5 j j PROPN cana-3154 98 6 β‰  β‰  PROPN cana-3154 98 7 0 0 NUM cana-3154 98 8 , , PUNCT cana-3154 98 9 such such ADJ cana-3154 98 10 that that SCONJ cana-3154 98 11 at at ADP cana-3154 98 12 least least ADJ cana-3154 98 13 of of ADP cana-3154 98 14 of of ADP cana-3154 98 15 them they PRON cana-3154 98 16 is be AUX cana-3154 98 17 principal principal ADJ cana-3154 98 18 , , PUNCT cana-3154 98 19 then then ADV cana-3154 98 20 i i PRON cana-3154 98 21 βˆ’ βˆ’ VERB cana-3154 98 22 j. j. PROPN cana-3154 98 23 thus thus ADV cana-3154 98 24 d(i d(i PROPN cana-3154 98 25 , , PUNCT cana-3154 98 26 j j PROPN cana-3154 98 27 ) ) PUNCT cana-3154 98 28 = = SYM cana-3154 99 1 1 1 X cana-3154 99 2 . . NOUN cana-3154 99 3 β€’ β€’ NOUN cana-3154 99 4 if if SCONJ cana-3154 99 5 i i PRON cana-3154 99 6 ∩ ∩ VERB cana-3154 99 7 j j PROPN cana-3154 99 8 β‰  β‰  PROPN cana-3154 99 9 0 0 NUM cana-3154 99 10 , , PUNCT cana-3154 100 1 i i PRON cana-3154 100 2 and and CCONJ cana-3154 100 3 j j PROPN cana-3154 100 4 both both DET cana-3154 100 5 none none NOUN cana-3154 100 6 principal principal NOUN cana-3154 100 7 , , PUNCT cana-3154 100 8 there there PRON cana-3154 100 9 are be VERB cana-3154 100 10 non non ADJ cana-3154 100 11 - - ADJ cana-3154 100 12 zero zero NUM cana-3154 100 13 elements element NOUN cana-3154 100 14 a a PRON cana-3154 100 15 and and CCONJ cana-3154 100 16 b b NOUN cana-3154 100 17 such such ADJ cana-3154 101 1 that that PRON cana-3154 101 2 iβˆ’ iβˆ’ PROPN cana-3154 101 3 cr cr PROPN cana-3154 101 4 βˆ’ βˆ’ PROPN cana-3154 101 5 j j PROPN cana-3154 101 6 with with ADP cana-3154 101 7 c c PROPN cana-3154 101 8 = = SYM cana-3154 101 9 ab ab PROPN cana-3154 101 10 . . PUNCT cana-3154 101 11 thus thus ADV cana-3154 101 12 d(i d(i PROPN cana-3154 101 13 , , PUNCT cana-3154 101 14 j j PROPN cana-3154 101 15 ) ) PUNCT cana-3154 101 16 = = SYM cana-3154 101 17 2 2 X cana-3154 101 18 . . NOUN cana-3154 101 19 β€’ β€’ NOUN cana-3154 101 20 if if SCONJ cana-3154 101 21 i i PRON cana-3154 101 22 ∩ ∩ VERB cana-3154 101 23 j j PROPN cana-3154 101 24 = = NOUN cana-3154 101 25 0 0 PROPN cana-3154 101 26 for for ADP cana-3154 101 27 all all DET cana-3154 101 28 nonzero nonzero NOUN cana-3154 101 29 a a DET cana-3154 101 30 ∈ ∈ PROPN cana-3154 102 1 i i PRON cana-3154 102 2 and and CCONJ cana-3154 102 3 b b PROPN cana-3154 102 4 ∈ ∈ PROPN cana-3154 102 5 j j PROPN cana-3154 102 6 , , PUNCT cana-3154 102 7 ar ar PROPN cana-3154 102 8 ∩ ∩ NOUN cana-3154 102 9 br br NOUN cana-3154 102 10 = = SYM cana-3154 102 11 0 0 NUM cana-3154 102 12 which which PRON cana-3154 102 13 contradicts contradict VERB cana-3154 102 14 theorem theorem ADJ cana-3154 102 15 3.2 3.2 NUM cana-3154 102 16 . . PUNCT cana-3154 103 1 hence hence ADV cana-3154 103 2 , , PUNCT cana-3154 103 3 diam(gc(r diam(gc(r NOUN cana-3154 103 4 ) ) PUNCT cana-3154 103 5 ) ) PUNCT cana-3154 103 6 ≀ ≀ ADV cana-3154 103 7 2 2 NUM cana-3154 103 8 communications communication NOUN cana-3154 103 9 on on ADP cana-3154 103 10 applied apply VERB cana-3154 103 11 nonlinear nonlinear ADJ cana-3154 103 12 analysis analysis NOUN cana-3154 103 13 issn issn NOUN cana-3154 103 14 : : PUNCT cana-3154 103 15 1074 1074 NUM cana-3154 103 16 - - PUNCT cana-3154 103 17 133x 133x NUM cana-3154 103 18 vol vol NOUN cana-3154 103 19 32 32 NUM cana-3154 103 20 no no NOUN cana-3154 103 21 . . PUNCT cana-3154 104 1 5s 5s NUM cana-3154 104 2 ( ( PUNCT cana-3154 104 3 2025 2025 NUM cana-3154 104 4 ) ) PUNCT cana-3154 104 5 484 484 NUM cana-3154 105 1 https://internationalpubls.com https://internationalpubls.com X cana-3154 105 2 proposition proposition NOUN cana-3154 105 3 3.4 3.4 NUM cana-3154 105 4 . . PUNCT cana-3154 106 1 let let VERB cana-3154 106 2 r r PRON cana-3154 106 3 be be AUX cana-3154 106 4 a a DET cana-3154 106 5 ring ring NOUN cana-3154 106 6 . . PUNCT cana-3154 107 1 the the DET cana-3154 107 2 following follow VERB cana-3154 107 3 statements statement NOUN cana-3154 107 4 are be AUX cana-3154 107 5 equivalents equivalent NOUN cana-3154 107 6 . . PUNCT cana-3154 108 1 1 1 X cana-3154 108 2 . . NUM cana-3154 108 3 gc(r gc(r NOUN cana-3154 108 4 ) ) PUNCT cana-3154 108 5 is be AUX cana-3154 108 6 a a DET cana-3154 108 7 complete complete ADJ cana-3154 108 8 graph graph NOUN cana-3154 108 9 ; ; PUNCT cana-3154 108 10 2 2 X cana-3154 108 11 . . X cana-3154 108 12 r r NOUN cana-3154 108 13 is be AUX cana-3154 108 14 essential essential ADJ cana-3154 108 15 and and CCONJ cana-3154 108 16 r r NOUN cana-3154 108 17 has have VERB cana-3154 108 18 at at ADP cana-3154 108 19 most most ADJ cana-3154 108 20 one one NUM cana-3154 108 21 non non ADJ cana-3154 108 22 - - ADJ cana-3154 108 23 principal principal ADJ cana-3154 108 24 ideal ideal NOUN cana-3154 108 25 . . PUNCT cana-3154 109 1 proof proof NOUN cana-3154 109 2 : : PUNCT cana-3154 109 3 β€’ β€’ X cana-3154 109 4 ( ( PUNCT cana-3154 109 5 1)β‡’ 1)β‡’ NUM cana-3154 109 6 ( ( PUNCT cana-3154 109 7 2 2 NUM cana-3154 109 8 ) ) PUNCT cana-3154 109 9 . . PUNCT cana-3154 110 1 assume assume VERB cana-3154 110 2 that that SCONJ cana-3154 110 3 gc(r gc(r NOUN cana-3154 110 4 ) ) PUNCT cana-3154 110 5 is be AUX cana-3154 110 6 complete complete ADJ cana-3154 110 7 and and CCONJ cana-3154 110 8 let let VERB cana-3154 110 9 i i PRON cana-3154 110 10 be be AUX cana-3154 110 11 a a DET cana-3154 110 12 proper proper ADJ cana-3154 110 13 ideal ideal NOUN cana-3154 110 14 of of ADP cana-3154 110 15 r. r. PROPN cana-3154 110 16 by by ADP cana-3154 110 17 definition definition NOUN cana-3154 110 18 , , PUNCT cana-3154 110 19 the the DET cana-3154 110 20 vertex vertex NOUN cana-3154 110 21 i i PRON cana-3154 110 22 is be AUX cana-3154 110 23 adjacent adjacent ADJ cana-3154 110 24 to to ADP cana-3154 110 25 any any DET cana-3154 110 26 others other NOUN cana-3154 110 27 vertex vertex NOUN cana-3154 110 28 , , PUNCT cana-3154 110 29 that that PRON cana-3154 110 30 is is ADV cana-3154 110 31 i i PRON cana-3154 110 32 is be AUX cana-3154 110 33 essential essential ADJ cana-3154 110 34 ideal ideal NOUN cana-3154 110 35 . . PUNCT cana-3154 111 1 then then ADV cana-3154 111 2 r r NOUN cana-3154 111 3 is be AUX cana-3154 111 4 essential essential ADJ cana-3154 111 5 ring ring NOUN cana-3154 111 6 . . PUNCT cana-3154 112 1 assume assume VERB cana-3154 112 2 again again ADV cana-3154 112 3 that that SCONJ cana-3154 112 4 r r NOUN cana-3154 112 5 has have VERB cana-3154 112 6 at at ADV cana-3154 112 7 least least ADV cana-3154 112 8 two two NUM cana-3154 112 9 proper proper ADJ cana-3154 112 10 ideals ideal NOUN cana-3154 112 11 which which PRON cana-3154 112 12 are be AUX cana-3154 112 13 not not PART cana-3154 112 14 principal principal ADJ cana-3154 112 15 . . PUNCT cana-3154 113 1 let let VERB cana-3154 113 2 i1 i1 PROPN cana-3154 113 3 and and CCONJ cana-3154 113 4 i2 i2 PROPN cana-3154 113 5 be be VERB cana-3154 113 6 two two NUM cana-3154 113 7 non non ADJ cana-3154 113 8 principal principal ADJ cana-3154 113 9 ideals ideal NOUN cana-3154 113 10 of of ADP cana-3154 113 11 r. r. PROPN cana-3154 113 12 the the DET cana-3154 113 13 vertices vertex NOUN cana-3154 113 14 i1 i1 PROPN cana-3154 113 15 and and CCONJ cana-3154 113 16 i2 i2 PROPN cana-3154 113 17 can can AUX cana-3154 113 18 not not PART cana-3154 113 19 be be AUX cana-3154 113 20 adjacent adjacent ADJ cana-3154 113 21 ; ; PUNCT cana-3154 113 22 that that PRON cana-3154 113 23 is is ADV cana-3154 113 24 gc(r gc(r NOUN cana-3154 113 25 ) ) PUNCT cana-3154 113 26 is be AUX cana-3154 113 27 not not PART cana-3154 113 28 complete complete ADJ cana-3154 113 29 . . PUNCT cana-3154 114 1 then then ADV cana-3154 114 2 r r NOUN cana-3154 114 3 has have VERB cana-3154 114 4 at at ADP cana-3154 114 5 most most ADJ cana-3154 114 6 one one NUM cana-3154 114 7 non non ADJ cana-3154 114 8 - - ADJ cana-3154 114 9 principal principal ADJ cana-3154 114 10 ideal ideal NOUN cana-3154 114 11 . . PUNCT cana-3154 115 1 β€’ β€’ NUM cana-3154 115 2 ( ( PUNCT cana-3154 115 3 2)β‡’ 2)β‡’ NUM cana-3154 115 4 ( ( PUNCT cana-3154 115 5 1 1 NUM cana-3154 115 6 ) ) PUNCT cana-3154 115 7 . . PUNCT cana-3154 116 1 let let VERB cana-3154 116 2 j j PROPN cana-3154 116 3 and and CCONJ cana-3154 116 4 k k PROPN cana-3154 116 5 be be AUX cana-3154 116 6 two two NUM cana-3154 116 7 vertices vertex NOUN cana-3154 116 8 of of ADP cana-3154 116 9 gc(r gc(r NOUN cana-3154 116 10 ) ) PUNCT cana-3154 116 11 . . PUNCT cana-3154 117 1 since since SCONJ cana-3154 117 2 r r NOUN cana-3154 117 3 is be AUX cana-3154 117 4 essential essential ADJ cana-3154 117 5 , , PUNCT cana-3154 117 6 then then ADV cana-3154 117 7 j j PROPN cana-3154 117 8 ∩ ∩ PROPN cana-3154 117 9 k k PROPN cana-3154 117 10 β‰  β‰  PROPN cana-3154 117 11 0 0 X cana-3154 117 12 . . PUNCT cana-3154 118 1 since since SCONJ cana-3154 118 2 r r NOUN cana-3154 118 3 has have VERB cana-3154 118 4 at at ADP cana-3154 118 5 most most ADJ cana-3154 118 6 one one NUM cana-3154 118 7 non non ADJ cana-3154 118 8 - - ADJ cana-3154 118 9 principal principal ADJ cana-3154 118 10 ideal ideal NOUN cana-3154 118 11 , , PUNCT cana-3154 118 12 we we PRON cana-3154 118 13 have have VERB cana-3154 118 14 two two NUM cana-3154 118 15 possible possible ADJ cana-3154 118 16 cases case NOUN cana-3154 118 17 : : PUNCT cana-3154 118 18 either either CCONJ cana-3154 118 19 j j PROPN cana-3154 118 20 and and CCONJ cana-3154 118 21 k k PROPN cana-3154 118 22 are be AUX cana-3154 118 23 principal principal ADJ cana-3154 118 24 , , PUNCT cana-3154 118 25 or or CCONJ cana-3154 118 26 exactly exactly ADV cana-3154 118 27 one one NUM cana-3154 118 28 between between ADP cana-3154 118 29 j j PROPN cana-3154 118 30 and and CCONJ cana-3154 118 31 k k PROPN cana-3154 118 32 is be AUX cana-3154 118 33 principal principal ADJ cana-3154 118 34 – – PUNCT cana-3154 118 35 if if SCONJ cana-3154 118 36 j j PROPN cana-3154 118 37 and and CCONJ cana-3154 118 38 k k PROPN cana-3154 118 39 are be AUX cana-3154 118 40 principal principal ADJ cana-3154 118 41 , , PUNCT cana-3154 118 42 j j PROPN cana-3154 118 43 and and CCONJ cana-3154 118 44 k k PROPN cana-3154 118 45 adjacent adjacent ADJ cana-3154 118 46 vertices vertex NOUN cana-3154 118 47 . . PUNCT cana-3154 119 1 – – PUNCT cana-3154 119 2 if if SCONJ cana-3154 119 3 one one NUM cana-3154 119 4 between between ADP cana-3154 119 5 j j PROPN cana-3154 119 6 and and CCONJ cana-3154 119 7 k k PROPN cana-3154 119 8 is be AUX cana-3154 119 9 principal principal ADJ cana-3154 119 10 , , PUNCT cana-3154 119 11 j j PROPN cana-3154 119 12 and and CCONJ cana-3154 119 13 k k PROPN cana-3154 119 14 are be AUX cana-3154 119 15 adjacent adjacent ADJ cana-3154 119 16 vertices vertex NOUN cana-3154 119 17 . . PUNCT cana-3154 120 1 β–« β–« VERB cana-3154 120 2 the the DET cana-3154 120 3 result result NOUN cana-3154 120 4 follows follow VERB cana-3154 120 5 . . PUNCT cana-3154 121 1 lemma lemma PROPN cana-3154 121 2 3.5 3.5 NUM cana-3154 121 3 . . PUNCT cana-3154 122 1 the the DET cana-3154 122 2 graph graph NOUN cana-3154 122 3 gc(r gc(r NOUN cana-3154 122 4 ) ) PUNCT cana-3154 122 5 of of ADP cana-3154 122 6 a a DET cana-3154 122 7 principal principal ADJ cana-3154 122 8 ideal ideal ADJ cana-3154 122 9 domain domain NOUN cana-3154 122 10 r r NOUN cana-3154 122 11 is be AUX cana-3154 122 12 a a DET cana-3154 122 13 complete complete ADJ cana-3154 122 14 graph graph NOUN cana-3154 122 15 . . PUNCT cana-3154 123 1 proof proof NOUN cana-3154 123 2 : : PUNCT cana-3154 123 3 let let VERB cana-3154 123 4 i i PRON cana-3154 123 5 and and CCONJ cana-3154 123 6 j j PROPN cana-3154 123 7 two two NUM cana-3154 123 8 proper proper ADJ cana-3154 123 9 ideals ideal NOUN cana-3154 123 10 of of ADP cana-3154 123 11 r. r. PROPN cana-3154 123 12 since since SCONJ cana-3154 123 13 i i PRON cana-3154 123 14 and and CCONJ cana-3154 123 15 j j PROPN cana-3154 123 16 nonzero nonzero PROPN cana-3154 123 17 ideals ideal NOUN cana-3154 123 18 , , PUNCT cana-3154 123 19 there there PRON cana-3154 123 20 is be VERB cana-3154 123 21 non non ADJ cana-3154 123 22 - - ADJ cana-3154 123 23 zero zero NUM cana-3154 123 24 elements element NOUN cana-3154 123 25 a a PRON cana-3154 123 26 and and CCONJ cana-3154 123 27 b b NOUN cana-3154 123 28 in in ADP cana-3154 123 29 r r NOUN cana-3154 123 30 such such ADJ cana-3154 123 31 that that SCONJ cana-3154 123 32 a a DET cana-3154 123 33 ∈ ∈ NOUN cana-3154 123 34 i i PRON cana-3154 123 35 and and CCONJ cana-3154 123 36 b b PROPN cana-3154 123 37 ∈ ∈ PROPN cana-3154 123 38 j. j. PROPN cana-3154 123 39 hence hence PROPN cana-3154 123 40 , , PUNCT cana-3154 123 41 0 0 NUM cana-3154 123 42 β‰  β‰  PROPN cana-3154 123 43 ab ab PROPN cana-3154 123 44 ∈ ∈ PROPN cana-3154 123 45 ar ar NOUN cana-3154 123 46 ∩ ∩ NOUN cana-3154 123 47 br br VERB cana-3154 123 48 βŠ† βŠ† NUM cana-3154 123 49 i i PROPN cana-3154 123 50 ∩ ∩ PROPN cana-3154 123 51 j j PROPN cana-3154 123 52 implies imply VERB cana-3154 123 53 that that SCONJ cana-3154 123 54 i i PRON cana-3154 123 55 and and CCONJ cana-3154 123 56 j j PROPN cana-3154 123 57 are be AUX cana-3154 123 58 adjacent adjacent ADJ cana-3154 123 59 . . PUNCT cana-3154 123 60 example example NOUN cana-3154 124 1 3.6 3.6 NUM cana-3154 124 2 . . PUNCT cana-3154 125 1 the the DET cana-3154 125 2 graph graph NOUN cana-3154 125 3 gc(β„€ gc(β„€ NOUN cana-3154 125 4 ) ) PUNCT cana-3154 125 5 is be AUX cana-3154 125 6 complete complete ADJ cana-3154 125 7 because because SCONJ cana-3154 125 8 for for SCONJ cana-3154 125 9 all all DET cana-3154 125 10 n n CCONJ cana-3154 125 11 , , PUNCT cana-3154 125 12 m m PROPN cana-3154 125 13 ∈ ∈ PROPN cana-3154 125 14 β„€ β„€ PROPN cana-3154 125 15 , , PUNCT cana-3154 125 16 nβ„€ nβ„€ PROPN cana-3154 125 17 ∩mβ„€ ∩mβ„€ NOUN cana-3154 125 18 β‰  β‰  PROPN cana-3154 125 19 0 0 X cana-3154 125 20 . . PUNCT cana-3154 125 21 corollary corollary ADJ cana-3154 125 22 3.7 3.7 NUM cana-3154 125 23 . . PUNCT cana-3154 126 1 if if SCONJ cana-3154 126 2 r r NOUN cana-3154 126 3 a a DET cana-3154 126 4 field field NOUN cana-3154 126 5 , , PUNCT cana-3154 126 6 then then ADV cana-3154 126 7 gc(r[x gc(r[x NOUN cana-3154 126 8 ] ] PUNCT cana-3154 126 9 ) ) PUNCT cana-3154 126 10 is be AUX cana-3154 126 11 a a DET cana-3154 126 12 complete complete ADJ cana-3154 126 13 graph graph NOUN cana-3154 126 14 . . PUNCT cana-3154 127 1 lemma lemma PROPN cana-3154 127 2 3.8 3.8 NUM cana-3154 127 3 . . PUNCT cana-3154 128 1 the the DET cana-3154 128 2 graph graph NOUN cana-3154 128 3 gc(r gc(r NOUN cana-3154 128 4 ) ) PUNCT cana-3154 128 5 of of ADP cana-3154 128 6 an an DET cana-3154 128 7 kΓΆthe kΓΆthe ADJ cana-3154 128 8 ring ring NOUN cana-3154 128 9 r r NOUN cana-3154 128 10 is be AUX cana-3154 128 11 a a DET cana-3154 128 12 complete complete ADJ cana-3154 128 13 graph graph NOUN cana-3154 128 14 . . PUNCT cana-3154 129 1 proof proof NOUN cana-3154 129 2 : : PUNCT cana-3154 129 3 let let VERB cana-3154 129 4 i i PRON cana-3154 129 5 and and CCONJ cana-3154 129 6 j j PROPN cana-3154 129 7 be be VERB cana-3154 129 8 two two NUM cana-3154 129 9 proper proper ADJ cana-3154 129 10 ideals ideal NOUN cana-3154 129 11 of of ADP cana-3154 129 12 r. r. PROPN cana-3154 129 13 since since SCONJ cana-3154 129 14 r r NOUN cana-3154 129 15 is be AUX cana-3154 129 16 a a DET cana-3154 129 17 kΓΆthe kΓΆthe ADJ cana-3154 129 18 ring ring NOUN cana-3154 129 19 , , PUNCT cana-3154 129 20 there there PRON cana-3154 129 21 is be VERB cana-3154 129 22 non non ADJ cana-3154 129 23 - - ADJ cana-3154 129 24 zero zero NUM cana-3154 129 25 elements element NOUN cana-3154 129 26 a a PRON cana-3154 129 27 and and CCONJ cana-3154 129 28 b b NOUN cana-3154 129 29 in in ADP cana-3154 129 30 r r NOUN cana-3154 129 31 such such ADJ cana-3154 129 32 that that SCONJ cana-3154 129 33 a a DET cana-3154 129 34 ∈ ∈ NOUN cana-3154 129 35 i i PRON cana-3154 129 36 and and CCONJ cana-3154 129 37 b b PROPN cana-3154 129 38 ∈ ∈ PROPN cana-3154 129 39 j. j. PROPN cana-3154 129 40 hence hence PROPN cana-3154 129 41 ar ar PROPN cana-3154 129 42 ∩ ∩ PROPN cana-3154 129 43 br br PROPN cana-3154 129 44 βŠ‚ βŠ‚ PROPN cana-3154 129 45 i i PROPN cana-3154 129 46 ∩ ∩ PROPN cana-3154 129 47 j j PROPN cana-3154 129 48 implies imply VERB cana-3154 129 49 that that SCONJ cana-3154 129 50 i i PRON cana-3154 129 51 and and CCONJ cana-3154 129 52 j j PROPN cana-3154 129 53 are be AUX cana-3154 129 54 adjacent adjacent ADJ cana-3154 129 55 . . PUNCT cana-3154 130 1 lemma lemma PROPN cana-3154 130 2 3.9 3.9 NUM cana-3154 130 3 . . NOUN cana-3154 130 4 1 1 NUM cana-3154 130 5 . . NUM cana-3154 130 6 gc(r gc(r NOUN cana-3154 130 7 ) ) PUNCT cana-3154 130 8 is be AUX cana-3154 130 9 a a DET cana-3154 130 10 complete complete ADJ cana-3154 130 11 graph graph NOUN cana-3154 130 12 if if SCONJ cana-3154 131 1 and and CCONJ cana-3154 131 2 only only ADV cana-3154 131 3 if if SCONJ cana-3154 131 4 r r NOUN cana-3154 131 5 is be AUX cana-3154 131 6 an an DET cana-3154 131 7 essential essential ADJ cana-3154 131 8 domain domain NOUN cana-3154 131 9 which which PRON cana-3154 131 10 has have VERB cana-3154 131 11 at at ADP cana-3154 131 12 most most ADV cana-3154 131 13 one one NUM cana-3154 131 14 nonprincipal nonprincipal NOUN cana-3154 131 15 ideal ideal NOUN cana-3154 131 16 . . PUNCT cana-3154 132 1 2 2 X cana-3154 132 2 . . X cana-3154 132 3 if if SCONJ cana-3154 132 4 r r NOUN cana-3154 132 5 has have VERB cana-3154 132 6 more more ADJ cana-3154 132 7 than than ADP cana-3154 132 8 one one NUM cana-3154 132 9 non non ADJ cana-3154 132 10 - - ADJ cana-3154 132 11 principal principal ADJ cana-3154 132 12 ideal ideal NOUN cana-3154 132 13 , , PUNCT cana-3154 132 14 then then ADV cana-3154 132 15 gc(r gc(r VERB cana-3154 132 16 ) ) PUNCT cana-3154 132 17 is be AUX cana-3154 132 18 a a DET cana-3154 132 19 disconnected disconnected ADJ cana-3154 132 20 graph graph NOUN cana-3154 132 21 . . PUNCT cana-3154 133 1 proof proof NOUN cana-3154 133 2 : : PUNCT cana-3154 134 1 1 1 X cana-3154 134 2 . . X cana-3154 135 1 if if SCONJ cana-3154 135 2 gc(r gc(r NOUN cana-3154 135 3 ) ) PUNCT cana-3154 135 4 is be AUX cana-3154 135 5 a a DET cana-3154 135 6 complete complete ADJ cana-3154 135 7 graph graph NOUN cana-3154 135 8 , , PUNCT cana-3154 135 9 by by ADP cana-3154 135 10 proposition proposition NOUN cana-3154 135 11 3.4 3.4 NUM cana-3154 135 12 , , PUNCT cana-3154 135 13 it it PRON cana-3154 135 14 has have VERB cana-3154 135 15 at at ADP cana-3154 135 16 most most ADJ cana-3154 135 17 one one NUM cana-3154 135 18 non non ADJ cana-3154 135 19 - - ADJ cana-3154 135 20 principal principal ADJ cana-3154 135 21 ideal ideal NOUN cana-3154 135 22 . . PUNCT cana-3154 136 1 for for ADP cana-3154 136 2 all all DET cana-3154 136 3 a a DET cana-3154 136 4 and and CCONJ cana-3154 136 5 b b NOUN cana-3154 136 6 two two NUM cana-3154 136 7 non non ADJ cana-3154 136 8 - - ADJ cana-3154 136 9 zero zero NUM cana-3154 136 10 elements element NOUN cana-3154 136 11 in in ADP cana-3154 136 12 r r NOUN cana-3154 136 13 , , PUNCT cana-3154 136 14 ar ar NOUN cana-3154 136 15 ∩ ∩ NOUN cana-3154 136 16 br br PROPN cana-3154 136 17 = = PUNCT cana-3154 136 18 abr abr PROPN cana-3154 136 19 . . PUNCT cana-3154 137 1 since since SCONJ cana-3154 137 2 gc(r gc(r PROPN cana-3154 137 3 ) ) PUNCT cana-3154 137 4 is be AUX cana-3154 137 5 a a DET cana-3154 137 6 complete complete ADJ cana-3154 137 7 , , PUNCT cana-3154 137 8 then then ADV cana-3154 137 9 ab ab PROPN cana-3154 137 10 β‰  β‰  PROPN cana-3154 137 11 0 0 X cana-3154 137 12 . . PUNCT cana-3154 138 1 conversely conversely ADV cana-3154 138 2 , , PUNCT cana-3154 138 3 if if SCONJ cana-3154 138 4 r r NOUN cana-3154 138 5 is be AUX cana-3154 138 6 an an DET cana-3154 138 7 essential essential ADJ cana-3154 138 8 domain domain NOUN cana-3154 138 9 which which PRON cana-3154 138 10 has have VERB cana-3154 138 11 at at ADP cana-3154 138 12 most most ADJ cana-3154 138 13 one one NUM cana-3154 138 14 non non ADJ cana-3154 138 15 - - ADJ cana-3154 138 16 principal principal ADJ cana-3154 138 17 ideal ideal NOUN cana-3154 138 18 , , PUNCT cana-3154 138 19 then then ADV cana-3154 138 20 gc(r gc(r VERB cana-3154 138 21 ) ) PUNCT cana-3154 138 22 is be AUX cana-3154 138 23 complete complete ADJ cana-3154 138 24 by by ADP cana-3154 138 25 proposition proposition NOUN cana-3154 138 26 3.4 3.4 NUM cana-3154 138 27 2 2 NUM cana-3154 138 28 . . PUNCT cana-3154 139 1 it it PRON cana-3154 139 2 is be AUX cana-3154 139 3 clear clear ADJ cana-3154 139 4 that that SCONJ cana-3154 139 5 two two NUM cana-3154 139 6 non non ADJ cana-3154 139 7 - - ADJ cana-3154 139 8 principal principal ADJ cana-3154 139 9 ideals ideal NOUN cana-3154 139 10 of of ADP cana-3154 139 11 r r NOUN cana-3154 139 12 can can AUX cana-3154 139 13 not not PART cana-3154 139 14 be be AUX cana-3154 139 15 adjacent adjacent ADJ cana-3154 139 16 vertices vertex NOUN cana-3154 139 17 of of ADP cana-3154 139 18 the the DET cana-3154 139 19 graph graph NOUN cana-3154 139 20 gc(r gc(r NOUN cana-3154 139 21 ) ) PUNCT cana-3154 139 22 . . PUNCT cana-3154 140 1 β–« β–« ADJ cana-3154 140 2 communications communication NOUN cana-3154 140 3 on on ADP cana-3154 140 4 applied apply VERB cana-3154 140 5 nonlinear nonlinear ADJ cana-3154 140 6 analysis analysis NOUN cana-3154 140 7 issn issn NOUN cana-3154 140 8 : : PUNCT cana-3154 140 9 1074 1074 NUM cana-3154 140 10 - - PUNCT cana-3154 140 11 133x 133x NUM cana-3154 140 12 vol vol NOUN cana-3154 140 13 32 32 NUM cana-3154 140 14 no no NOUN cana-3154 140 15 . . PUNCT cana-3154 141 1 5s 5s NUM cana-3154 141 2 ( ( PUNCT cana-3154 141 3 2025 2025 NUM cana-3154 141 4 ) ) PUNCT cana-3154 141 5 485 485 NUM cana-3154 141 6 https://internationalpubls.com https://internationalpubls.com X cana-3154 141 7 theorem theorem VERB cana-3154 141 8 3.10 3.10 NUM cana-3154 141 9 . . PUNCT cana-3154 142 1 let let VERB cana-3154 142 2 r r PRON cana-3154 142 3 be be AUX cana-3154 142 4 a a DET cana-3154 142 5 bezout bezout NOUN cana-3154 142 6 ring ring NOUN cana-3154 142 7 . . PUNCT cana-3154 143 1 the the DET cana-3154 143 2 followings following NOUN cana-3154 143 3 statements statement NOUN cana-3154 143 4 are be AUX cana-3154 143 5 equivalents equivalent NOUN cana-3154 143 6 : : PUNCT cana-3154 143 7 1 1 X cana-3154 143 8 . . NUM cana-3154 143 9 gc(r gc(r NOUN cana-3154 143 10 ) ) PUNCT cana-3154 143 11 is be AUX cana-3154 143 12 a a DET cana-3154 143 13 complete complete ADJ cana-3154 143 14 graph graph NOUN cana-3154 143 15 ; ; PUNCT cana-3154 143 16 2 2 X cana-3154 143 17 . . X cana-3154 143 18 r r NOUN cana-3154 143 19 is be AUX cana-3154 143 20 a a DET cana-3154 143 21 principal principal ADJ cana-3154 143 22 ideal ideal ADJ cana-3154 143 23 domain domain NOUN cana-3154 143 24 . . PUNCT cana-3154 144 1 proof proof NOUN cana-3154 144 2 : : PUNCT cana-3154 144 3 1 1 X cana-3154 144 4 . . PUNCT cana-3154 144 5 β‡’(2 β‡’(2 PROPN cana-3154 144 6 ) ) PUNCT cana-3154 144 7 . . PUNCT cana-3154 145 1 since since SCONJ cana-3154 145 2 gc(r gc(r NOUN cana-3154 145 3 ) ) PUNCT cana-3154 145 4 is be AUX cana-3154 145 5 a a DET cana-3154 145 6 complete complete ADJ cana-3154 145 7 graph graph NOUN cana-3154 145 8 , , PUNCT cana-3154 145 9 there there PRON cana-3154 145 10 is be VERB cana-3154 145 11 at at ADP cana-3154 145 12 most most ADJ cana-3154 145 13 one one NUM cana-3154 145 14 non non ADJ cana-3154 145 15 - - ADJ cana-3154 145 16 principal principal ADJ cana-3154 145 17 ideal ideal NOUN cana-3154 145 18 . . PUNCT cana-3154 146 1 let let VERB cana-3154 146 2 i i PRON cana-3154 146 3 this this DET cana-3154 146 4 ideal ideal NOUN cana-3154 146 5 . . PUNCT cana-3154 147 1 for for ADP cana-3154 147 2 all all DET cana-3154 147 3 a1 a1 NOUN cana-3154 147 4 ∈ ∈ PROPN cana-3154 147 5 i i PROPN cana-3154 147 6 , , PUNCT cana-3154 147 7 i1 i1 PROPN cana-3154 147 8 = = PUNCT cana-3154 147 9 ra1 ra1 PROPN cana-3154 147 10 is be AUX cana-3154 147 11 adjacent adjacent ADJ cana-3154 147 12 i i PRON cana-3154 147 13 , , PUNCT cana-3154 147 14 that that PRON cana-3154 147 15 is be AUX cana-3154 147 16 i1 i1 PROPN cana-3154 147 17 ∩ ∩ NOUN cana-3154 147 18 i i PRON cana-3154 147 19 β‰  β‰  PROPN cana-3154 147 20 0 0 X cana-3154 147 21 . . PUNCT cana-3154 148 1 if if SCONJ cana-3154 148 2 i i PRON cana-3154 148 3 = = PROPN cana-3154 148 4 i1 i1 PROPN cana-3154 148 5 , , PUNCT cana-3154 148 6 i i PRON cana-3154 148 7 is be AUX cana-3154 148 8 principal principal ADJ cana-3154 148 9 . . PUNCT cana-3154 149 1 otherwise otherwise ADV cana-3154 149 2 , , PUNCT cana-3154 149 3 there there PRON cana-3154 149 4 exist exist VERB cana-3154 149 5 a2 a2 PROPN cana-3154 149 6 ∈ ∈ PROPN cana-3154 150 1 i i PRON cana-3154 150 2 βˆ– βˆ– PROPN cana-3154 150 3 i1 i1 PROPN cana-3154 150 4 and and CCONJ cana-3154 150 5 let let VERB cana-3154 150 6 i2 i2 PROPN cana-3154 150 7 = = SYM cana-3154 150 8 ra1 ra1 PROPN cana-3154 150 9 + + CCONJ cana-3154 151 1 ra2 ra2 PROPN cana-3154 151 2 . . PUNCT cana-3154 152 1 if if SCONJ cana-3154 152 2 i i PRON cana-3154 152 3 β‰  β‰  PROPN cana-3154 152 4 i2 i2 NOUN cana-3154 152 5 = = SYM cana-3154 152 6 ra1 ra1 PROPN cana-3154 152 7 + + CCONJ cana-3154 152 8 ra2 ra2 PROPN cana-3154 152 9 , , PUNCT cana-3154 152 10 there there PRON cana-3154 152 11 exists exist VERB cana-3154 152 12 a3 a3 NOUN cana-3154 152 13 ∈ ∈ PROPN cana-3154 153 1 i i PRON cana-3154 153 2 βˆ– βˆ– NOUN cana-3154 153 3 i2 i2 PROPN cana-3154 153 4 and and CCONJ cana-3154 153 5 let let VERB cana-3154 153 6 i3 i3 NOUN cana-3154 153 7 = = SYM cana-3154 153 8 ra1 ra1 PROPN cana-3154 153 9 + + CCONJ cana-3154 153 10 a2 a2 PROPN cana-3154 153 11 + + CCONJ cana-3154 153 12 ra3 ra3 PROPN cana-3154 153 13 . . PUNCT cana-3154 154 1 inductively inductively ADV cana-3154 154 2 , , PUNCT cana-3154 154 3 let let VERB cana-3154 154 4 in in ADP cana-3154 154 5 = = PUNCT cana-3154 154 6 ra1 ra1 PROPN cana-3154 154 7 + + NOUN cana-3154 154 8 β‹―+ β‹―+ NOUN cana-3154 154 9 ran run VERB cana-3154 154 10 . . PUNCT cana-3154 155 1 if if SCONJ cana-3154 155 2 i i PRON cana-3154 155 3 β‰  β‰  PROPN cana-3154 155 4 in in ADP cana-3154 155 5 , , PUNCT cana-3154 155 6 we we PRON cana-3154 155 7 choose choose VERB cana-3154 155 8 an+1 an+1 NOUN cana-3154 155 9 ∈ ∈ PROPN cana-3154 155 10 i i PRON cana-3154 155 11 βˆ– βˆ– VERB cana-3154 155 12 in in ADP cana-3154 155 13 . . PUNCT cana-3154 156 1 since since SCONJ cana-3154 156 2 gc(r gc(r NOUN cana-3154 156 3 ) ) PUNCT cana-3154 156 4 is be AUX cana-3154 156 5 complete complete ADJ cana-3154 156 6 , , PUNCT cana-3154 156 7 the the DET cana-3154 156 8 chain chain NOUN cana-3154 156 9 i1 i1 PROPN cana-3154 156 10 βˆ’ βˆ’ PROPN cana-3154 156 11 i2 i2 PROPN cana-3154 156 12 βˆ’β‹―βˆ’ βˆ’β‹―βˆ’ ADP cana-3154 156 13 in in ADV cana-3154 156 14 must must AUX cana-3154 156 15 be be AUX cana-3154 156 16 finite finite ADJ cana-3154 156 17 . . PUNCT cana-3154 157 1 moreover moreover ADV cana-3154 157 2 , , PUNCT cana-3154 157 3 the the DET cana-3154 157 4 ideal ideal NOUN cana-3154 157 5 in in ADP cana-3154 157 6 = = PUNCT cana-3154 157 7 ra1 ra1 PROPN cana-3154 158 1 + + NOUN cana-3154 158 2 β‹―+ β‹―+ NOUN cana-3154 158 3 ran run VERB cana-3154 158 4 is be AUX cana-3154 158 5 principal principal ADJ cana-3154 158 6 because because SCONJ cana-3154 158 7 r r NOUN cana-3154 158 8 is be AUX cana-3154 158 9 bezout bezout NOUN cana-3154 158 10 domain domain NOUN cana-3154 158 11 . . PUNCT cana-3154 159 1 this this PRON cana-3154 159 2 is be AUX cana-3154 159 3 a a DET cana-3154 159 4 contradiction contradiction NOUN cana-3154 159 5 . . PUNCT cana-3154 160 1 then then ADV cana-3154 160 2 r r NOUN cana-3154 160 3 is be AUX cana-3154 160 4 a a DET cana-3154 160 5 principal principal ADJ cana-3154 160 6 ideal ideal ADJ cana-3154 160 7 domain domain NOUN cana-3154 160 8 . . PUNCT cana-3154 161 1 2 2 X cana-3154 161 2 . . X cana-3154 161 3 β‡’(1 β‡’(1 NOUN cana-3154 161 4 ) ) PUNCT cana-3154 161 5 follows follow VERB cana-3154 161 6 from from ADP cana-3154 161 7 lemma lemma PROPN cana-3154 161 8 3.5 3.5 NUM cana-3154 161 9 corollary corollary NOUN cana-3154 161 10 3.11 3.11 NUM cana-3154 161 11 . . PUNCT cana-3154 162 1 let let VERB cana-3154 162 2 r r PRON cana-3154 162 3 be be AUX cana-3154 162 4 a a DET cana-3154 162 5 bezout bezout NOUN cana-3154 162 6 domain domain NOUN cana-3154 162 7 . . PUNCT cana-3154 163 1 the the DET cana-3154 163 2 followings following NOUN cana-3154 163 3 statements statement NOUN cana-3154 163 4 are be AUX cana-3154 163 5 equivalents equivalent NOUN cana-3154 163 6 : : PUNCT cana-3154 163 7 1 1 X cana-3154 163 8 . . NUM cana-3154 163 9 gc(r gc(r NOUN cana-3154 163 10 ) ) PUNCT cana-3154 163 11 is be AUX cana-3154 163 12 a a DET cana-3154 163 13 complete complete ADJ cana-3154 163 14 graph graph NOUN cana-3154 163 15 ; ; PUNCT cana-3154 163 16 2 2 X cana-3154 163 17 . . X cana-3154 163 18 r r NOUN cana-3154 163 19 is be AUX cana-3154 163 20 a a DET cana-3154 163 21 principal principal ADJ cana-3154 163 22 ideal ideal ADJ cana-3154 163 23 domain domain NOUN cana-3154 163 24 . . PUNCT cana-3154 164 1 3 3 X cana-3154 164 2 . . X cana-3154 164 3 for for ADP cana-3154 164 4 two two NUM cana-3154 164 5 ideals ideal NOUN cana-3154 164 6 i i PRON cana-3154 164 7 and and CCONJ cana-3154 164 8 j j PROPN cana-3154 164 9 , , PUNCT cana-3154 164 10 i i PROPN cana-3154 164 11 ∩ ∩ NOUN cana-3154 164 12 j j PROPN cana-3154 164 13 = = SYM cana-3154 164 14 0 0 PROPN cana-3154 164 15 implies imply VERB cana-3154 164 16 i i NOUN cana-3154 164 17 = = SYM cana-3154 164 18 0 0 NUM cana-3154 164 19 or or CCONJ cana-3154 164 20 j j PROPN cana-3154 164 21 = = SYM cana-3154 164 22 0 0 PROPN cana-3154 164 23 . . NOUN cana-3154 165 1 4 4 NUM cana-3154 165 2 . . X cana-3154 165 3 for for ADP cana-3154 165 4 all all PRON cana-3154 165 5 ( ( PUNCT cana-3154 165 6 a a PRON cana-3154 165 7 ; ; PUNCT cana-3154 165 8 b b X cana-3154 165 9 ) ) PUNCT cana-3154 165 10 ∈ ∈ PROPN cana-3154 165 11 r2 r2 NOUN cana-3154 165 12 , , PUNCT cana-3154 165 13 ar ar NOUN cana-3154 165 14 ∩ ∩ NOUN cana-3154 165 15 br br NOUN cana-3154 165 16 = = SYM cana-3154 165 17 0 0 NUM cana-3154 165 18 implies imply VERB cana-3154 165 19 a a DET cana-3154 165 20 = = SYM cana-3154 165 21 0 0 NUM cana-3154 165 22 or or CCONJ cana-3154 165 23 b b NOUN cana-3154 165 24 = = SYM cana-3154 165 25 0 0 NUM cana-3154 165 26 . . NOUN cana-3154 165 27 5 5 NUM cana-3154 165 28 . . X cana-3154 166 1 every every DET cana-3154 166 2 non non ADJ cana-3154 166 3 - - ADJ cana-3154 166 4 zero zero NUM cana-3154 166 5 ideal ideal NOUN cana-3154 166 6 of of ADP cana-3154 166 7 r r NOUN cana-3154 166 8 is be AUX cana-3154 166 9 indecomposable indecomposable ADJ cana-3154 166 10 . . PUNCT cana-3154 167 1 remark remark PROPN cana-3154 167 2 3.12 3.12 NUM cana-3154 167 3 . . NOUN cana-3154 168 1 1 1 NUM cana-3154 168 2 . . PUNCT cana-3154 169 1 if if SCONJ cana-3154 169 2 r r NOUN cana-3154 169 3 is be AUX cana-3154 169 4 bezout bezout NOUN cana-3154 169 5 domain domain NOUN cana-3154 169 6 , , PUNCT cana-3154 169 7 for for ADP cana-3154 169 8 all all DET cana-3154 169 9 vertices vertex NOUN cana-3154 169 10 i1 i1 PROPN cana-3154 169 11 , , PUNCT cana-3154 169 12 i2 i2 PROPN cana-3154 169 13 , , PUNCT cana-3154 169 14 β‹― β‹― PROPN cana-3154 169 15 , , PUNCT cana-3154 169 16 in in ADP cana-3154 169 17 ∈ ∈ PROPN cana-3154 169 18 gc(r gc(r PRON cana-3154 169 19 ) ) PUNCT cana-3154 169 20 , , PUNCT cana-3154 169 21 i1 i1 PROPN cana-3154 169 22 βˆ’ βˆ’ PROPN cana-3154 169 23 i2 i2 PROPN cana-3154 169 24 βˆ’β‹―βˆ’ βˆ’β‹―βˆ’ ADV cana-3154 169 25 in in ADP cana-3154 169 26 βˆ’ βˆ’ PROPN cana-3154 169 27 i1 i1 PROPN cana-3154 169 28 is be AUX cana-3154 169 29 a a DET cana-3154 169 30 cycle cycle NOUN cana-3154 169 31 . . PUNCT cana-3154 170 1 2 2 X cana-3154 170 2 . . NUM cana-3154 170 3 girth(gc(r girth(gc(r NOUN cana-3154 170 4 ) ) PUNCT cana-3154 170 5 ) ) PUNCT cana-3154 171 1 = = SYM cana-3154 171 2 3 3 NUM cana-3154 171 3 proposition proposition NOUN cana-3154 171 4 3.13 3.13 NUM cana-3154 171 5 . . PUNCT cana-3154 172 1 if if SCONJ cana-3154 172 2 r r NOUN cana-3154 172 3 is be AUX cana-3154 172 4 bezout bezout NOUN cana-3154 172 5 domain domain NOUN cana-3154 172 6 , , PUNCT cana-3154 172 7 n n PROPN cana-3154 172 8 and and CCONJ cana-3154 172 9 k k PROPN cana-3154 172 10 two two NUM cana-3154 172 11 vertices vertex NOUN cana-3154 172 12 of of ADP cana-3154 172 13 gc(r gc(r NOUN cana-3154 172 14 ) ) PUNCT cana-3154 172 15 such such ADJ cana-3154 172 16 that that SCONJ cana-3154 172 17 k k PROPN cana-3154 172 18 βŠ‚ βŠ‚ PROPN cana-3154 172 19 n n CCONJ cana-3154 172 20 , , PUNCT cana-3154 172 21 then then ADV cana-3154 172 22 deg(k deg(k PROPN cana-3154 172 23 ) ) PUNCT cana-3154 172 24 ≀ ≀ NOUN cana-3154 172 25 deg(n deg(n NOUN cana-3154 172 26 ) ) PUNCT cana-3154 172 27 . . PUNCT cana-3154 173 1 proof proof NOUN cana-3154 173 2 : : PUNCT cana-3154 173 3 let let VERB cana-3154 173 4 n n PRON cana-3154 173 5 and and CCONJ cana-3154 173 6 k k X cana-3154 173 7 two two NUM cana-3154 173 8 vertices vertex NOUN cana-3154 173 9 of of ADP cana-3154 173 10 gc(r gc(r NOUN cana-3154 173 11 ) ) PUNCT cana-3154 173 12 such such ADJ cana-3154 173 13 that that SCONJ cana-3154 173 14 k k PROPN cana-3154 173 15 βŠ‚ βŠ‚ PROPN cana-3154 173 16 n. n. PROPN cana-3154 173 17 if if SCONJ cana-3154 173 18 j j PROPN cana-3154 173 19 is be AUX cana-3154 173 20 another another DET cana-3154 173 21 vertex vertex NOUN cana-3154 173 22 of of ADP cana-3154 173 23 gc(r gc(r PROPN cana-3154 173 24 ) ) PUNCT cana-3154 173 25 then then ADV cana-3154 173 26 j j PROPN cana-3154 173 27 ∩ ∩ PROPN cana-3154 173 28 k k PROPN cana-3154 173 29 β‰  β‰  PROPN cana-3154 173 30 0 0 X cana-3154 173 31 . . PUNCT cana-3154 174 1 since since SCONJ cana-3154 174 2 r r NOUN cana-3154 174 3 is be AUX cana-3154 174 4 bezout bezout ADJ cana-3154 174 5 principal principal ADJ cana-3154 174 6 ideal ideal ADJ cana-3154 174 7 domain domain NOUN cana-3154 174 8 and and CCONJ cana-3154 174 9 j j PROPN cana-3154 174 10 ∩ ∩ PROPN cana-3154 174 11 k k PROPN cana-3154 174 12 βŠ‚ βŠ‚ PROPN cana-3154 174 13 j j PROPN cana-3154 174 14 ∩ ∩ PROPN cana-3154 174 15 j j PROPN cana-3154 174 16 ∩ ∩ PROPN cana-3154 174 17 n n CCONJ cana-3154 174 18 , , PUNCT cana-3154 174 19 then then ADV cana-3154 174 20 j j PROPN cana-3154 174 21 ∩ ∩ PROPN cana-3154 174 22 n n CCONJ cana-3154 174 23 β‰  β‰  PROPN cana-3154 174 24 0 0 NUM cana-3154 174 25 . . PUNCT cana-3154 174 26 theorem theorem VERB cana-3154 174 27 3.14 3.14 NUM cana-3154 174 28 . . PUNCT cana-3154 175 1 the the DET cana-3154 175 2 followings following NOUN cana-3154 175 3 statements statement NOUN cana-3154 175 4 are be AUX cana-3154 175 5 equivalents equivalent NOUN cana-3154 175 6 in in ADP cana-3154 175 7 a a DET cana-3154 175 8 bezout bezout NOUN cana-3154 175 9 domain domain NOUN cana-3154 175 10 r. r. PROPN cana-3154 175 11 1 1 NUM cana-3154 175 12 . . PUNCT cana-3154 175 13 gc(r gc(r PROPN cana-3154 175 14 ) ) PUNCT cana-3154 175 15 is be AUX cana-3154 175 16 a a DET cana-3154 175 17 complete complete ADJ cana-3154 175 18 ; ; PUNCT cana-3154 175 19 2 2 X cana-3154 175 20 . . X cana-3154 176 1 r r NOUN cana-3154 176 2 is be AUX cana-3154 176 3 an an DET cana-3154 176 4 integral integral ADJ cana-3154 176 5 domain domain NOUN cana-3154 176 6 and and CCONJ cana-3154 176 7 has have VERB cana-3154 176 8 at at ADP cana-3154 176 9 most most ADJ cana-3154 176 10 one one NUM cana-3154 176 11 non non ADJ cana-3154 176 12 - - ADJ cana-3154 176 13 principal principal ADJ cana-3154 176 14 ideal ideal NOUN cana-3154 176 15 ; ; PUNCT cana-3154 176 16 3 3 X cana-3154 176 17 . . X cana-3154 177 1 r r NOUN cana-3154 177 2 is be AUX cana-3154 177 3 a a DET cana-3154 177 4 principal principal ADJ cana-3154 177 5 ideal ideal ADJ cana-3154 177 6 domain domain NOUN cana-3154 177 7 . . PUNCT cana-3154 178 1 proof proof NOUN cana-3154 178 2 : : PUNCT cana-3154 178 3 β€’ β€’ X cana-3154 178 4 ( ( PUNCT cana-3154 178 5 1)⇔ 1)⇔ NUM cana-3154 178 6 ( ( PUNCT cana-3154 178 7 2 2 NUM cana-3154 178 8 ) ) PUNCT cana-3154 178 9 follows follow VERB cana-3154 178 10 from from ADP cana-3154 178 11 lemma lemma PROPN cana-3154 178 12 3.9 3.9 NUM cana-3154 178 13 β€’ β€’ NOUN cana-3154 178 14 ( ( PUNCT cana-3154 178 15 3)⇔ 3)⇔ NUM cana-3154 178 16 ( ( PUNCT cana-3154 178 17 1 1 NUM cana-3154 178 18 ) ) PUNCT cana-3154 178 19 follows follow VERB cana-3154 178 20 from from ADP cana-3154 178 21 theorem theorem ADJ cana-3154 178 22 3.10 3.10 NUM cana-3154 178 23 communications communication NOUN cana-3154 178 24 on on ADP cana-3154 178 25 applied apply VERB cana-3154 178 26 nonlinear nonlinear ADJ cana-3154 178 27 analysis analysis NOUN cana-3154 178 28 issn issn NOUN cana-3154 178 29 : : PUNCT cana-3154 178 30 1074 1074 NUM cana-3154 178 31 - - PUNCT cana-3154 178 32 133x 133x NUM cana-3154 178 33 vol vol NOUN cana-3154 178 34 32 32 NUM cana-3154 178 35 no no NOUN cana-3154 178 36 . . PUNCT cana-3154 179 1 5s 5s NUM cana-3154 179 2 ( ( PUNCT cana-3154 179 3 2025 2025 NUM cana-3154 179 4 ) ) PUNCT cana-3154 179 5 486 486 NUM cana-3154 179 6 https://internationalpubls.com https://internationalpubls.com X cana-3154 179 7 corollary corollary NOUN cana-3154 179 8 3.15 3.15 NUM cana-3154 179 9 . . PUNCT cana-3154 180 1 the the DET cana-3154 180 2 graph graph NOUN cana-3154 180 3 gc(r gc(r NOUN cana-3154 180 4 ) ) PUNCT cana-3154 180 5 of of ADP cana-3154 180 6 a a DET cana-3154 180 7 bezout bezout NOUN cana-3154 180 8 domain domain NOUN cana-3154 180 9 is be AUX cana-3154 180 10 a a DET cana-3154 180 11 regular regular ADJ cana-3154 180 12 graph graph NOUN cana-3154 180 13 . . PUNCT cana-3154 181 1 proof proof NOUN cana-3154 181 2 : : PUNCT cana-3154 181 3 since since SCONJ cana-3154 181 4 r r NOUN cana-3154 181 5 is be AUX cana-3154 181 6 a a DET cana-3154 181 7 bezout bezout NOUN cana-3154 181 8 domain domain NOUN cana-3154 181 9 , , PUNCT cana-3154 181 10 gc(r gc(r X cana-3154 181 11 ) ) PUNCT cana-3154 181 12 is be AUX cana-3154 181 13 complete complete ADJ cana-3154 181 14 in in ADP cana-3154 181 15 view view NOUN cana-3154 181 16 of of ADP cana-3154 181 17 theorem theorem NOUN cana-3154 181 18 3.14 3.14 NUM cana-3154 181 19 ; ; PUNCT cana-3154 181 20 then then ADV cana-3154 181 21 gc(r gc(r VERB cana-3154 181 22 ) ) PUNCT cana-3154 181 23 is be AUX cana-3154 181 24 regular regular ADJ cana-3154 181 25 graph graph NOUN cana-3154 181 26 . . PUNCT cana-3154 182 1 proposition proposition NOUN cana-3154 182 2 3.16 3.16 NUM cana-3154 182 3 . . PUNCT cana-3154 183 1 if if SCONJ cana-3154 183 2 r r NOUN cana-3154 183 3 is be AUX cana-3154 183 4 an an DET cana-3154 183 5 ore ore NOUN cana-3154 183 6 domain domain NOUN cana-3154 183 7 with with ADP cana-3154 183 8 k k PROPN cana-3154 183 9 proper proper ADJ cana-3154 183 10 principal principal ADJ cana-3154 183 11 ideals ideal NOUN cana-3154 183 12 , , PUNCT cana-3154 183 13 then then ADV cana-3154 183 14 the the DET cana-3154 183 15 clique clique ADJ cana-3154 183 16 number number NOUN cana-3154 183 17 w(gc(r w(gc(r PROPN cana-3154 183 18 ) ) PUNCT cana-3154 183 19 ) ) PUNCT cana-3154 184 1 = = PUNCT cana-3154 185 1 k. k. PROPN cana-3154 185 2 proof proof NOUN cana-3154 185 3 : : PUNCT cana-3154 185 4 let let VERB cana-3154 185 5 i1 i1 PROPN cana-3154 185 6 , , PUNCT cana-3154 185 7 i2 i2 PROPN cana-3154 185 8 , , PUNCT cana-3154 185 9 … … PUNCT cana-3154 185 10 , , PUNCT cana-3154 185 11 ik ik PROPN cana-3154 185 12 the the DET cana-3154 185 13 k k PROPN cana-3154 185 14 proper proper ADJ cana-3154 185 15 principal principal ADJ cana-3154 185 16 ideals ideal NOUN cana-3154 185 17 , , PUNCT cana-3154 185 18 ik+1 ik+1 NOUN cana-3154 185 19 , , PUNCT cana-3154 185 20 ik+2 ik+2 NOUN cana-3154 185 21 , , PUNCT cana-3154 185 22 … … PUNCT cana-3154 185 23 , , PUNCT cana-3154 185 24 in in ADP cana-3154 185 25 the the DET cana-3154 185 26 nβˆ’ nβˆ’ NOUN cana-3154 185 27 k k PROPN cana-3154 185 28 proper proper ADJ cana-3154 185 29 non non ADJ cana-3154 185 30 - - ADJ cana-3154 185 31 principal principal ADJ cana-3154 185 32 ideals ideal NOUN cana-3154 185 33 of of ADP cana-3154 185 34 r. r. PROPN cana-3154 185 35 since since SCONJ cana-3154 185 36 r r NOUN cana-3154 185 37 is be AUX cana-3154 185 38 an an DET cana-3154 185 39 ore ore NOUN cana-3154 185 40 domain domain NOUN cana-3154 185 41 , , PUNCT cana-3154 185 42 for for ADP cana-3154 185 43 every every DET cana-3154 185 44 vertex vertex NOUN cana-3154 185 45 ii ii NOUN cana-3154 185 46 for for ADP cana-3154 185 47 i i PRON cana-3154 185 48 ∈ ∈ PROPN cana-3154 185 49 { { PUNCT cana-3154 185 50 1,2 1,2 NUM cana-3154 185 51 , , PUNCT cana-3154 185 52 … … PUNCT cana-3154 185 53 , , PUNCT cana-3154 185 54 k k ADJ cana-3154 185 55 } } PUNCT cana-3154 185 56 ii ii PROPN cana-3154 185 57 ∩ ∩ NOUN cana-3154 185 58 ij ij NOUN cana-3154 185 59 β‰  β‰  PROPN cana-3154 185 60 0 0 NUM cana-3154 185 61 for for ADP cana-3154 185 62 j j PROPN cana-3154 185 63 > > X cana-3154 185 64 k. k. PROPN cana-3154 186 1 that that PRON cana-3154 186 2 is be AUX cana-3154 186 3 ii ii NOUN cana-3154 186 4 for for ADP cana-3154 186 5 i i PRON cana-3154 186 6 ∈ ∈ PROPN cana-3154 186 7 { { PUNCT cana-3154 186 8 1,2 1,2 NUM cana-3154 186 9 , , PUNCT cana-3154 186 10 … … PUNCT cana-3154 186 11 , , PUNCT cana-3154 186 12 k k NOUN cana-3154 186 13 } } PUNCT cana-3154 186 14 adjacent adjacent ADJ cana-3154 186 15 to to ADP cana-3154 186 16 each each DET cana-3154 186 17 other other ADJ cana-3154 186 18 vertex vertex NOUN cana-3154 186 19 in in ADP cana-3154 186 20 the the DET cana-3154 186 21 graph graph NOUN cana-3154 186 22 . . PUNCT cana-3154 187 1 then then ADV cana-3154 187 2 the the DET cana-3154 187 3 graph graph NOUN cana-3154 187 4 induced induce VERB cana-3154 187 5 by by ADP cana-3154 187 6 the the DET cana-3154 187 7 path path NOUN cana-3154 187 8 { { PUNCT cana-3154 187 9 ik+1 ik+1 PROPN cana-3154 187 10 , , PUNCT cana-3154 187 11 ik+2 ik+2 NOUN cana-3154 187 12 , , PUNCT cana-3154 187 13 … … PUNCT cana-3154 187 14 , , PUNCT cana-3154 187 15 in in SCONJ cana-3154 187 16 } } PUNCT cana-3154 187 17 is be AUX cana-3154 187 18 complete complete ADJ cana-3154 187 19 . . PUNCT cana-3154 188 1 thus thus ADV cana-3154 188 2 w(gc(r w(gc(r NOUN cana-3154 188 3 ) ) PUNCT cana-3154 188 4 ) ) PUNCT cana-3154 189 1 = = PUNCT cana-3154 190 1 k. k. PROPN cana-3154 190 2 here here ADV cana-3154 190 3 we we PRON cana-3154 190 4 recall recall VERB cana-3154 190 5 a a DET cana-3154 190 6 result result NOUN cana-3154 190 7 from from ADP cana-3154 190 8 [ [ X cana-3154 190 9 5 5 NUM cana-3154 190 10 ] ] PUNCT cana-3154 190 11 theorem theorem VERB cana-3154 190 12 3.17 3.17 NUM cana-3154 190 13 . . PUNCT cana-3154 191 1 let let VERB cana-3154 191 2 r r NOUN cana-3154 191 3 a a DET cana-3154 191 4 domain domain NOUN cana-3154 191 5 with with ADP cana-3154 191 6 k k PROPN cana-3154 191 7 proper proper ADJ cana-3154 191 8 non non ADJ cana-3154 191 9 - - ADJ cana-3154 191 10 principal principal ADJ cana-3154 191 11 ideals ideal NOUN cana-3154 191 12 and and CCONJ cana-3154 191 13 kβ€² kβ€² X cana-3154 191 14 proper proper ADJ cana-3154 191 15 principal principal ADJ cana-3154 191 16 ideals ideal NOUN cana-3154 191 17 of of ADP cana-3154 191 18 r. r. PROPN cana-3154 191 19 the the DET cana-3154 191 20 followings following NOUN cana-3154 191 21 statements statement NOUN cana-3154 191 22 are be AUX cana-3154 191 23 equivalents equivalent NOUN cana-3154 191 24 . . PUNCT cana-3154 192 1 1 1 X cana-3154 192 2 . . NUM cana-3154 192 3 gc(r gc(r NOUN cana-3154 192 4 ) ) PUNCT cana-3154 192 5 is be AUX cana-3154 192 6 a a DET cana-3154 192 7 simple simple ADJ cana-3154 192 8 graph graph NOUN cana-3154 192 9 and and CCONJ cana-3154 192 10 kβ€² kβ€² PROPN cana-3154 192 11 β‰₯ β‰₯ NUM cana-3154 192 12 k k NOUN cana-3154 192 13 ; ; PUNCT cana-3154 192 14 2 2 X cana-3154 192 15 . . NUM cana-3154 192 16 gc(r gc(r NOUN cana-3154 192 17 ) ) PUNCT cana-3154 192 18 is be AUX cana-3154 192 19 hamiltonian hamiltonian ADJ cana-3154 192 20 graph graph NOUN cana-3154 192 21 . . PUNCT cana-3154 193 1 proof proof NOUN cana-3154 193 2 : : PUNCT cana-3154 193 3 ( ( PUNCT cana-3154 193 4 1 1 X cana-3154 193 5 ) ) PUNCT cana-3154 193 6 β‡’ β‡’ NOUN cana-3154 193 7 ( ( PUNCT cana-3154 193 8 2 2 NUM cana-3154 193 9 ) ) PUNCT cana-3154 193 10 . . PUNCT cana-3154 194 1 the the DET cana-3154 194 2 order order NOUN cana-3154 194 3 of of ADP cana-3154 194 4 gc(r gc(r PROPN cana-3154 194 5 ) ) PUNCT cana-3154 194 6 is be AUX cana-3154 194 7 n n NOUN cana-3154 194 8 = = PUNCT cana-3154 194 9 kβ€²+ kβ€²+ PROPN cana-3154 194 10 k. k. PROPN cana-3154 194 11 let let VERB cana-3154 194 12 lk lk PROPN cana-3154 194 13 the the DET cana-3154 194 14 set set NOUN cana-3154 194 15 of of ADP cana-3154 194 16 non non ADJ cana-3154 194 17 - - ADJ cana-3154 194 18 principal principal ADJ cana-3154 194 19 ideals ideal NOUN cana-3154 194 20 , , PUNCT cana-3154 194 21 lkβ€² lkβ€² VERB cana-3154 194 22 the the DET cana-3154 194 23 set set NOUN cana-3154 194 24 of of ADP cana-3154 194 25 principal principal ADJ cana-3154 194 26 ideals ideal NOUN cana-3154 194 27 , , PUNCT cana-3154 194 28 ( ( PUNCT cana-3154 194 29 i i PROPN cana-3154 194 30 , , PUNCT cana-3154 194 31 j j PROPN cana-3154 194 32 ) ) PUNCT cana-3154 194 33 a a DET cana-3154 194 34 pair pair NOUN cana-3154 194 35 of of ADP cana-3154 194 36 non non ADJ cana-3154 194 37 - - ADJ cana-3154 194 38 adjacent adjacent ADJ cana-3154 194 39 vertices vertex NOUN cana-3154 194 40 of of ADP cana-3154 194 41 gc(r gc(r NOUN cana-3154 194 42 ) ) PUNCT cana-3154 194 43 . . PUNCT cana-3154 195 1 three three NUM cana-3154 195 2 cases case NOUN cana-3154 195 3 are be AUX cana-3154 195 4 possibles possible NOUN cana-3154 195 5 : : PUNCT cana-3154 195 6 – – PUNCT cana-3154 195 7 case case NOUN cana-3154 195 8 1 1 NUM cana-3154 195 9 : : PUNCT cana-3154 195 10 if if SCONJ cana-3154 195 11 i i PRON cana-3154 195 12 and and CCONJ cana-3154 195 13 j j PROPN cana-3154 195 14 are be AUX cana-3154 195 15 non non X cana-3154 195 16 principal principal ADJ cana-3154 195 17 ideals ideal NOUN cana-3154 195 18 , , PUNCT cana-3154 195 19 then then ADV cana-3154 195 20 deg(i deg(i PROPN cana-3154 195 21 ) ) PUNCT cana-3154 195 22 + + CCONJ cana-3154 195 23 deg(j deg(j ADJ cana-3154 195 24 ) ) PUNCT cana-3154 195 25 = = SYM cana-3154 195 26 2kβ€² 2kβ€² NUM cana-3154 195 27 β‰₯ β‰₯ NOUN cana-3154 195 28 kβ€²+ kβ€²+ NOUN cana-3154 195 29 k k PROPN cana-3154 195 30 = = PUNCT cana-3154 195 31 n. n. PROPN cana-3154 195 32 – – PUNCT cana-3154 195 33 case case NOUN cana-3154 195 34 2 2 NUM cana-3154 195 35 : : PUNCT cana-3154 195 36 if if SCONJ cana-3154 195 37 i i PRON cana-3154 195 38 and and CCONJ cana-3154 195 39 j j PROPN cana-3154 195 40 are be AUX cana-3154 195 41 principal principal ADJ cana-3154 195 42 ideals ideal NOUN cana-3154 195 43 , , PUNCT cana-3154 195 44 then then ADV cana-3154 195 45 deg(i deg(i PROPN cana-3154 195 46 ) ) PUNCT cana-3154 196 1 + + CCONJ cana-3154 196 2 deg(j deg(j ADJ cana-3154 196 3 ) ) PUNCT cana-3154 196 4 = = PUNCT cana-3154 197 1 2(nβˆ’ 2(nβˆ’ NUM cana-3154 197 2 1 1 NUM cana-3154 197 3 ) ) PUNCT cana-3154 197 4 β‰₯ β‰₯ NOUN cana-3154 197 5 kβ€²+ kβ€²+ NOUN cana-3154 197 6 k k PROPN cana-3154 197 7 = = PUNCT cana-3154 197 8 n. n. ADJ cana-3154 197 9 – – PUNCT cana-3154 197 10 case case NOUN cana-3154 197 11 3 3 NUM cana-3154 197 12 : : PUNCT cana-3154 197 13 if if SCONJ cana-3154 197 14 exactly exactly ADV cana-3154 197 15 one one NUM cana-3154 197 16 ideal ideal NOUN cana-3154 197 17 between between ADP cana-3154 197 18 i i PRON cana-3154 197 19 or or CCONJ cana-3154 197 20 j j PROPN cana-3154 197 21 is be AUX cana-3154 197 22 principal principal ADJ cana-3154 197 23 , , PUNCT cana-3154 197 24 deg(i deg(i PROPN cana-3154 197 25 ) ) PUNCT cana-3154 197 26 + + CCONJ cana-3154 198 1 deg(j deg(j ADJ cana-3154 198 2 ) ) PUNCT cana-3154 198 3 = = VERB cana-3154 198 4 kβ€²+ kβ€²+ NOUN cana-3154 198 5 2(nβˆ’ 2(nβˆ’ NUM cana-3154 198 6 1 1 NUM cana-3154 198 7 ) ) PUNCT cana-3154 198 8 β‰₯ β‰₯ NOUN cana-3154 198 9 n. n. VERB cana-3154 198 10 by by ADP cana-3154 198 11 ore ore NOUN cana-3154 198 12 theorem theorem NOUN cana-3154 198 13 , , PUNCT cana-3154 198 14 gc(r gc(r X cana-3154 198 15 ) ) PUNCT cana-3154 198 16 is be AUX cana-3154 198 17 a a DET cana-3154 198 18 hamiltonian hamiltonian ADJ cana-3154 198 19 graph graph NOUN cana-3154 198 20 . . PUNCT cana-3154 199 1 ( ( PUNCT cana-3154 199 2 2 2 X cana-3154 199 3 ) ) PUNCT cana-3154 199 4 β‡’ β‡’ NOUN cana-3154 199 5 ( ( PUNCT cana-3154 199 6 1 1 NUM cana-3154 199 7 ) ) PUNCT cana-3154 199 8 . . PUNCT cana-3154 200 1 assume assume VERB cana-3154 200 2 that that SCONJ cana-3154 200 3 gc(r gc(r NOUN cana-3154 200 4 ) ) PUNCT cana-3154 200 5 is be AUX cana-3154 200 6 not not PART cana-3154 200 7 a a DET cana-3154 200 8 simple simple ADJ cana-3154 200 9 graph graph NOUN cana-3154 200 10 or or CCONJ cana-3154 200 11 kβ€² kβ€² PROPN cana-3154 200 12 < < X cana-3154 200 13 k k X cana-3154 200 14 ; ; PUNCT cana-3154 200 15 – – PUNCT cana-3154 200 16 if if SCONJ cana-3154 200 17 gc(r gc(r NOUN cana-3154 200 18 ) ) PUNCT cana-3154 200 19 is be AUX cana-3154 200 20 not not PART cana-3154 200 21 a a DET cana-3154 200 22 simple simple ADJ cana-3154 200 23 graph graph NOUN cana-3154 200 24 clearly clearly ADV cana-3154 200 25 gc(r gc(r VERB cana-3154 200 26 ) ) PUNCT cana-3154 200 27 is be AUX cana-3154 200 28 not not PART cana-3154 200 29 hamiltonian hamiltonian ADJ cana-3154 200 30 graph graph NOUN cana-3154 200 31 . . PUNCT cana-3154 201 1 – – PUNCT cana-3154 201 2 if if SCONJ cana-3154 201 3 kβ€² kβ€² PROPN cana-3154 201 4 < < X cana-3154 201 5 k k X cana-3154 201 6 , , PUNCT cana-3154 201 7 since since SCONJ cana-3154 201 8 n n NOUN cana-3154 201 9 = = VERB cana-3154 201 10 k+ k+ X cana-3154 201 11 kβ€² kβ€² PROPN cana-3154 201 12 there there PRON cana-3154 201 13 is be VERB cana-3154 201 14 no no DET cana-3154 201 15 cycle cycle NOUN cana-3154 201 16 containing contain VERB cana-3154 201 17 every every DET cana-3154 201 18 vertex vertex NOUN cana-3154 201 19 of of ADP cana-3154 201 20 gc(r gc(r NOUN cana-3154 201 21 ) ) PUNCT cana-3154 201 22 . . PUNCT cana-3154 202 1 that that PRON cana-3154 202 2 is be AUX cana-3154 202 3 , , PUNCT cana-3154 202 4 there there PRON cana-3154 202 5 is be VERB cana-3154 202 6 no no DET cana-3154 202 7 hamiltonian hamiltonian ADJ cana-3154 202 8 cycle cycle NOUN cana-3154 202 9 . . PUNCT cana-3154 203 1 then then ADV cana-3154 203 2 gc(r gc(r VERB cana-3154 203 3 ) ) PUNCT cana-3154 203 4 is be AUX cana-3154 203 5 not not PART cana-3154 203 6 hamiltonian hamiltonian ADJ cana-3154 203 7 . . PUNCT cana-3154 204 1 acknowledgements acknowledgement NOUN cana-3154 204 2 the the DET cana-3154 204 3 authors author NOUN cana-3154 204 4 would would AUX cana-3154 204 5 like like VERB cana-3154 204 6 to to PART cana-3154 204 7 express express VERB cana-3154 204 8 their their PRON cana-3154 204 9 sincere sincere ADJ cana-3154 204 10 thanks thank NOUN cana-3154 204 11 for for ADP cana-3154 204 12 the the DET cana-3154 204 13 referee referee NOUN cana-3154 204 14 for for ADP cana-3154 204 15 his his PRON cana-3154 204 16 / / SYM cana-3154 204 17 her her PRON cana-3154 204 18 helpful helpful ADJ cana-3154 204 19 suggestions suggestion NOUN cana-3154 204 20 and and CCONJ cana-3154 204 21 comments comment NOUN cana-3154 204 22 . . PUNCT cana-3154 205 1 compliance compliance NOUN cana-3154 205 2 with with ADP cana-3154 205 3 ethical ethical ADJ cana-3154 205 4 standards standard NOUN cana-3154 205 5 conflict conflict NOUN cana-3154 205 6 of of ADP cana-3154 205 7 interest interest NOUN cana-3154 205 8 on on ADP cana-3154 205 9 behalf behalf NOUN cana-3154 205 10 of of ADP cana-3154 205 11 all all DET cana-3154 205 12 authors author NOUN cana-3154 205 13 , , PUNCT cana-3154 205 14 the the DET cana-3154 205 15 corresponding corresponding ADJ cana-3154 205 16 author author NOUN cana-3154 205 17 states state VERB cana-3154 205 18 that that SCONJ cana-3154 205 19 there there PRON cana-3154 205 20 is be VERB cana-3154 205 21 no no DET cana-3154 205 22 conflict conflict NOUN cana-3154 205 23 of of ADP cana-3154 205 24 interest interest NOUN cana-3154 205 25 references reference NOUN cana-3154 205 26 [ [ X cana-3154 205 27 1 1 NUM cana-3154 205 28 ] ] X cana-3154 205 29 ahmed ahmed PROPN cana-3154 205 30 h. h. PROPN cana-3154 205 31 alwan alwan PROPN cana-3154 205 32 . . PUNCT cana-3154 206 1 semisimple semisimple NOUN cana-3154 206 2 - - PUNCT cana-3154 206 3 ntersection ntersection NOUN cana-3154 206 4 graph graph NOUN cana-3154 206 5 of of ADP cana-3154 206 6 ideals ideal NOUN cana-3154 206 7 of of ADP cana-3154 206 8 rings ring NOUN cana-3154 206 9 , , PUNCT cana-3154 206 10 communications communication NOUN cana-3154 206 11 in in ADP cana-3154 206 12 combinatorics combinatoric NOUN cana-3154 206 13 , , PUNCT cana-3154 206 14 cryptography cryptography NOUN cana-3154 206 15 and and CCONJ cana-3154 206 16 computer computer NOUN cana-3154 206 17 science science NOUN cana-3154 206 18 , , PUNCT cana-3154 206 19 2 2 NUM cana-3154 206 20 ( ( PUNCT cana-3154 206 21 2023 2023 NUM cana-3154 206 22 ) ) PUNCT cana-3154 206 23 , , PUNCT cana-3154 206 24 140 140 NUM cana-3154 206 25 - - SYM cana-3154 206 26 148 148 NUM cana-3154 206 27 [ [ X cana-3154 206 28 2 2 NUM cana-3154 206 29 ] ] PUNCT cana-3154 206 30 f. f. PROPN cana-3154 206 31 w. w. PROPN cana-3154 206 32 anderson anderson PROPN cana-3154 206 33 , , PUNCT cana-3154 206 34 k. k. PROPN cana-3154 206 35 r. r. PROPN cana-3154 206 36 fuller fuller PROPN cana-3154 206 37 . . PUNCT cana-3154 207 1 rings ring NOUN cana-3154 207 2 and and CCONJ cana-3154 207 3 categories category NOUN cana-3154 207 4 of of ADP cana-3154 207 5 modules module NOUN cana-3154 207 6 . . PUNCT cana-3154 208 1 second second ADJ cana-3154 208 2 edition edition PROPN cana-3154 208 3 . . PUNCT cana-3154 209 1 springer springer NOUN cana-3154 209 2 - - PUNCT cana-3154 209 3 verlag verlag PROPN cana-3154 209 4 , , PUNCT cana-3154 209 5 1991 1991 NUM cana-3154 209 6 . . PUNCT cana-3154 210 1 communications communication NOUN cana-3154 210 2 on on ADP cana-3154 210 3 applied apply VERB cana-3154 210 4 nonlinear nonlinear ADJ cana-3154 210 5 analysis analysis NOUN cana-3154 210 6 issn issn NOUN cana-3154 210 7 : : PUNCT cana-3154 210 8 1074 1074 NUM cana-3154 210 9 - - PUNCT cana-3154 210 10 133x 133x NUM cana-3154 210 11 vol vol NOUN cana-3154 210 12 32 32 NUM cana-3154 210 13 no no NOUN cana-3154 210 14 . . PUNCT cana-3154 211 1 5s 5s NUM cana-3154 211 2 ( ( PUNCT cana-3154 211 3 2025 2025 NUM cana-3154 211 4 ) ) PUNCT cana-3154 211 5 487 487 NUM cana-3154 211 6 https://internationalpubls.com https://internationalpubls.com X cana-3154 212 1 [ [ X cana-3154 212 2 3 3 X cana-3154 212 3 ] ] X cana-3154 212 4 h. h. PROPN cana-3154 212 5 ansari ansari PROPN cana-3154 212 6 - - PUNCT cana-3154 212 7 toroghy toroghy ADJ cana-3154 212 8 , , PUNCT cana-3154 212 9 f. f. PROPN cana-3154 212 10 farshadifar farshadifar PROPN cana-3154 212 11 , , PUNCT cana-3154 212 12 and and CCONJ cana-3154 212 13 f. f. PROPN cana-3154 212 14 mahboobi mahboobi PROPN cana-3154 212 15 - - PUNCT cana-3154 212 16 abkenar abkenar ADJ cana-3154 212 17 , , PUNCT cana-3154 212 18 small small ADJ cana-3154 212 19 intersection intersection NOUN cana-3154 212 20 graph graph NOUN cana-3154 212 21 of of ADP cana-3154 212 22 multiplicative multiplicative ADJ cana-3154 212 23 modules module NOUN cana-3154 212 24 , , PUNCT cana-3154 212 25 journal journal NOUN cana-3154 212 26 of of ADP cana-3154 212 27 algebra algebra PROPN cana-3154 212 28 and and CCONJ cana-3154 212 29 related related ADJ cana-3154 212 30 topics topic NOUN cana-3154 212 31 , , PUNCT cana-3154 212 32 vol vol NOUN cana-3154 212 33 . . PUNCT cana-3154 213 1 4(2016 4(2016 NUM cana-3154 213 2 ) ) PUNCT cana-3154 213 3 , , PUNCT cana-3154 213 4 pp pp PROPN cana-3154 213 5 . . PUNCT cana-3154 214 1 2 2 NUM cana-3154 214 2 - - SYM cana-3154 214 3 32 32 NUM cana-3154 214 4 . . PUNCT cana-3154 215 1 [ [ X cana-3154 215 2 4 4 X cana-3154 215 3 ] ] PUNCT cana-3154 215 4 m. m. PROPN cana-3154 215 5 f. f. PROPN cana-3154 215 6 atiyah atiyah PROPN cana-3154 215 7 , , PUNCT cana-3154 215 8 i. i. PROPN cana-3154 215 9 g. g. PROPN cana-3154 215 10 macdonald macdonald PROPN cana-3154 215 11 . . PUNCT cana-3154 216 1 introduction introduction NOUN cana-3154 216 2 to to ADP cana-3154 216 3 commutative commutative ADJ cana-3154 216 4 algebra algebra PROPN cana-3154 216 5 . . PUNCT cana-3154 217 1 addison addison PROPN cana-3154 217 2 wesley wesley PROPN cana-3154 217 3 , , PUNCT cana-3154 217 4 reading reading NOUN cana-3154 217 5 , , PUNCT cana-3154 217 6 mass mass PROPN cana-3154 217 7 , , PUNCT cana-3154 217 8 1969 1969 NUM cana-3154 217 9 . . PUNCT cana-3154 218 1 [ [ X cana-3154 218 2 5 5 X cana-3154 218 3 ] ] PUNCT cana-3154 218 4 j. j. PROPN cana-3154 218 5 a. a. PROPN cana-3154 218 6 bondy bondy PROPN cana-3154 218 7 and and CCONJ cana-3154 218 8 u. u. PROPN cana-3154 218 9 s. s. PROPN cana-3154 218 10 r. r. PROPN cana-3154 218 11 murty murty PROPN cana-3154 218 12 , , PUNCT cana-3154 218 13 graph graph NOUN cana-3154 218 14 theory theory NOUN cana-3154 218 15 , , PUNCT cana-3154 218 16 gradute gradute ADJ cana-3154 218 17 text text NOUN cana-3154 218 18 in in ADP cana-3154 218 19 mathematics mathematic NOUN cana-3154 218 20 , , PUNCT cana-3154 218 21 244 244 NUM cana-3154 218 22 , , PUNCT cana-3154 218 23 springer springer NOUN cana-3154 218 24 , , PUNCT cana-3154 218 25 new new PROPN cana-3154 218 26 york york PROPN cana-3154 218 27 ( ( PUNCT cana-3154 218 28 1964 1964 NUM cana-3154 218 29 ) ) PUNCT cana-3154 218 30 199125 199125 NUM cana-3154 218 31 . . PUNCT cana-3154 219 1 [ [ X cana-3154 219 2 6 6 NUM cana-3154 219 3 ] ] PUNCT cana-3154 219 4 j. j. PROPN cana-3154 219 5 bosak bosak PROPN cana-3154 219 6 , , PUNCT cana-3154 219 7 the the DET cana-3154 219 8 graph graph NOUN cana-3154 219 9 of of ADP cana-3154 219 10 semigroups semigroup NOUN cana-3154 219 11 , , PUNCT cana-3154 219 12 in in ADP cana-3154 219 13 theory theory NOUN cana-3154 219 14 of of ADP cana-3154 219 15 graphs graph NOUN cana-3154 219 16 and and CCONJ cana-3154 219 17 its its PRON cana-3154 219 18 applications application NOUN cana-3154 219 19 , , PUNCT cana-3154 219 20 ( ( PUNCT cana-3154 219 21 academic academic ADJ cana-3154 219 22 press press NOUN cana-3154 219 23 , , PUNCT cana-3154 219 24 new new PROPN cana-3154 219 25 york york PROPN cana-3154 219 26 , , PUNCT cana-3154 219 27 1964 1964 NUM cana-3154 219 28 ) ) PUNCT cana-3154 219 29 , , PUNCT cana-3154 219 30 pp pp ADP cana-3154 219 31 . . PUNCT cana-3154 220 1 119 119 NUM cana-3154 220 2 - - SYM cana-3154 220 3 125 125 NUM cana-3154 220 4 . . PUNCT cana-3154 221 1 [ [ X cana-3154 221 2 7 7 X cana-3154 221 3 ] ] X cana-3154 221 4 i. i. PROPN cana-3154 221 5 chakrabarty chakrabarty PROPN cana-3154 221 6 , , PUNCT cana-3154 221 7 s. s. PROPN cana-3154 221 8 ghosh ghosh PROPN cana-3154 221 9 , , PUNCT cana-3154 221 10 t. t. PROPN cana-3154 221 11 k. k. PROPN cana-3154 221 12 mukherjee mukherjee PROPN cana-3154 221 13 and and CCONJ cana-3154 221 14 m. m. PROPN cana-3154 221 15 k. k. PROPN cana-3154 221 16 sen sen PROPN cana-3154 221 17 , , PUNCT cana-3154 221 18 intersection intersection NOUN cana-3154 221 19 graph graph NOUN cana-3154 221 20 of of ADP cana-3154 221 21 ideals ideal NOUN cana-3154 221 22 of of ADP cana-3154 221 23 rings ring NOUN cana-3154 221 24 , , PUNCT cana-3154 221 25 discrete discrete ADJ cana-3154 221 26 math math NOUN cana-3154 221 27 . . PUNCT cana-3154 221 28 , , PUNCT cana-3154 221 29 23 23 NUM cana-3154 221 30 ( ( PUNCT cana-3154 221 31 2005 2005 NUM cana-3154 221 32 ) ) PUNCT cana-3154 221 33 23 23 NUM cana-3154 221 34 - - SYM cana-3154 221 35 32 32 NUM cana-3154 221 36 . . PUNCT cana-3154 222 1 [ [ X cana-3154 222 2 8 8 NUM cana-3154 222 3 ] ] X cana-3154 222 4 w. w. PROPN cana-3154 222 5 k. k. PROPN cana-3154 222 6 nicholson nicholson PROPN cana-3154 222 7 , , PUNCT cana-3154 222 8 m. m. PROPN cana-3154 222 9 f. f. PROPN cana-3154 222 10 yousif yousif PROPN cana-3154 222 11 ; ; PUNCT cana-3154 222 12 quasi quasi ADJ cana-3154 222 13 - - ADJ cana-3154 222 14 robenius robenius NOUN cana-3154 222 15 rings ring NOUN cana-3154 222 16 , , PUNCT cana-3154 222 17 cambridge cambridge PROPN cana-3154 222 18 university university PROPN cana-3154 222 19 press press NOUN cana-3154 222 20 , , PUNCT cana-3154 222 21 new new PROPN cana-3154 222 22 york york PROPN cana-3154 222 23 2003 2003 NUM cana-3154 222 24 . . PUNCT cana-3154 223 1 [ [ X cana-3154 223 2 9 9 NUM cana-3154 223 3 ] ] PUNCT cana-3154 223 4 m. m. NOUN cana-3154 223 5 j. j. PROPN cana-3154 223 6 nikmehr nikmehr PROPN cana-3154 223 7 , , PUNCT cana-3154 223 8 b. b. PROPN cana-3154 223 9 soleymanzadeh soleymanzadeh PROPN cana-3154 223 10 ; ; PUNCT cana-3154 223 11 the the DET cana-3154 223 12 prime prime ADJ cana-3154 223 13 intersection intersection NOUN cana-3154 223 14 graph graph NOUN cana-3154 223 15 of of ADP cana-3154 223 16 ideal ideal NOUN cana-3154 223 17 of of ADP cana-3154 223 18 a a DET cana-3154 223 19 ring ring NOUN cana-3154 223 20 ; ; PUNCT cana-3154 223 21 comm comm NOUN cana-3154 223 22 . . PUNCT cana-3154 223 23 math math NOUN cana-3154 223 24 . . PUNCT cana-3154 224 1 univ univ PROPN cana-3154 224 2 . . PUNCT cana-3154 224 3 carol carol PROPN cana-3154 224 4 . . PUNCT cana-3154 225 1 ( ( PUNCT cana-3154 225 2 2017 2017 NUM cana-3154 225 3 ) ) PUNCT cana-3154 225 4 137 137 NUM cana-3154 225 5 - - SYM cana-3154 225 6 145 145 NUM cana-3154 225 7 . . PUNCT cana-3154 226 1 [ [ X cana-3154 226 2 10 10 NUM cana-3154 226 3 ] ] X cana-3154 226 4 r. r. PROPN cana-3154 226 5 wisbauer wisbauer NOUN cana-3154 226 6 , , PUNCT cana-3154 226 7 foundations foundation NOUN cana-3154 226 8 of of ADP cana-3154 226 9 module module NOUN cana-3154 226 10 and and CCONJ cana-3154 226 11 ring ring NOUN cana-3154 226 12 theory theory NOUN cana-3154 226 13 , , PUNCT cana-3154 226 14 a a DET cana-3154 226 15 hendbook hendbook NOUN cana-3154 226 16 for for ADP cana-3154 226 17 study study NOUN cana-3154 226 18 and and CCONJ cana-3154 226 19 research research NOUN cana-3154 226 20 , , PUNCT cana-3154 226 21 gordon gordon PROPN cana-3154 226 22 and and CCONJ cana-3154 226 23 breach breach VERB cana-3154 226 24 science science NOUN cana-3154 226 25 publishers publisher NOUN cana-3154 226 26 , , PUNCT cana-3154 226 27 reading reading NOUN cana-3154 226 28 . . PUNCT cana-3154 227 1 1991 1991 NUM cana-3154 227 2 . . PUNCT cana-3154 228 1 [ [ X cana-3154 228 2 11 11 NUM cana-3154 228 3 ] ] X cana-3154 228 4 f. f. PROPN cana-3154 228 5 yaraneri yaraneri PROPN cana-3154 228 6 , , PUNCT cana-3154 228 7 intersection intersection NOUN cana-3154 228 8 graph graph NOUN cana-3154 228 9 of of ADP cana-3154 228 10 a a DET cana-3154 228 11 module module NOUN cana-3154 228 12 , , PUNCT cana-3154 228 13 journal journal NOUN cana-3154 228 14 of of ADP cana-3154 228 15 algebra algebra NOUN cana-3154 228 16 and and CCONJ cana-3154 228 17 applications application NOUN cana-3154 228 18 , , PUNCT cana-3154 228 19 12 12 NUM cana-3154 228 20 ( ( PUNCT cana-3154 228 21 2013 2013 NUM cana-3154 228 22 ) ) PUNCT cana-3154 228 23 125 125 NUM cana-3154 228 24 208 208 NUM cana-3154 228 25 . . PUNCT