id sid tid token lemma pos cana-3900 1 1 communications communication NOUN cana-3900 1 2 on on ADP cana-3900 1 3 applied apply VERB cana-3900 1 4 nonlinear nonlinear ADJ cana-3900 1 5 analysis analysis NOUN cana-3900 1 6 issn issn NOUN cana-3900 1 7 : : PUNCT cana-3900 1 8 1074 1074 NUM cana-3900 1 9 - - PUNCT cana-3900 1 10 133x 133x NUM cana-3900 1 11 vol vol NOUN cana-3900 1 12 32 32 NUM cana-3900 1 13 no no NOUN cana-3900 1 14 . . PUNCT cana-3900 2 1 9s 9s NUM cana-3900 2 2 ( ( PUNCT cana-3900 2 3 2025 2025 NUM cana-3900 2 4 ) ) PUNCT cana-3900 2 5 345 345 NUM cana-3900 2 6 https://internationalpubls.com https://internationalpubls.com X cana-3900 3 1 some some DET cana-3900 3 2 special special ADJ cana-3900 3 3 structures structure NOUN cana-3900 3 4 of of ADP cana-3900 3 5 𝜹𝟏 𝜹𝟏 NOUN cana-3900 3 6 near near ADJ cana-3900 3 7 - - PUNCT cana-3900 3 8 rings ring NOUN cana-3900 3 9 1s.sivanthi 1s.sivanthi NUM cana-3900 3 10 , , PUNCT cana-3900 3 11 2g.sugantha 2g.sugantha NUM cana-3900 3 12 , , PUNCT cana-3900 3 13 3 3 NUM cana-3900 3 14 m.amirthakodi m.amirthakodi NOUN cana-3900 3 15 1research 1research NUM cana-3900 3 16 scholar scholar NOUN cana-3900 3 17 of of ADP cana-3900 3 18 mathematics mathematic NOUN cana-3900 3 19 ( ( PUNCT cana-3900 3 20 part part NOUN cana-3900 3 21 time time NOUN cana-3900 3 22 ) ) PUNCT cana-3900 3 23 , , PUNCT cana-3900 3 24 reg reg VERB cana-3900 3 25 no no PRON cana-3900 3 26 : : PUNCT cana-3900 3 27 21122102092007 21122102092007 NUM cana-3900 3 28 , , PUNCT cana-3900 3 29 email email NOUN cana-3900 3 30 : : PUNCT cana-3900 4 1 sssivanthi@gmail.com sssivanthi@gmail.com X cana-3900 4 2 pg pg NOUN cana-3900 4 3 and and CCONJ cana-3900 4 4 research research PROPN cana-3900 4 5 department department PROPN cana-3900 4 6 of of ADP cana-3900 4 7 mathematics mathematic NOUN cana-3900 4 8 , , PUNCT cana-3900 4 9 kamaraj kamaraj ADJ cana-3900 4 10 college college NOUN cana-3900 4 11 , , PUNCT cana-3900 4 12 thoothukudi thoothukudi PROPN cana-3900 4 13 – – PUNCT cana-3900 4 14 628003 628003 NUM cana-3900 4 15 ( ( PUNCT cana-3900 4 16 affiliated affiliate VERB cana-3900 4 17 to to ADP cana-3900 4 18 manonmaniam manonmaniam PROPN cana-3900 4 19 sundaranar sundaranar PROPN cana-3900 4 20 university university PROPN cana-3900 4 21 , , PUNCT cana-3900 4 22 abishekapatti abishekapatti ADJ cana-3900 4 23 , , PUNCT cana-3900 4 24 tirunelveli tirunelveli ADJ cana-3900 4 25 – – PUNCT cana-3900 4 26 627012 627012 NUM cana-3900 4 27 ) ) PUNCT cana-3900 5 1 2assistant 2assistant NUM cana-3900 5 2 professor professor NOUN cana-3900 5 3 of of ADP cana-3900 5 4 mathematics mathematics PROPN cana-3900 5 5 , , PUNCT cana-3900 5 6 pope pope PROPN cana-3900 5 7 ’s ’s PART cana-3900 5 8 college college PROPN cana-3900 5 9 ( ( PUNCT cana-3900 5 10 autonomous autonomous ADJ cana-3900 5 11 ) ) PUNCT cana-3900 5 12 , , PUNCT cana-3900 5 13 sawyerpuram sawyerpuram NOUN cana-3900 5 14 , , PUNCT cana-3900 5 15 tamil tamil PROPN cana-3900 5 16 nadu nadu NOUN cana-3900 5 17 627 627 NUM cana-3900 5 18 251 251 NUM cana-3900 5 19 , , PUNCT cana-3900 5 20 india india PROPN cana-3900 5 21 . . PUNCT cana-3900 6 1 e.mail e.mail ADV cana-3900 6 2 : : PUNCT cana-3900 7 1 sugi.trini@gmail.com sugi.trini@gmail.com X cana-3900 7 2 ( ( PUNCT cana-3900 7 3 affiliated affiliate VERB cana-3900 7 4 to to ADP cana-3900 7 5 manonmaniam manonmaniam PROPN cana-3900 7 6 sundaranar sundaranar PROPN cana-3900 7 7 university university PROPN cana-3900 7 8 , , PUNCT cana-3900 7 9 abishekapatti abishekapatti ADJ cana-3900 7 10 , , PUNCT cana-3900 7 11 tirunelveli tirunelveli ADJ cana-3900 7 12 – – PUNCT cana-3900 7 13 627012 627012 NUM cana-3900 7 14 ) ) PUNCT cana-3900 8 1 3assistant 3assistant NUM cana-3900 8 2 professor professor NOUN cana-3900 8 3 of of ADP cana-3900 8 4 mathematics mathematic NOUN cana-3900 8 5 , , PUNCT cana-3900 8 6 pg pg NOUN cana-3900 8 7 and and CCONJ cana-3900 8 8 research research PROPN cana-3900 8 9 department department PROPN cana-3900 8 10 of of ADP cana-3900 8 11 mathematics mathematic NOUN cana-3900 8 12 , , PUNCT cana-3900 8 13 kamaraj kamaraj ADJ cana-3900 8 14 college college NOUN cana-3900 8 15 , , PUNCT cana-3900 8 16 thoothukudi thoothukudi PROPN cana-3900 8 17 – – PUNCT cana-3900 8 18 628003 628003 NUM cana-3900 8 19 . . PUNCT cana-3900 9 1 e.mail:amirthakodim@yahoo.com e.mail:amirthakodim@yahoo.com X cana-3900 9 2 ( ( PUNCT cana-3900 9 3 affiliated affiliate VERB cana-3900 9 4 to to ADP cana-3900 9 5 manonmaniam manonmaniam PROPN cana-3900 9 6 sundaranar sundaranar PROPN cana-3900 9 7 university university PROPN cana-3900 9 8 , , PUNCT cana-3900 9 9 abishekapatti abishekapatti ADJ cana-3900 9 10 , , PUNCT cana-3900 9 11 tirunelveli tirunelveli ADJ cana-3900 9 12 – – PUNCT cana-3900 9 13 627012 627012 NUM cana-3900 9 14 ) ) PUNCT cana-3900 9 15 article article NOUN cana-3900 9 16 history history NOUN cana-3900 9 17 : : PUNCT cana-3900 9 18 received receive VERB cana-3900 9 19 : : PUNCT cana-3900 9 20 14 14 NUM cana-3900 9 21 - - SYM cana-3900 9 22 11 11 NUM cana-3900 9 23 - - PUNCT cana-3900 9 24 2024 2024 NUM cana-3900 9 25 revised revise VERB cana-3900 9 26 : : PUNCT cana-3900 9 27 25 25 NUM cana-3900 9 28 - - SYM cana-3900 9 29 12 12 NUM cana-3900 9 30 - - PUNCT cana-3900 9 31 2024 2024 NUM cana-3900 9 32 accepted:09 accepted:09 NOUN cana-3900 9 33 - - PUNCT cana-3900 9 34 01 01 NUM cana-3900 9 35 - - PUNCT cana-3900 9 36 2025 2025 NUM cana-3900 9 37 abstract abstract NOUN cana-3900 9 38 : : PUNCT cana-3900 9 39 if if SCONJ cana-3900 9 40 , , PUNCT cana-3900 9 41 in in ADP cana-3900 9 42 a a DET cana-3900 9 43 ring ring NOUN cana-3900 9 44 ( ( PUNCT cana-3900 9 45 n n CCONJ cana-3900 9 46 , , PUNCT cana-3900 9 47 + + ADV cana-3900 9 48 , , PUNCT cana-3900 9 49 βˆ™ βˆ™ PROPN cana-3900 9 50 ) ) PUNCT cana-3900 9 51 we we PRON cana-3900 9 52 ignore ignore VERB cana-3900 9 53 the the DET cana-3900 9 54 commutativity commutativity NOUN cana-3900 9 55 of of ADP cana-3900 9 56 β€˜ ' PUNCT cana-3900 9 57 + + NOUN cana-3900 9 58 ’ ' PUNCT cana-3900 9 59 and and CCONJ cana-3900 9 60 one one NUM cana-3900 9 61 of of ADP cana-3900 9 62 the the DET cana-3900 9 63 distributive distributive ADJ cana-3900 9 64 laws law NOUN cana-3900 9 65 , , PUNCT cana-3900 9 66 ( ( PUNCT cana-3900 9 67 n n CCONJ cana-3900 9 68 , , PUNCT cana-3900 9 69 + + ADV cana-3900 9 70 , , PUNCT cana-3900 9 71 βˆ™ βˆ™ PROPN cana-3900 9 72 ) ) PUNCT cana-3900 9 73 becomes become VERB cana-3900 9 74 a a DET cana-3900 9 75 near near ADJ cana-3900 9 76 - - PUNCT cana-3900 9 77 ring ring NOUN cana-3900 9 78 . . PUNCT cana-3900 10 1 if if SCONJ cana-3900 10 2 we we PRON cana-3900 10 3 do do AUX cana-3900 10 4 not not PART cana-3900 10 5 stipulate stipulate VERB cana-3900 10 6 the the DET cana-3900 10 7 left left ADJ cana-3900 10 8 distributive distributive ADJ cana-3900 10 9 law law NOUN cana-3900 10 10 , , PUNCT cana-3900 10 11 ( ( PUNCT cana-3900 10 12 n n CCONJ cana-3900 10 13 , , PUNCT cana-3900 10 14 + + ADV cana-3900 10 15 , , PUNCT cana-3900 10 16 βˆ™ βˆ™ PROPN cana-3900 10 17 ) ) PUNCT cana-3900 10 18 is be AUX cana-3900 10 19 a a DET cana-3900 10 20 right right ADJ cana-3900 10 21 nearring nearring NOUN cana-3900 10 22 . . PUNCT cana-3900 11 1 this this DET cana-3900 11 2 research research NOUN cana-3900 11 3 aims aim VERB cana-3900 11 4 to to PART cana-3900 11 5 introduce introduce VERB cana-3900 11 6 the the DET cana-3900 11 7 concept concept NOUN cana-3900 11 8 of of ADP cana-3900 11 9 Ξ΄_1 Ξ΄_1 NOUN cana-3900 11 10 near near ADP cana-3900 11 11 - - PUNCT cana-3900 11 12 ring ring NOUN cana-3900 11 13 . . PUNCT cana-3900 12 1 for for ADP cana-3900 12 2 every every DET cana-3900 12 3 x x PROPN cana-3900 12 4 , , PUNCT cana-3900 12 5 y y PROPN cana-3900 12 6 in in ADP cana-3900 12 7 n n CCONJ cana-3900 12 8 , , PUNCT cana-3900 12 9 xny xny PROPN cana-3900 12 10 = = NOUN cana-3900 12 11 nx^2 nx^2 NOUN cana-3900 12 12 y^2 y^2 PROPN cana-3900 12 13 is be AUX cana-3900 12 14 called call VERB cana-3900 12 15 Ξ΄_1near Ξ΄_1near NOUN cana-3900 12 16 - - NOUN cana-3900 12 17 ring ring NOUN cana-3900 12 18 . . PUNCT cana-3900 13 1 the the DET cana-3900 13 2 element element ADJ cana-3900 13 3 wise wise ADJ cana-3900 13 4 characterization characterization NOUN cana-3900 13 5 forγ€– forγ€– PROPN cana-3900 13 6 Ξ΄γ€—_1 Ξ΄γ€—_1 NOUN cana-3900 13 7 nearring nearre VERB cana-3900 13 8 will will AUX cana-3900 13 9 be be AUX cana-3900 13 10 investigated investigate VERB cana-3900 13 11 and and CCONJ cana-3900 13 12 shall shall AUX cana-3900 13 13 establish establish VERB cana-3900 13 14 theorems theorem NOUN cana-3900 13 15 and and CCONJ cana-3900 13 16 properties property NOUN cana-3900 13 17 in in ADP cana-3900 13 18 this this DET cana-3900 13 19 near near NOUN cana-3900 13 20 - - PUNCT cana-3900 13 21 ring ring NOUN cana-3900 13 22 . . PUNCT cana-3900 14 1 mathematics mathematic NOUN cana-3900 14 2 subject subject ADJ cana-3900 14 3 classification classification NOUN cana-3900 14 4 : : PUNCT cana-3900 14 5 16y30 16y30 NUM cana-3900 14 6 keywords keyword NOUN cana-3900 14 7 : : PUNCT cana-3900 14 8 Ξ΄_1near Ξ΄_1near NOUN cana-3900 14 9 - - NOUN cana-3900 14 10 ring ring NOUN cana-3900 14 11 , , PUNCT cana-3900 14 12 near near ADJ cana-3900 14 13 - - PUNCT cana-3900 14 14 field field NOUN cana-3900 14 15 . . PUNCT cana-3900 15 1 1 1 NUM cana-3900 15 2 introduction introduction NOUN cana-3900 15 3 a a DET cana-3900 15 4 right right ADJ cana-3900 15 5 near near NOUN cana-3900 15 6 - - PUNCT cana-3900 15 7 ring ring NOUN cana-3900 15 8 ( ( PUNCT cana-3900 15 9 n n CCONJ cana-3900 15 10 , , PUNCT cana-3900 15 11 + + ADV cana-3900 15 12 , , PUNCT cana-3900 15 13 βˆ™ βˆ™ PROPN cana-3900 15 14 ) ) PUNCT cana-3900 15 15 is be AUX cana-3900 15 16 an an DET cana-3900 15 17 algebraic algebraic ADJ cana-3900 15 18 system system NOUN cana-3900 15 19 with with ADP cana-3900 15 20 two two NUM cana-3900 15 21 binary binary ADJ cana-3900 15 22 operations operation NOUN cana-3900 15 23 such such ADJ cana-3900 15 24 that that SCONJ cana-3900 15 25 ( ( PUNCT cana-3900 15 26 i i NOUN cana-3900 15 27 ) ) PUNCT cana-3900 15 28 ( ( PUNCT cana-3900 15 29 n n X cana-3900 15 30 , , PUNCT cana-3900 15 31 + + PUNCT cana-3900 15 32 ) ) PUNCT cana-3900 15 33 is be AUX cana-3900 15 34 a a DET cana-3900 15 35 group group NOUN cana-3900 15 36 - - PUNCT cana-3900 15 37 not not PART cana-3900 15 38 necessarily necessarily ADV cana-3900 15 39 abelian abelian ADJ cana-3900 15 40 - - PUNCT cana-3900 15 41 with with ADP cana-3900 15 42 0 0 NUM cana-3900 15 43 as as ADP cana-3900 15 44 its its PRON cana-3900 15 45 identity identity NOUN cana-3900 15 46 element element NOUN cana-3900 15 47 , , PUNCT cana-3900 15 48 ( ( PUNCT cana-3900 15 49 ii ii NOUN cana-3900 15 50 ) ) PUNCT cana-3900 15 51 ( ( PUNCT cana-3900 15 52 n n X cana-3900 15 53 , , PUNCT cana-3900 15 54 βˆ™ βˆ™ PROPN cana-3900 15 55 ) ) PUNCT cana-3900 15 56 is be AUX cana-3900 15 57 a a DET cana-3900 15 58 semigroup semigroup NOUN cana-3900 15 59 [ [ X cana-3900 15 60 we we PRON cana-3900 15 61 write write VERB cana-3900 15 62 xy xy PROPN cana-3900 15 63 for for ADP cana-3900 15 64 x.y x.y PROPN cana-3900 15 65 for for ADP cana-3900 15 66 all all DET cana-3900 15 67 x x NOUN cana-3900 15 68 , , PUNCT cana-3900 15 69 y y PROPN cana-3900 15 70 in in ADP cana-3900 15 71 n n CCONJ cana-3900 15 72 ] ] PUNCT cana-3900 15 73 and and CCONJ cana-3900 15 74 ( ( PUNCT cana-3900 15 75 iii iii NOUN cana-3900 15 76 ) ) PUNCT cana-3900 15 77 ( ( PUNCT cana-3900 15 78 π‘₯ π‘₯ X cana-3900 16 1 + + CCONJ cana-3900 16 2 𝑦)𝑧 𝑦)𝑧 NOUN cana-3900 16 3 = = SYM cana-3900 16 4 π‘₯𝑧 π‘₯𝑧 PART cana-3900 16 5 + + CCONJ cana-3900 16 6 𝑦𝑧 𝑦𝑧 NOUN cana-3900 16 7 for for ADP cana-3900 16 8 all all DET cana-3900 16 9 x x NOUN cana-3900 16 10 , , PUNCT cana-3900 16 11 y y PROPN cana-3900 16 12 , , PUNCT cana-3900 16 13 z z NOUN cana-3900 16 14 in in ADP cana-3900 16 15 n. n. NOUN cana-3900 16 16 because because SCONJ cana-3900 16 17 of of ADP cana-3900 16 18 ( ( PUNCT cana-3900 16 19 iii iii NOUN cana-3900 16 20 ) ) PUNCT cana-3900 16 21 0n 0n NOUN cana-3900 17 1 = = NOUN cana-3900 17 2 0 0 NUM cana-3900 17 3 for for ADP cana-3900 17 4 all all DET cana-3900 17 5 n n NOUN cana-3900 17 6 in in ADP cana-3900 17 7 n. n. NOUN cana-3900 17 8 as as SCONJ cana-3900 17 9 we we PRON cana-3900 17 10 do do AUX cana-3900 17 11 not not PART cana-3900 17 12 stipulate stipulate VERB cana-3900 17 13 the the DET cana-3900 17 14 left left ADJ cana-3900 17 15 distributive distributive ADJ cana-3900 17 16 law law NOUN cana-3900 17 17 , , PUNCT cana-3900 17 18 " " PUNCT cana-3900 17 19 n0 n0 X cana-3900 17 20 = = SYM cana-3900 17 21 0 0 NUM cana-3900 17 22 ” " PUNCT cana-3900 17 23 need need AUX cana-3900 17 24 not not PART cana-3900 17 25 hold hold VERB cana-3900 17 26 good good ADJ cana-3900 17 27 for for ADP cana-3900 17 28 all all DET cana-3900 17 29 n n NOUN cana-3900 17 30 in in ADP cana-3900 17 31 n. n. NOUN cana-3900 17 32 we we PRON cana-3900 17 33 say say VERB cana-3900 17 34 that that SCONJ cana-3900 17 35 n n NOUN cana-3900 17 36 is be AUX cana-3900 17 37 zero zero NUM cana-3900 17 38 - - PUNCT cana-3900 17 39 symmetric symmetric NOUN cana-3900 17 40 if if SCONJ cana-3900 17 41 𝑛0 𝑛0 VERB cana-3900 17 42 = = NOUN cana-3900 17 43 0 0 NUM cana-3900 17 44 for for ADP cana-3900 17 45 all all DET cana-3900 17 46 n n NOUN cana-3900 17 47 in in ADP cana-3900 17 48 n. n. NOUN cana-3900 17 49 n n NUM cana-3900 17 50 is be AUX cana-3900 17 51 called call VERB cana-3900 17 52 an an DET cana-3900 17 53 s s NOUN cana-3900 17 54 - - PUNCT cana-3900 17 55 near near ADP cana-3900 17 56 - - PUNCT cana-3900 17 57 ring ring NOUN cana-3900 17 58 or or CCONJ cana-3900 17 59 an an DET cana-3900 17 60 s'-near s'-near NOUN cana-3900 17 61 - - PUNCT cana-3900 17 62 ring ring NOUN cana-3900 17 63 according accord VERB cana-3900 17 64 as as ADP cana-3900 17 65 π‘₯ π‘₯ PRON cana-3900 17 66 ∈ ∈ PROPN cana-3900 17 67 𝑁π‘₯ 𝑁π‘₯ PROPN cana-3900 17 68 or or CCONJ cana-3900 17 69 π‘₯ π‘₯ PRON cana-3900 17 70 ∈ ∈ NOUN cana-3900 17 71 π‘₯𝑁 π‘₯𝑁 NOUN cana-3900 17 72 for for ADP cana-3900 17 73 all all DET cana-3900 17 74 π‘₯ π‘₯ PRON cana-3900 17 75 ∈ ∈ NOUN cana-3900 17 76 𝑁. 𝑁. PROPN cana-3900 17 77 a a DET cana-3900 17 78 subgroup subgroup NOUN cana-3900 17 79 m m NOUN cana-3900 17 80 of of ADP cana-3900 17 81 n n PROPN cana-3900 17 82 is be AUX cana-3900 17 83 called call VERB cana-3900 17 84 an an DET cana-3900 17 85 nsubgroup nsubgroup NOUN cana-3900 17 86 if if SCONJ cana-3900 17 87 𝑁𝑀 𝑁𝑀 PROPN cana-3900 17 88 βŠ‚ βŠ‚ PROPN cana-3900 17 89 𝑀 𝑀 PROPN cana-3900 17 90 and and CCONJ cana-3900 17 91 an an DET cana-3900 17 92 invariant invariant ADJ cana-3900 17 93 n n CCONJ cana-3900 17 94 - - PUNCT cana-3900 17 95 subgroup subgroup NOUN cana-3900 17 96 if if SCONJ cana-3900 17 97 , , PUNCT cana-3900 17 98 in in ADP cana-3900 17 99 addition addition NOUN cana-3900 17 100 , , PUNCT cana-3900 17 101 𝑀𝑁 𝑀𝑁 PROPN cana-3900 17 102 βŠ‚ βŠ‚ PROPN cana-3900 17 103 𝑀. 𝑀. PROPN cana-3900 17 104 an an DET cana-3900 17 105 ideal ideal NOUN cana-3900 17 106 i i PRON cana-3900 17 107 of of ADP cana-3900 17 108 n n PROPN cana-3900 17 109 is be AUX cana-3900 17 110 called call VERB cana-3900 17 111 a a DET cana-3900 17 112 semi semi ADJ cana-3900 17 113 prime prime ADJ cana-3900 17 114 ideal ideal NOUN cana-3900 17 115 if if SCONJ cana-3900 17 116 for for ADP cana-3900 17 117 all all DET cana-3900 17 118 ideals ideal NOUN cana-3900 17 119 j j PROPN cana-3900 17 120 of of ADP cana-3900 17 121 n. n. PROPN cana-3900 17 122 𝐽2 𝐽2 PROPN cana-3900 18 1 βŠ‚ βŠ‚ PROPN cana-3900 18 2 𝐼 𝐼 ADP cana-3900 18 3 β‡’ β‡’ NOUN cana-3900 18 4 𝐽 𝐽 PROPN cana-3900 18 5 βŠ‚ βŠ‚ NOUN cana-3900 18 6 𝐼 𝐼 PROPN cana-3900 18 7 . . PUNCT cana-3900 19 1 if if SCONJ cana-3900 19 2 { { PUNCT cana-3900 19 3 0 0 X cana-3900 19 4 } } PUNCT cana-3900 19 5 is be AUX cana-3900 19 6 a a DET cana-3900 19 7 semiprime semiprime NOUN cana-3900 19 8 ideal ideal NOUN cana-3900 19 9 , , PUNCT cana-3900 19 10 then then ADV cana-3900 19 11 n n CCONJ cana-3900 19 12 is be AUX cana-3900 19 13 called call VERB cana-3900 19 14 a a DET cana-3900 19 15 semi semi ADJ cana-3900 19 16 prime prime ADJ cana-3900 19 17 near near ADP cana-3900 19 18 - - PUNCT cana-3900 19 19 ring ring NOUN cana-3900 19 20 . . PUNCT cana-3900 20 1 an an DET cana-3900 20 2 ideal ideal ADJ cana-3900 20 3 i i PRON cana-3900 20 4 of of ADP cana-3900 20 5 n n PROPN cana-3900 20 6 is be AUX cana-3900 20 7 called call VERB cana-3900 20 8 completely completely ADV cana-3900 20 9 semi semi ADV cana-3900 20 10 prime prime ADJ cana-3900 20 11 if if SCONJ cana-3900 20 12 x x PROPN cana-3900 20 13 ∈ ∈ PROPN cana-3900 20 14 i i PRON cana-3900 20 15 wheneverπ‘₯2 wheneverπ‘₯2 NOUN cana-3900 20 16 ∈ ∈ PROPN cana-3900 20 17 𝐼. 𝐼. PROPN cana-3900 20 18 n n PRON cana-3900 20 19 is be AUX cana-3900 20 20 called call VERB cana-3900 20 21 a a DET cana-3900 20 22 strictly strictly ADV cana-3900 20 23 prime prime ADJ cana-3900 20 24 near near ADP cana-3900 20 25 - - PUNCT cana-3900 20 26 ring ring NOUN cana-3900 20 27 if if SCONJ cana-3900 20 28 { { PUNCT cana-3900 20 29 0 0 NUM cana-3900 20 30 } } PUNCT cana-3900 20 31 is be AUX cana-3900 20 32 a a DET cana-3900 20 33 strictly strictly ADV cana-3900 20 34 prime prime ADJ cana-3900 20 35 ideal ideal NOUN cana-3900 20 36 i.e. i.e. X cana-3900 20 37 if if SCONJ cana-3900 20 38 a a PRON cana-3900 20 39 and and CCONJ cana-3900 20 40 b b NOUN cana-3900 20 41 are be AUX cana-3900 20 42 n n PRON cana-3900 20 43 - - PUNCT cana-3900 20 44 subgroups subgroup NOUN cana-3900 20 45 of of ADP cana-3900 20 46 n n PRON cana-3900 20 47 such such ADJ cana-3900 20 48 that that DET cana-3900 20 49 𝐴𝐡 𝐴𝐡 NOUN cana-3900 20 50 = = PRON cana-3900 20 51 { { PUNCT cana-3900 20 52 0 0 NUM cana-3900 20 53 } } PUNCT cana-3900 20 54 , , PUNCT cana-3900 20 55 then then ADV cana-3900 20 56 either either CCONJ cana-3900 20 57 𝐴 𝐴 PROPN cana-3900 20 58 = = SYM cana-3900 20 59 { { PUNCT cana-3900 20 60 0 0 NUM cana-3900 20 61 } } PUNCT cana-3900 20 62 or or CCONJ cana-3900 20 63 b b X cana-3900 20 64 = = SYM cana-3900 20 65 { { PUNCT cana-3900 20 66 0 0 NUM cana-3900 20 67 } } PUNCT cana-3900 20 68 . . PUNCT cana-3900 21 1 a a DET cana-3900 21 2 near near ADV cana-3900 21 3 - - PUNCT cana-3900 21 4 ring ring NOUN cana-3900 21 5 n n PRON cana-3900 21 6 has have VERB cana-3900 21 7 property property NOUN cana-3900 21 8 p4 p4 ADJ cana-3900 21 9 if if SCONJ cana-3900 21 10 for for ADP cana-3900 21 11 all all DET cana-3900 21 12 ideals ideal NOUN cana-3900 21 13 i i PRON cana-3900 21 14 of of ADP cana-3900 21 15 n n CCONJ cana-3900 21 16 , , PUNCT cana-3900 21 17 π‘₯𝑦 π‘₯𝑦 X cana-3900 21 18 ∈ ∈ NOUN cana-3900 21 19 𝐼 𝐼 PROPN cana-3900 21 20 β‡’ β‡’ NOUN cana-3900 21 21 𝑦π‘₯ 𝑦π‘₯ PROPN cana-3900 21 22 ∈ ∈ PROPN cana-3900 21 23 𝐼. 𝐼. PROPN cana-3900 21 24 from from ADP cana-3900 21 25 p.289 p.289 PROPN cana-3900 21 26 of of ADP cana-3900 21 27 pilz pilz PROPN cana-3900 22 1 [ [ X cana-3900 22 2 3 3 X cana-3900 22 3 ] ] PUNCT cana-3900 22 4 the the DET cana-3900 22 5 concept concept NOUN cana-3900 22 6 of of ADP cana-3900 22 7 a a DET cana-3900 22 8 mate mate NOUN cana-3900 22 9 function function NOUN cana-3900 22 10 in in ADP cana-3900 22 11 n n NUM cana-3900 22 12 has have AUX cana-3900 22 13 been be AUX cana-3900 22 14 introduced introduce VERB cana-3900 22 15 in in ADP cana-3900 22 16 [ [ X cana-3900 22 17 4 4 NUM cana-3900 22 18 ] ] PUNCT cana-3900 22 19 with with ADP cana-3900 22 20 a a DET cana-3900 22 21 view view NOUN cana-3900 22 22 to to PART cana-3900 22 23 handle handle VERB cana-3900 22 24 the the DET cana-3900 22 25 regularity regularity NOUN cana-3900 22 26 structure structure NOUN cana-3900 22 27 in in ADP cana-3900 22 28 a a DET cana-3900 22 29 near near NOUN cana-3900 22 30 - - PUNCT cana-3900 22 31 ring ring NOUN cana-3900 22 32 with with ADP cana-3900 22 33 considerable considerable ADJ cana-3900 22 34 ease ease NOUN cana-3900 22 35 . . PUNCT cana-3900 23 1 a a DET cana-3900 23 2 map map NOUN cana-3900 23 3 𝑓 𝑓 ADV cana-3900 23 4 from from ADP cana-3900 23 5 n n PROPN cana-3900 23 6 into into ADP cana-3900 23 7 n n PROPN cana-3900 23 8 is be AUX cana-3900 23 9 called call VERB cana-3900 23 10 a a DET cana-3900 23 11 mate mate NOUN cana-3900 23 12 function function NOUN cana-3900 23 13 for for ADP cana-3900 23 14 n n CCONJ cana-3900 23 15 , , PUNCT cana-3900 23 16 if if SCONJ cana-3900 23 17 π‘₯ π‘₯ PRON cana-3900 23 18 = = PUNCT cana-3900 23 19 π‘₯𝑓(π‘₯)π‘₯ π‘₯𝑓(π‘₯)π‘₯ VERB cana-3900 23 20 for for ADP cana-3900 23 21 all all DET cana-3900 23 22 x x NOUN cana-3900 23 23 in in ADP cana-3900 23 24 n. n. NOUN cana-3900 23 25 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3900 23 26 ) ) PUNCT cana-3900 23 27 is be AUX cana-3900 23 28 called call VERB cana-3900 23 29 a a DET cana-3900 23 30 mate mate NOUN cana-3900 23 31 of of ADP cana-3900 23 32 x. x. NOUN cana-3900 23 33 a a DET cana-3900 23 34 map map NOUN cana-3900 23 35 𝑓 𝑓 ADV cana-3900 23 36 from from ADP cana-3900 23 37 n n PROPN cana-3900 23 38 into into ADP cana-3900 23 39 n n PROPN cana-3900 23 40 is be AUX cana-3900 23 41 called call VERB cana-3900 23 42 a a DET cana-3900 23 43 p3 p3 NOUN cana-3900 23 44 mate mate NOUN cana-3900 23 45 function function NOUN cana-3900 23 46 for for ADP cana-3900 23 47 n n CCONJ cana-3900 23 48 , , PUNCT cana-3900 23 49 if if SCONJ cana-3900 23 50 π‘₯ π‘₯ PRON cana-3900 23 51 = = PUNCT cana-3900 23 52 π‘₯𝑓(π‘₯)π‘₯ π‘₯𝑓(π‘₯)π‘₯ PROPN cana-3900 23 53 and and CCONJ cana-3900 23 54 π‘₯𝑓(π‘₯ π‘₯𝑓(π‘₯ NOUN cana-3900 23 55 ) ) PUNCT cana-3900 23 56 = = SYM cana-3900 24 1 𝑓(π‘₯)π‘₯ 𝑓(π‘₯)π‘₯ VERB cana-3900 24 2 for for ADP cana-3900 24 3 all all DET cana-3900 24 4 x x NOUN cana-3900 24 5 in in ADP cana-3900 24 6 n. n. PROPN cana-3900 24 7 mailto:sssivanthi@gmail.com mailto:sssivanthi@gmail.com PROPN cana-3900 24 8 communications communication NOUN cana-3900 24 9 on on ADP cana-3900 24 10 applied apply VERB cana-3900 24 11 nonlinear nonlinear ADJ cana-3900 24 12 analysis analysis NOUN cana-3900 24 13 issn issn NOUN cana-3900 24 14 : : PUNCT cana-3900 24 15 1074 1074 NUM cana-3900 24 16 - - PUNCT cana-3900 24 17 133x 133x NUM cana-3900 24 18 vol vol NOUN cana-3900 24 19 32 32 NUM cana-3900 25 1 no no NOUN cana-3900 25 2 . . PUNCT cana-3900 26 1 9s 9s NUM cana-3900 26 2 ( ( PUNCT cana-3900 26 3 2025 2025 NUM cana-3900 26 4 ) ) PUNCT cana-3900 26 5 346 346 NUM cana-3900 26 6 https://internationalpubls.com https://internationalpubls.com X cana-3900 26 7 basic basic ADJ cana-3900 26 8 concepts concept NOUN cana-3900 26 9 and and CCONJ cana-3900 26 10 terms term NOUN cana-3900 26 11 used use VERB cana-3900 26 12 but but CCONJ cana-3900 26 13 not not PART cana-3900 26 14 defined define VERB cana-3900 26 15 in in ADP cana-3900 26 16 this this DET cana-3900 26 17 paper paper NOUN cana-3900 26 18 can can AUX cana-3900 26 19 be be AUX cana-3900 26 20 found find VERB cana-3900 26 21 in in ADP cana-3900 26 22 pilz pilz PROPN cana-3900 26 23 [ [ X cana-3900 26 24 3 3 NUM cana-3900 26 25 ] ] PUNCT cana-3900 26 26 . . PUNCT cana-3900 27 1 throughout throughout ADP cana-3900 27 2 this this DET cana-3900 27 3 paper paper NOUN cana-3900 27 4 n n PRON cana-3900 27 5 stands stand VERB cana-3900 27 6 for for ADP cana-3900 27 7 a a DET cana-3900 27 8 near near ADJ cana-3900 27 9 - - PUNCT cana-3900 27 10 ring ring NOUN cana-3900 27 11 – – PUNCT cana-3900 27 12 more more ADV cana-3900 27 13 precisely precisely ADV cana-3900 27 14 a a DET cana-3900 27 15 right right ADJ cana-3900 27 16 near near NOUN cana-3900 27 17 - - PUNCT cana-3900 27 18 ring ring NOUN cana-3900 27 19 – – PUNCT cana-3900 27 20 with with ADP cana-3900 27 21 at at ADV cana-3900 27 22 least least ADV cana-3900 27 23 two two NUM cana-3900 27 24 elements element NOUN cana-3900 27 25 . . PUNCT cana-3900 28 1 as as ADP cana-3900 28 2 in in ADP cana-3900 28 3 p.249 p.249 PROPN cana-3900 28 4 pilz pilz PROPN cana-3900 28 5 [ [ X cana-3900 28 6 3 3 NUM cana-3900 28 7 ] ] PUNCT cana-3900 28 8 , , PUNCT cana-3900 28 9 β€œ " PUNCT cana-3900 28 10 if if SCONJ cana-3900 28 11 n n PRON cana-3900 28 12 is be AUX cana-3900 28 13 a a DET cana-3900 28 14 near near ADJ cana-3900 28 15 - - PUNCT cana-3900 28 16 field field NOUN cana-3900 28 17 then then ADV cana-3900 28 18 either either CCONJ cana-3900 28 19 n n ADV cana-3900 28 20 is be AUX cana-3900 28 21 isomorphic isomorphic ADJ cana-3900 28 22 to to ADP cana-3900 28 23 𝑀𝑐 𝑀𝑐 PROPN cana-3900 28 24 ( ( PUNCT cana-3900 28 25 𝑍2 𝑍2 PROPN cana-3900 28 26 ) ) PUNCT cana-3900 28 27 or or CCONJ cana-3900 28 28 n n PROPN cana-3900 28 29 is be AUX cana-3900 28 30 zerosymmetric zerosymmetric ADJ cana-3900 28 31 ” " PUNCT cana-3900 28 32 ( ( PUNCT cana-3900 28 33 for for ADP cana-3900 28 34 the the DET cana-3900 28 35 concept concept NOUN cana-3900 28 36 of of ADP cana-3900 28 37 𝑀𝑐 𝑀𝑐 PROPN cana-3900 28 38 ( ( PUNCT cana-3900 28 39 𝑍2 𝑍2 PROPN cana-3900 28 40 ) ) PUNCT cana-3900 28 41 one one PRON cana-3900 28 42 may may AUX cana-3900 28 43 refer refer VERB cana-3900 28 44 to to ADP cana-3900 28 45 example example NOUN cana-3900 28 46 1.4(a 1.4(a NUM cana-3900 28 47 ) ) PUNCT cana-3900 28 48 , , PUNCT cana-3900 28 49 p.8 p.8 NOUN cana-3900 28 50 and and CCONJ cana-3900 28 51 1.15 1.15 NUM cana-3900 28 52 , , PUNCT cana-3900 28 53 p.12 p.12 NOUN cana-3900 28 54 of of ADP cana-3900 28 55 pilz pilz PROPN cana-3900 29 1 [ [ X cana-3900 29 2 3 3 NUM cana-3900 29 3 ] ] PUNCT cana-3900 29 4 . . PUNCT cana-3900 30 1 obviously obviously ADV cana-3900 30 2 𝑀𝑐 𝑀𝑐 PROPN cana-3900 30 3 ( ( PUNCT cana-3900 30 4 𝑍2 𝑍2 PROPN cana-3900 30 5 ) ) PUNCT cana-3900 30 6 is be AUX cana-3900 30 7 a a DET cana-3900 30 8 near near ADJ cana-3900 30 9 - - PUNCT cana-3900 30 10 field field NOUN cana-3900 30 11 of of ADP cana-3900 30 12 order order NOUN cana-3900 30 13 2 2 NUM cana-3900 30 14 and and CCONJ cana-3900 30 15 is be AUX cana-3900 30 16 not not PART cana-3900 30 17 zero zero NUM cana-3900 30 18 - - PUNCT cana-3900 30 19 symmetric symmetric ADJ cana-3900 30 20 ) ) PUNCT cana-3900 30 21 . . PUNCT cana-3900 31 1 all all DET cana-3900 31 2 the the DET cana-3900 31 3 near near ADJ cana-3900 31 4 - - PUNCT cana-3900 31 5 fields field NOUN cana-3900 31 6 in in ADP cana-3900 31 7 this this DET cana-3900 31 8 paper paper NOUN cana-3900 31 9 are be AUX cana-3900 31 10 zero zero NUM cana-3900 31 11 - - PUNCT cana-3900 31 12 symmetric symmetric ADJ cana-3900 31 13 . . PUNCT cana-3900 31 14 2 2 NUM cana-3900 31 15 notations notation NOUN cana-3900 31 16 ( ( PUNCT cana-3900 31 17 i i NOUN cana-3900 31 18 ) ) PUNCT cana-3900 31 19 e e NOUN cana-3900 31 20 denotes denote VERB cana-3900 31 21 the the DET cana-3900 31 22 set set NOUN cana-3900 31 23 of of ADP cana-3900 31 24 all all DET cana-3900 31 25 idempotent idempotent NOUN cana-3900 31 26 of of ADP cana-3900 31 27 n. n. NOUN cana-3900 31 28 [ [ X cana-3900 31 29 e e NOUN cana-3900 31 30 in in ADP cana-3900 31 31 n n PROPN cana-3900 31 32 is be AUX cana-3900 31 33 called call VERB cana-3900 31 34 an an DET cana-3900 31 35 idempotent idempotent NOUN cana-3900 31 36 if if SCONJ cana-3900 31 37 𝑒2 𝑒2 PROPN cana-3900 31 38 = = SYM cana-3900 31 39 𝑒 𝑒 X cana-3900 31 40 ] ] X cana-3900 31 41 ( ( PUNCT cana-3900 31 42 ii ii NOUN cana-3900 31 43 ) ) PUNCT cana-3900 31 44 l l NOUN cana-3900 31 45 denotes denote VERB cana-3900 31 46 the the DET cana-3900 31 47 set set NOUN cana-3900 31 48 of of ADP cana-3900 31 49 all all DET cana-3900 31 50 nilpotent nilpotent NOUN cana-3900 31 51 of of ADP cana-3900 31 52 n. n. NOUN cana-3900 31 53 [ [ X cana-3900 31 54 a a PRON cana-3900 31 55 in in ADP cana-3900 31 56 n n PRON cana-3900 31 57 is be AUX cana-3900 31 58 nilpotent nilpotent ADJ cana-3900 31 59 if if SCONJ cana-3900 31 60 ak ak PROPN cana-3900 31 61 = = PROPN cana-3900 31 62 0 0 PROPN cana-3900 31 63 for for ADP cana-3900 31 64 some some DET cana-3900 31 65 positive positive ADJ cana-3900 31 66 integer integer NOUN cana-3900 31 67 k. k. PROPN cana-3900 31 68 ] ] PUNCT cana-3900 32 1 ( ( PUNCT cana-3900 32 2 iii iii X cana-3900 32 3 ) ) PUNCT cana-3900 32 4 𝑁0 𝑁0 ADJ cana-3900 32 5 = = PUNCT cana-3900 32 6 { { PUNCT cana-3900 32 7 𝑛 𝑛 PRON cana-3900 32 8 ∈ ∈ NOUN cana-3900 32 9 𝑁 𝑁 PROPN cana-3900 32 10 / / SYM cana-3900 32 11 𝑛0 𝑛0 VERB cana-3900 32 12 = = NOUN cana-3900 32 13 0 0 NUM cana-3900 32 14 } } PUNCT cana-3900 32 15 zero zero NUM cana-3900 32 16 - - PUNCT cana-3900 32 17 symmetric symmetric ADJ cana-3900 32 18 part part NOUN cana-3900 32 19 of of ADP cana-3900 32 20 n. n. NOUN cana-3900 32 21 ( ( PUNCT cana-3900 32 22 iv iv NUM cana-3900 32 23 ) ) PUNCT cana-3900 32 24 . . PUNCT cana-3900 33 1 𝑁𝑑 𝑁𝑑 ADV cana-3900 33 2 = = SYM cana-3900 33 3 { { PUNCT cana-3900 33 4 𝑛 𝑛 PRON cana-3900 33 5 ∈ ∈ NOUN cana-3900 33 6 𝑁 𝑁 PROPN cana-3900 33 7 / / SYM cana-3900 33 8 𝑛(π‘₯ 𝑛(π‘₯ PROPN cana-3900 33 9 + + CCONJ cana-3900 33 10 𝑦 𝑦 X cana-3900 33 11 ) ) PUNCT cana-3900 33 12 = = SYM cana-3900 33 13 𝑛π‘₯ 𝑛π‘₯ PROPN cana-3900 33 14 + + CCONJ cana-3900 33 15 𝑛𝑦 𝑛𝑦 PROPN cana-3900 33 16 for for ADP cana-3900 33 17 all all DET cana-3900 33 18 π‘₯ π‘₯ PROPN cana-3900 33 19 , , PUNCT cana-3900 33 20 𝑦 𝑦 NOUN cana-3900 33 21 in in ADP cana-3900 33 22 𝑁 𝑁 NOUN cana-3900 33 23 } } PUNCT cana-3900 33 24 – – PUNCT cana-3900 33 25 set set NOUN cana-3900 33 26 of of ADP cana-3900 33 27 all all DET cana-3900 33 28 distributive distributive ADJ cana-3900 33 29 element element NOUN cana-3900 33 30 of of ADP cana-3900 33 31 n. n. NOUN cana-3900 33 32 ( ( PUNCT cana-3900 33 33 v v NOUN cana-3900 33 34 ) ) PUNCT cana-3900 33 35 𝐢(𝑁 𝐢(𝑁 NUM cana-3900 33 36 ) ) PUNCT cana-3900 33 37 = = PRON cana-3900 33 38 { { PUNCT cana-3900 33 39 𝑛 𝑛 PRON cana-3900 33 40 ∈ ∈ NOUN cana-3900 33 41 𝑁 𝑁 PROPN cana-3900 33 42 / / SYM cana-3900 33 43 𝑛π‘₯ 𝑛π‘₯ PROPN cana-3900 33 44 = = PUNCT cana-3900 33 45 π‘₯𝑛 π‘₯𝑛 PROPN cana-3900 33 46 for for ADP cana-3900 33 47 all all DET cana-3900 33 48 π‘₯ π‘₯ PRON cana-3900 33 49 in in ADP cana-3900 33 50 𝑁 𝑁 PROPN cana-3900 33 51 } } PUNCT cana-3900 33 52 center center NOUN cana-3900 33 53 of of ADP cana-3900 33 54 n. n. NOUN cana-3900 33 55 3 3 NUM cana-3900 33 56 . . PUNCT cana-3900 33 57 preliminary preliminary ADJ cana-3900 33 58 results result NOUN cana-3900 33 59 we we PRON cana-3900 33 60 freely freely ADV cana-3900 33 61 make make VERB cana-3900 33 62 use use NOUN cana-3900 33 63 of of ADP cana-3900 33 64 the the DET cana-3900 33 65 following follow VERB cana-3900 33 66 results result NOUN cana-3900 33 67 from from ADP cana-3900 33 68 [ [ X cana-3900 33 69 4 4 NUM cana-3900 33 70 ] ] PUNCT cana-3900 33 71 , , PUNCT cana-3900 33 72 [ [ X cana-3900 33 73 3 3 NUM cana-3900 33 74 ] ] PUNCT cana-3900 33 75 and and CCONJ cana-3900 33 76 [ [ X cana-3900 33 77 2 2 NUM cana-3900 33 78 ] ] PUNCT cana-3900 33 79 and and CCONJ cana-3900 33 80 designate designate VERB cana-3900 33 81 them they PRON cana-3900 33 82 as as ADP cana-3900 33 83 k k PROPN cana-3900 33 84 ( ( PUNCT cana-3900 33 85 1 1 NUM cana-3900 33 86 ) ) PUNCT cana-3900 33 87 , , PUNCT cana-3900 33 88 k k PROPN cana-3900 33 89 ( ( PUNCT cana-3900 33 90 2 2 NUM cana-3900 33 91 ) ) PUNCT cana-3900 33 92 etc etc X cana-3900 33 93 . . X cana-3900 34 1 ( ( PUNCT cana-3900 34 2 k k X cana-3900 34 3 for for ADP cana-3900 34 4 β€˜ ' PUNCT cana-3900 34 5 known know VERB cana-3900 34 6 results result NOUN cana-3900 34 7 ’ ' PUNCT cana-3900 34 8 ) ) PUNCT cana-3900 34 9 . . PUNCT cana-3900 35 1 k k X cana-3900 35 2 ( ( PUNCT cana-3900 35 3 1 1 NUM cana-3900 35 4 ): ): PUNCT cana-3900 35 5 if if SCONJ cana-3900 35 6 n n PRON cana-3900 35 7 has have VERB cana-3900 35 8 a a DET cana-3900 35 9 mate mate NOUN cana-3900 35 10 function function NOUN cana-3900 35 11 m m NOUN cana-3900 35 12 , , PUNCT cana-3900 35 13 then then ADV cana-3900 35 14 for for ADP cana-3900 35 15 every every DET cana-3900 35 16 π‘₯ π‘₯ PRON cana-3900 35 17 ∈ ∈ PROPN cana-3900 35 18 𝑁 𝑁 PROPN cana-3900 35 19 , , PUNCT cana-3900 35 20 π‘₯𝑓(π‘₯ π‘₯𝑓(π‘₯ NOUN cana-3900 35 21 ) ) PUNCT cana-3900 35 22 , , PUNCT cana-3900 35 23 𝑓(π‘₯)π‘₯ 𝑓(π‘₯)π‘₯ VERB cana-3900 35 24 ∈ ∈ NOUN cana-3900 35 25 e e NOUN cana-3900 35 26 and and CCONJ cana-3900 35 27 𝑁π‘₯ 𝑁π‘₯ PROPN cana-3900 35 28 = = SYM cana-3900 35 29 𝑁𝑓(π‘₯)π‘₯ 𝑁𝑓(π‘₯)π‘₯ PROPN cana-3900 35 30 and and CCONJ cana-3900 35 31 π‘₯𝑁 π‘₯𝑁 NOUN cana-3900 35 32 = = ADJ cana-3900 35 33 π‘₯𝑓(π‘₯)𝑁. π‘₯𝑓(π‘₯)𝑁. X cana-3900 35 34 ( ( PUNCT cana-3900 35 35 lemma lemma PROPN cana-3900 35 36 3.2 3.2 NUM cana-3900 35 37 of of ADP cana-3900 35 38 [ [ X cana-3900 35 39 4 4 NUM cana-3900 35 40 ] ] NUM cana-3900 35 41 ) ) PUNCT cana-3900 35 42 . . PUNCT cana-3900 36 1 k(2 k(2 NOUN cana-3900 36 2 ): ): PUNCT cana-3900 36 3 if if SCONJ cana-3900 36 4 l l NOUN cana-3900 36 5 = = X cana-3900 36 6 { { PUNCT cana-3900 36 7 0}and 0}and NUM cana-3900 36 8 , , PUNCT cana-3900 36 9 n= n= PROPN cana-3900 36 10 n0 n0 X cana-3900 36 11 then then ADV cana-3900 36 12 ( ( PUNCT cana-3900 36 13 i i NOUN cana-3900 36 14 ) ) PUNCT cana-3900 36 15 π‘₯𝑦 π‘₯𝑦 PROPN cana-3900 36 16 = = SYM cana-3900 36 17 0 0 NUM cana-3900 36 18 β‡’ β‡’ NOUN cana-3900 36 19 𝑦π‘₯ 𝑦π‘₯ NOUN cana-3900 36 20 = = SYM cana-3900 36 21 0 0 PUNCT cana-3900 37 1 ( ( PUNCT cana-3900 37 2 for for ADP cana-3900 37 3 π‘₯ π‘₯ PROPN cana-3900 37 4 , , PUNCT cana-3900 37 5 𝑦 𝑦 NOUN cana-3900 37 6 in in ADP cana-3900 37 7 n n CCONJ cana-3900 37 8 ) ) PUNCT cana-3900 37 9 and and CCONJ cana-3900 37 10 ( ( PUNCT cana-3900 37 11 ii ii NOUN cana-3900 37 12 ) ) PUNCT cana-3900 37 13 n n VERB cana-3900 37 14 has have AUX cana-3900 37 15 " " PUNCT cana-3900 37 16 insertion insertion NOUN cana-3900 37 17 of of ADP cana-3900 37 18 factors factor NOUN cana-3900 37 19 property property NOUN cana-3900 37 20 " " PUNCT cana-3900 37 21 – – PUNCT cana-3900 37 22 ifp ifp NOUN cana-3900 37 23 for for ADP cana-3900 37 24 short short ADJ cana-3900 37 25 – – PUNCT cana-3900 37 26 i.e. i.e. X cana-3900 37 27 for for ADP cana-3900 37 28 π‘₯ π‘₯ PROPN cana-3900 37 29 , , PUNCT cana-3900 37 30 𝑦 𝑦 NOUN cana-3900 37 31 in in ADP cana-3900 37 32 n n CCONJ cana-3900 37 33 , , PUNCT cana-3900 37 34 π‘₯𝑦 π‘₯𝑦 PROPN cana-3900 37 35 = = SYM cana-3900 37 36 0 0 NUM cana-3900 37 37 β‡’ β‡’ NOUN cana-3900 37 38 π‘₯𝑛𝑦 π‘₯𝑛𝑦 ADP cana-3900 37 39 = = SYM cana-3900 37 40 0 0 NUM cana-3900 37 41 for for ADP cana-3900 37 42 all all DET cana-3900 37 43 n n NOUN cana-3900 37 44 in in ADP cana-3900 37 45 n. n. NOUN cana-3900 37 46 ( ( PUNCT cana-3900 37 47 in in ADP cana-3900 37 48 this this DET cana-3900 37 49 paper paper NOUN cana-3900 37 50 we we PRON cana-3900 37 51 write write VERB cana-3900 37 52 that that SCONJ cana-3900 37 53 n n PRON cana-3900 37 54 has have VERB cana-3900 37 55 ( ( PUNCT cana-3900 37 56 * * PROPN cana-3900 37 57 , , PUNCT cana-3900 37 58 ifp ifp NOUN cana-3900 37 59 ) ) PUNCT cana-3900 37 60 if if SCONJ cana-3900 37 61 n n PRON cana-3900 37 62 has have VERB cana-3900 37 63 both both PRON cana-3900 37 64 ( ( PUNCT cana-3900 37 65 i i NOUN cana-3900 37 66 ) ) PUNCT cana-3900 37 67 and and CCONJ cana-3900 37 68 ( ( PUNCT cana-3900 37 69 ii ii NOUN cana-3900 37 70 ) ) PUNCT cana-3900 37 71 ) ) PUNCT cana-3900 38 1 ( ( PUNCT cana-3900 38 2 lemma lemma PROPN cana-3900 38 3 2.3 2.3 NUM cana-3900 38 4 of of ADP cana-3900 38 5 [ [ X cana-3900 38 6 4 4 NUM cana-3900 38 7 ] ] NUM cana-3900 38 8 ) ) PUNCT cana-3900 38 9 . . PUNCT cana-3900 39 1 k k X cana-3900 39 2 ( ( PUNCT cana-3900 39 3 3 3 NUM cana-3900 39 4 ): ): PUNCT cana-3900 39 5 a a DET cana-3900 39 6 zero zero NUM cana-3900 39 7 - - PUNCT cana-3900 39 8 symmetric symmetric ADJ cana-3900 39 9 near near ADP cana-3900 39 10 - - PUNCT cana-3900 39 11 ring ring NOUN cana-3900 39 12 n n PRON cana-3900 39 13 has have AUX cana-3900 39 14 ifp ifp NOUN cana-3900 39 15 if if SCONJ cana-3900 39 16 and and CCONJ cana-3900 39 17 only only ADV cana-3900 39 18 if if SCONJ cana-3900 39 19 ( ( PUNCT cana-3900 39 20 0 0 NUM cana-3900 39 21 : : PUNCT cana-3900 39 22 s s X cana-3900 39 23 ) ) PUNCT cana-3900 39 24 is be AUX cana-3900 39 25 an an DET cana-3900 39 26 ideal ideal NOUN cana-3900 39 27 , , PUNCT cana-3900 39 28 where where SCONJ cana-3900 39 29 s s NOUN cana-3900 39 30 is be AUX cana-3900 39 31 any any DET cana-3900 39 32 non non ADJ cana-3900 39 33 - - ADJ cana-3900 39 34 empty empty ADJ cana-3900 39 35 subset subset NOUN cana-3900 39 36 of of ADP cana-3900 39 37 n. n. PROPN cana-3900 39 38 ( ( PUNCT cana-3900 39 39 9.3 9.3 NUM cana-3900 39 40 , , PUNCT cana-3900 39 41 p.289 p.289 NUM cana-3900 39 42 of of ADP cana-3900 39 43 [ [ X cana-3900 39 44 3 3 NUM cana-3900 39 45 ] ] NUM cana-3900 39 46 ) ) PUNCT cana-3900 39 47 . . PUNCT cana-3900 40 1 k k X cana-3900 40 2 ( ( PUNCT cana-3900 40 3 4 4 NUM cana-3900 40 4 ): ): PUNCT cana-3900 40 5 a a DET cana-3900 40 6 near near ADV cana-3900 40 7 - - PUNCT cana-3900 40 8 ring ring NOUN cana-3900 40 9 n n PRON cana-3900 40 10 has have VERB cana-3900 40 11 no no DET cana-3900 40 12 non non ADJ cana-3900 40 13 - - ADJ cana-3900 40 14 zero zero ADJ cana-3900 40 15 nilpotent nilpotent ADJ cana-3900 40 16 elements element NOUN cana-3900 40 17 if if SCONJ cana-3900 40 18 and and CCONJ cana-3900 40 19 only only ADV cana-3900 40 20 if if SCONJ cana-3900 40 21 π‘₯2 π‘₯2 NOUN cana-3900 40 22 = = SYM cana-3900 40 23 0 0 NUM cana-3900 40 24 β‡’ β‡’ NOUN cana-3900 40 25 π‘₯ π‘₯ X cana-3900 40 26 = = SYM cana-3900 40 27 0 0 NUM cana-3900 40 28 for for ADP cana-3900 40 29 all all DET cana-3900 40 30 x x NOUN cana-3900 40 31 in in ADP cana-3900 40 32 n n PROPN cana-3900 40 33 ( ( PUNCT cana-3900 40 34 prob prob NOUN cana-3900 40 35 . . PROPN cana-3900 40 36 14 14 NUM cana-3900 40 37 , , PUNCT cana-3900 40 38 p.9 p.9 NOUN cana-3900 40 39 of of ADP cana-3900 40 40 [ [ X cana-3900 40 41 2 2 NUM cana-3900 40 42 ] ] NUM cana-3900 40 43 ) ) PUNCT cana-3900 40 44 . . PUNCT cana-3900 41 1 3.1 3.1 NUM cana-3900 41 2 definition definition NOUN cana-3900 41 3 and and CCONJ cana-3900 41 4 examples example NOUN cana-3900 41 5 in in ADP cana-3900 41 6 the the DET cana-3900 41 7 section section NOUN cana-3900 41 8 , , PUNCT cana-3900 41 9 we we PRON cana-3900 41 10 introduce introduce VERB cana-3900 41 11 the the DET cana-3900 41 12 notion notion NOUN cana-3900 41 13 of of ADP cana-3900 41 14 𝛿1near 𝛿1near NOUN cana-3900 41 15 - - PUNCT cana-3900 41 16 ring ring NOUN cana-3900 41 17 and and CCONJ cana-3900 41 18 furnish furnish VERB cana-3900 41 19 examples example NOUN cana-3900 41 20 to to PART cana-3900 41 21 illustrate illustrate VERB cana-3900 41 22 it it PRON cana-3900 41 23 . . PUNCT cana-3900 42 1 to to PART cana-3900 42 2 start start VERB cana-3900 42 3 with with ADP cana-3900 42 4 we we PRON cana-3900 42 5 have have VERB cana-3900 42 6 the the DET cana-3900 42 7 following follow VERB cana-3900 42 8 definition definition NOUN cana-3900 42 9 . . PUNCT cana-3900 43 1 definition definition NOUN cana-3900 43 2 3.1.1 3.1.1 NOUN cana-3900 43 3 let let VERB cana-3900 43 4 n n PRON cana-3900 43 5 be be AUX cana-3900 43 6 a a DET cana-3900 43 7 right right ADJ cana-3900 43 8 near near NOUN cana-3900 43 9 - - PUNCT cana-3900 43 10 ring ring NOUN cana-3900 43 11 . . PUNCT cana-3900 44 1 if if SCONJ cana-3900 44 2 for for ADP cana-3900 44 3 every every DET cana-3900 44 4 π‘₯ π‘₯ PROPN cana-3900 44 5 , , PUNCT cana-3900 44 6 𝑦 𝑦 NOUN cana-3900 44 7 in in ADP cana-3900 44 8 𝑁 𝑁 NOUN cana-3900 44 9 , , PUNCT cana-3900 44 10 π‘₯𝑁𝑦 π‘₯𝑁𝑦 NOUN cana-3900 44 11 = = PUNCT cana-3900 44 12 𝑁π‘₯2𝑦2 𝑁π‘₯2𝑦2 NOUN cana-3900 44 13 then then ADV cana-3900 44 14 we we PRON cana-3900 44 15 say say VERB cana-3900 44 16 n n PRON cana-3900 44 17 is be AUX cana-3900 44 18 a a DET cana-3900 44 19 𝛿1near 𝛿1near NOUN cana-3900 44 20 - - PUNCT cana-3900 44 21 ring ring NOUN cana-3900 44 22 . . PUNCT cana-3900 45 1 examples example NOUN cana-3900 45 2 3.1.2 3.1.2 NUM cana-3900 45 3 ( ( PUNCT cana-3900 45 4 i i NOUN cana-3900 45 5 ) ) PUNCT cana-3900 45 6 let let VERB cana-3900 45 7 ( ( PUNCT cana-3900 45 8 𝑁 𝑁 PROPN cana-3900 45 9 , , PUNCT cana-3900 45 10 + + NOUN cana-3900 45 11 ) ) PUNCT cana-3900 45 12 be be VERB cana-3900 45 13 the the DET cana-3900 45 14 klein klein PROPN cana-3900 45 15 ’s ’s PART cana-3900 45 16 four four NUM cana-3900 45 17 group group NOUN cana-3900 45 18 { { PUNCT cana-3900 45 19 0 0 NUM cana-3900 45 20 , , PUNCT cana-3900 45 21 π‘Ž π‘Ž NOUN cana-3900 45 22 , , PUNCT cana-3900 45 23 𝑏 𝑏 NOUN cana-3900 45 24 , , PUNCT cana-3900 45 25 𝑐 𝑐 NOUN cana-3900 45 26 } } PUNCT cana-3900 45 27 . . PUNCT cana-3900 46 1 the the DET cana-3900 46 2 near near ADV cana-3900 46 3 - - PUNCT cana-3900 46 4 ring ring NOUN cana-3900 46 5 ( ( PUNCT cana-3900 46 6 𝑁 𝑁 PROPN cana-3900 46 7 , , PUNCT cana-3900 46 8 + + NOUN cana-3900 46 9 , , PUNCT cana-3900 46 10 Β· Β· PUNCT cana-3900 46 11 ) ) PUNCT cana-3900 46 12 where where SCONJ cana-3900 46 13 β€˜ ' PUNCT cana-3900 46 14 Β· Β· PUNCT cana-3900 46 15 ’ ' PUNCT cana-3900 46 16 is be AUX cana-3900 46 17 defined define VERB cana-3900 46 18 as as ADP cana-3900 46 19 per per ADP cana-3900 46 20 scheme scheme NOUN cana-3900 46 21 12 12 NUM cana-3900 46 22 , , PUNCT cana-3900 46 23 p.408 p.408 ADV cana-3900 46 24 of of ADP cana-3900 46 25 pilz pilz PROPN cana-3900 46 26 [ [ X cana-3900 46 27 33 33 NUM cana-3900 46 28 ] ] PUNCT cana-3900 46 29 . . PUNCT cana-3900 47 1 βˆ™ βˆ™ NOUN cana-3900 47 2 0 0 PUNCT cana-3900 48 1 a a DET cana-3900 48 2 b b PROPN cana-3900 48 3 c c NOUN cana-3900 48 4 communications communication NOUN cana-3900 48 5 on on ADP cana-3900 48 6 applied apply VERB cana-3900 48 7 nonlinear nonlinear ADJ cana-3900 48 8 analysis analysis NOUN cana-3900 48 9 issn issn NOUN cana-3900 48 10 : : PUNCT cana-3900 48 11 1074 1074 NUM cana-3900 48 12 - - PUNCT cana-3900 48 13 133x 133x NUM cana-3900 48 14 vol vol NOUN cana-3900 48 15 32 32 NUM cana-3900 48 16 no no NOUN cana-3900 48 17 . . PUNCT cana-3900 49 1 9s 9s NUM cana-3900 49 2 ( ( PUNCT cana-3900 49 3 2025 2025 NUM cana-3900 49 4 ) ) PUNCT cana-3900 49 5 347 347 NUM cana-3900 49 6 https://internationalpubls.com https://internationalpubls.com X cana-3900 49 7 0 0 PUNCT cana-3900 49 8 0 0 NUM cana-3900 49 9 0 0 NUM cana-3900 49 10 0 0 NUM cana-3900 49 11 0 0 NUM cana-3900 50 1 a a DET cana-3900 50 2 0 0 NUM cana-3900 50 3 a a DET cana-3900 50 4 0 0 NUM cana-3900 50 5 a a DET cana-3900 50 6 b b NOUN cana-3900 50 7 0 0 NUM cana-3900 50 8 0 0 NUM cana-3900 50 9 0 0 NUM cana-3900 50 10 0 0 NUM cana-3900 51 1 c c NOUN cana-3900 51 2 0 0 NUM cana-3900 52 1 a a DET cana-3900 52 2 0 0 NUM cana-3900 52 3 a a PRON cana-3900 52 4 is be AUX cana-3900 52 5 a a DET cana-3900 52 6 𝛿1near 𝛿1near NOUN cana-3900 52 7 - - PUNCT cana-3900 52 8 ring ring NOUN cana-3900 52 9 . . PUNCT cana-3900 53 1 ( ( PUNCT cana-3900 53 2 ii ii NOUN cana-3900 53 3 ) ) PUNCT cana-3900 53 4 let let VERB cana-3900 53 5 ( ( PUNCT cana-3900 53 6 𝑁 𝑁 PROPN cana-3900 53 7 , , PUNCT cana-3900 53 8 + + NOUN cana-3900 53 9 ) ) PUNCT cana-3900 53 10 be be VERB cana-3900 53 11 the the DET cana-3900 53 12 klein klein PROPN cana-3900 53 13 ’s ’s PART cana-3900 53 14 four four NUM cana-3900 53 15 group group NOUN cana-3900 53 16 { { PUNCT cana-3900 53 17 0 0 NUM cana-3900 53 18 , , PUNCT cana-3900 53 19 π‘Ž π‘Ž NOUN cana-3900 53 20 , , PUNCT cana-3900 53 21 𝑏 𝑏 NOUN cana-3900 53 22 , , PUNCT cana-3900 53 23 𝑐 𝑐 NOUN cana-3900 53 24 } } PUNCT cana-3900 53 25 . . PUNCT cana-3900 54 1 the the DET cana-3900 54 2 near near ADV cana-3900 54 3 - - PUNCT cana-3900 54 4 ring ring NOUN cana-3900 54 5 ( ( PUNCT cana-3900 54 6 𝑁 𝑁 PROPN cana-3900 54 7 , , PUNCT cana-3900 54 8 + + NOUN cana-3900 54 9 , , PUNCT cana-3900 54 10 Β· Β· PUNCT cana-3900 54 11 ) ) PUNCT cana-3900 54 12 where where SCONJ cana-3900 54 13 β€˜ ' PUNCT cana-3900 54 14 Β· Β· PUNCT cana-3900 54 15 ’ ' PUNCT cana-3900 54 16 is be AUX cana-3900 54 17 defined define VERB cana-3900 54 18 as as ADP cana-3900 54 19 per per ADP cana-3900 54 20 scheme scheme NOUN cana-3900 54 21 11 11 NUM cana-3900 54 22 , , PUNCT cana-3900 54 23 p.408 p.408 ADV cana-3900 54 24 of of ADP cana-3900 54 25 pilz pilz PROPN cana-3900 54 26 [ [ X cana-3900 54 27 33 33 NUM cana-3900 54 28 ] ] PUNCT cana-3900 54 29 . . PUNCT cana-3900 55 1 βˆ™ βˆ™ NOUN cana-3900 55 2 0 0 PUNCT cana-3900 56 1 a a DET cana-3900 56 2 b b X cana-3900 56 3 c c NOUN cana-3900 56 4 0 0 NUM cana-3900 56 5 0 0 NUM cana-3900 56 6 0 0 NUM cana-3900 56 7 0 0 NUM cana-3900 56 8 0 0 NUM cana-3900 56 9 a a DET cana-3900 56 10 0 0 NUM cana-3900 56 11 a a DET cana-3900 56 12 b b NOUN cana-3900 56 13 a a DET cana-3900 56 14 b b NOUN cana-3900 56 15 0 0 NUM cana-3900 56 16 0 0 NUM cana-3900 56 17 0 0 NUM cana-3900 56 18 0 0 NUM cana-3900 56 19 c c NOUN cana-3900 56 20 0 0 NUM cana-3900 57 1 a a DET cana-3900 57 2 b b NOUN cana-3900 57 3 a a PRON cana-3900 57 4 is be AUX cana-3900 57 5 a a DET cana-3900 57 6 not not PART cana-3900 57 7 𝛿1near 𝛿1near NOUN cana-3900 57 8 - - PUNCT cana-3900 57 9 ring ring NOUN cana-3900 57 10 . . PUNCT cana-3900 58 1 since since SCONJ cana-3900 58 2 π‘Žπ‘π‘ π‘Žπ‘π‘ ADV cana-3900 58 3 β‰  β‰  PROPN cana-3900 58 4 π‘π‘Ž2𝑏2 π‘π‘Ž2𝑏2 NUM cana-3900 58 5 3.2 3.2 NUM cana-3900 58 6 properties property NOUN cana-3900 58 7 of of ADP cana-3900 58 8 𝜹𝟏 𝜹𝟏 NOUN cana-3900 58 9 near near ADP cana-3900 58 10 - - PUNCT cana-3900 58 11 ring ring NOUN cana-3900 58 12 we we PRON cana-3900 58 13 shall shall AUX cana-3900 58 14 obtain obtain VERB cana-3900 58 15 a a DET cana-3900 58 16 complete complete ADJ cana-3900 58 17 characterization characterization NOUN cana-3900 58 18 for for ADP cana-3900 58 19 𝛿1near 𝛿1near NOUN cana-3900 58 20 - - PUNCT cana-3900 58 21 ring ring NOUN cana-3900 58 22 and and CCONJ cana-3900 58 23 obtain obtain VERB cana-3900 58 24 structure structure NOUN cana-3900 58 25 theorem theorem NOUN cana-3900 58 26 for for ADP cana-3900 58 27 such such ADJ cana-3900 58 28 near near ADP cana-3900 58 29 - - PUNCT cana-3900 58 30 ring ring NOUN cana-3900 58 31 – – PUNCT cana-3900 58 32 under under ADP cana-3900 58 33 certain certain ADJ cana-3900 58 34 conditions condition NOUN cana-3900 58 35 . . PUNCT cana-3900 59 1 proposition proposition NOUN cana-3900 59 2 3.2.1 3.2.1 NUM cana-3900 59 3 let let VERB cana-3900 59 4 n n PRON cana-3900 59 5 be be AUX cana-3900 59 6 a a DET cana-3900 59 7 𝛿1near 𝛿1near NOUN cana-3900 59 8 - - PUNCT cana-3900 59 9 ring ring NOUN cana-3900 59 10 with with ADP cana-3900 59 11 identity identity NOUN cana-3900 59 12 . . PUNCT cana-3900 60 1 ( ( PUNCT cana-3900 60 2 i i NOUN cana-3900 60 3 ) ) PUNCT cana-3900 60 4 n n PROPN cana-3900 60 5 is be AUX cana-3900 60 6 zero zero NUM cana-3900 60 7 symmetric symmetric ADJ cana-3900 60 8 ( ( PUNCT cana-3900 60 9 ii ii NOUN cana-3900 60 10 ) ) PUNCT cana-3900 60 11 if if SCONJ cana-3900 60 12 n n PRON cana-3900 60 13 has have VERB cana-3900 60 14 no no DET cana-3900 60 15 non non ADJ cana-3900 60 16 - - ADJ cana-3900 60 17 zero zero ADJ cana-3900 60 18 nilpotent nilpotent ADJ cana-3900 60 19 elements element NOUN cana-3900 60 20 then then ADV cana-3900 60 21 n n PRON cana-3900 60 22 is be AUX cana-3900 60 23 an an DET cana-3900 60 24 s s NOUN cana-3900 60 25 - - PUNCT cana-3900 60 26 near near ADP cana-3900 60 27 - - PUNCT cana-3900 60 28 ring ring NOUN cana-3900 60 29 . . PUNCT cana-3900 61 1 ( ( PUNCT cana-3900 61 2 iii iii X cana-3900 61 3 ) ) PUNCT cana-3900 61 4 if if SCONJ cana-3900 61 5 n n PRON cana-3900 61 6 is be AUX cana-3900 61 7 an an DET cana-3900 61 8 s'-near s'-near NOUN cana-3900 61 9 - - PUNCT cana-3900 61 10 ring ring NOUN cana-3900 61 11 then then ADV cana-3900 61 12 n n PRON cana-3900 61 13 has have VERB cana-3900 61 14 no no DET cana-3900 61 15 non non ADJ cana-3900 61 16 - - ADJ cana-3900 61 17 zero zero ADJ cana-3900 61 18 nilpotent nilpotent ADJ cana-3900 61 19 elements element NOUN cana-3900 61 20 . . PUNCT cana-3900 62 1 proof proof NOUN cana-3900 62 2 . . PUNCT cana-3900 63 1 ( ( PUNCT cana-3900 63 2 i i NOUN cana-3900 63 3 ) ) PUNCT cana-3900 63 4 let let VERB cana-3900 63 5 n n PRON cana-3900 63 6 be be AUX cana-3900 63 7 a a DET cana-3900 63 8 𝛿1near 𝛿1near NOUN cana-3900 63 9 ring.then ring.then ADV cana-3900 63 10 for for ADP cana-3900 63 11 all all DET cana-3900 63 12 x x NOUN cana-3900 63 13 , , PUNCT cana-3900 63 14 y y PROPN cana-3900 63 15 in in ADP cana-3900 63 16 n n CCONJ cana-3900 63 17 , , PUNCT cana-3900 63 18 π‘₯𝑁𝑦 π‘₯𝑁𝑦 NOUN cana-3900 63 19 = = PUNCT cana-3900 63 20 𝑁π‘₯2𝑦2 𝑁π‘₯2𝑦2 NOUN cana-3900 63 21 … … PUNCT cana-3900 63 22 … … PUNCT cana-3900 63 23 … … PUNCT cana-3900 63 24 … … SYM cana-3900 63 25 . . NUM cana-3900 63 26 … … PUNCT cana-3900 63 27 .(1 .(1 NUM cana-3900 63 28 ) ) PUNCT cana-3900 64 1 putting put VERB cana-3900 64 2 y=1 y=1 PRON cana-3900 64 3 , , PUNCT cana-3900 64 4 we we PRON cana-3900 64 5 get get VERB cana-3900 64 6 π‘₯𝑁. π‘₯𝑁. ADP cana-3900 64 7 1 1 NUM cana-3900 64 8 = = NOUN cana-3900 64 9 𝑁π‘₯2 𝑁π‘₯2 ADJ cana-3900 64 10 . . PUNCT cana-3900 65 1 1 1 NUM cana-3900 65 2 for for ADP cana-3900 65 3 all all DET cana-3900 65 4 x x NOUN cana-3900 65 5 in in ADP cana-3900 65 6 n. n. NOUN cana-3900 65 7 β‡’ β‡’ NOUN cana-3900 65 8 π‘₯𝑁 π‘₯𝑁 NOUN cana-3900 65 9 = = NOUN cana-3900 65 10 𝑁π‘₯2 𝑁π‘₯2 NOUN cana-3900 65 11 for for ADP cana-3900 65 12 all all DET cana-3900 65 13 x x NOUN cana-3900 65 14 in in ADP cana-3900 65 15 n. n. NOUN cana-3900 65 16 when when SCONJ cana-3900 65 17 π‘₯ π‘₯ PROPN cana-3900 65 18 = = SYM cana-3900 65 19 0 0 NUM cana-3900 65 20 , , PUNCT cana-3900 65 21 0𝑁 0𝑁 NOUN cana-3900 65 22 = = PUNCT cana-3900 65 23 𝑁0 𝑁0 ADJ cana-3900 65 24 = = PUNCT cana-3900 65 25 { { PUNCT cana-3900 65 26 0 0 NUM cana-3900 65 27 } } PUNCT cana-3900 65 28 . . PUNCT cana-3900 66 1 it it PRON cana-3900 66 2 follows follow VERB cana-3900 66 3 that that SCONJ cana-3900 66 4 n n PRON cana-3900 66 5 is be AUX cana-3900 66 6 zero zero NUM cana-3900 66 7 symmetric symmetric NOUN cana-3900 66 8 . . PUNCT cana-3900 67 1 ( ( PUNCT cana-3900 67 2 ii ii NOUN cana-3900 67 3 ) ) PUNCT cana-3900 67 4 putting put VERB cana-3900 67 5 y=1 y=1 NOUN cana-3900 67 6 in in ADP cana-3900 67 7 equation equation NOUN cana-3900 67 8 ( ( PUNCT cana-3900 67 9 1 1 NUM cana-3900 67 10 ) ) PUNCT cana-3900 67 11 , , PUNCT cana-3900 67 12 we we PRON cana-3900 67 13 get get VERB cana-3900 67 14 π‘₯𝑁 π‘₯𝑁 NOUN cana-3900 67 15 = = NOUN cana-3900 67 16 𝑁π‘₯2 𝑁π‘₯2 NOUN cana-3900 67 17 for for ADP cana-3900 67 18 all all DET cana-3900 67 19 x x NOUN cana-3900 67 20 in in ADP cana-3900 67 21 n n PRON cana-3900 67 22 … … PUNCT cana-3900 67 23 … … PUNCT cana-3900 67 24 … … PUNCT cana-3900 67 25 … … PUNCT cana-3900 67 26 …… …… NOUN cana-3900 67 27 …… …… NOUN cana-3900 67 28 ... ... PUNCT cana-3900 67 29 ( ( PUNCT cana-3900 67 30 2 2 NUM cana-3900 67 31 ) ) PUNCT cana-3900 67 32 . . PUNCT cana-3900 68 1 now now ADV cana-3900 68 2 π‘₯2 π‘₯2 PROPN cana-3900 68 3 ∈ ∈ PROPN cana-3900 68 4 π‘₯𝑁 π‘₯𝑁 NOUN cana-3900 68 5 for for ADP cana-3900 68 6 all all DET cana-3900 68 7 x x NOUN cana-3900 68 8 in in ADP cana-3900 68 9 n n PRON cana-3900 68 10 β‡’ β‡’ NOUN cana-3900 68 11 π‘₯2 π‘₯2 NOUN cana-3900 68 12 ∈ ∈ PROPN cana-3900 68 13 𝑁π‘₯2 𝑁π‘₯2 NOUN cana-3900 69 1 [ [ X cana-3900 69 2 by by ADP cana-3900 69 3 equation equation NOUN cana-3900 69 4 ( ( PUNCT cana-3900 69 5 2 2 NUM cana-3900 69 6 ) ) PUNCT cana-3900 69 7 ] ] PUNCT cana-3900 69 8 . . PUNCT cana-3900 70 1 we we PRON cana-3900 70 2 have have VERB cana-3900 70 3 π‘₯2 π‘₯2 NOUN cana-3900 70 4 = = SYM cana-3900 70 5 𝑧π‘₯2 𝑧π‘₯2 NOUN cana-3900 70 6 for for ADP cana-3900 70 7 some some DET cana-3900 70 8 z z NOUN cana-3900 70 9 in in ADP cana-3900 70 10 n. n. NOUN cana-3900 70 11 therefore therefore ADV cana-3900 70 12 ( ( PUNCT cana-3900 70 13 π‘₯ π‘₯ NOUN cana-3900 70 14 βˆ’ βˆ’ PROPN cana-3900 70 15 𝑧π‘₯)π‘₯ 𝑧π‘₯)π‘₯ PROPN cana-3900 70 16 = = SYM cana-3900 70 17 0 0 PROPN cana-3900 70 18 . . PUNCT cana-3900 71 1 by by ADP cana-3900 71 2 k k PROPN cana-3900 71 3 ( ( PUNCT cana-3900 71 4 2 2 NUM cana-3900 71 5 ) ) PUNCT cana-3900 71 6 , , PUNCT cana-3900 71 7 this this PRON cana-3900 71 8 implies imply VERB cana-3900 71 9 that that SCONJ cana-3900 71 10 π‘₯(π‘₯ π‘₯(π‘₯ PROPN cana-3900 71 11 βˆ’ βˆ’ PROPN cana-3900 71 12 𝑧π‘₯ 𝑧π‘₯ NOUN cana-3900 71 13 ) ) PUNCT cana-3900 71 14 = = SYM cana-3900 71 15 0 0 NUM cana-3900 71 16 and and CCONJ cana-3900 71 17 𝑧π‘₯(π‘₯ 𝑧π‘₯(π‘₯ NOUN cana-3900 71 18 βˆ’ βˆ’ PROPN cana-3900 71 19 𝑧π‘₯ 𝑧π‘₯ NOUN cana-3900 71 20 ) ) PUNCT cana-3900 71 21 = = SYM cana-3900 72 1 0 0 X cana-3900 72 2 . . PUNCT cana-3900 73 1 consequently consequently ADV cana-3900 73 2 ( ( PUNCT cana-3900 73 3 π‘₯ π‘₯ NOUN cana-3900 73 4 βˆ’ βˆ’ PROPN cana-3900 73 5 𝑧π‘₯ 𝑧π‘₯ NOUN cana-3900 73 6 ) ) PUNCT cana-3900 73 7 2 2 NUM cana-3900 73 8 = = SYM cana-3900 73 9 0 0 NUM cana-3900 73 10 . . PUNCT cana-3900 74 1 by by ADP cana-3900 74 2 assumption assumption NOUN cana-3900 74 3 𝐿 𝐿 PROPN cana-3900 74 4 = = SYM cana-3900 74 5 { { PUNCT cana-3900 74 6 0 0 NUM cana-3900 74 7 } } PUNCT cana-3900 74 8 therefore therefore ADV cana-3900 74 9 π‘₯ π‘₯ X cana-3900 74 10 – – PUNCT cana-3900 74 11 𝑧π‘₯ 𝑧π‘₯ NOUN cana-3900 74 12 = = SYM cana-3900 74 13 0 0 NUM cana-3900 74 14 forcing force VERB cana-3900 74 15 π‘₯ π‘₯ X cana-3900 74 16 = = PUNCT cana-3900 74 17 𝑧π‘₯. 𝑧π‘₯. NOUN cana-3900 74 18 thus thus ADV cana-3900 74 19 π‘₯ π‘₯ PRON cana-3900 74 20 ∈ ∈ PROPN cana-3900 74 21 𝑁π‘₯ 𝑁π‘₯ PROPN cana-3900 74 22 i.e. i.e. X cana-3900 74 23 n n PRON cana-3900 74 24 is be AUX cana-3900 74 25 an an DET cana-3900 74 26 s s NOUN cana-3900 74 27 - - PUNCT cana-3900 74 28 near near ADP cana-3900 74 29 - - PUNCT cana-3900 74 30 ring ring NOUN cana-3900 74 31 . . PUNCT cana-3900 75 1 ( ( PUNCT cana-3900 75 2 iii iii X cana-3900 75 3 ) ) PUNCT cana-3900 75 4 if if SCONJ cana-3900 75 5 n n PRON cana-3900 75 6 is be AUX cana-3900 75 7 an an DET cana-3900 75 8 s'-near s'-near NOUN cana-3900 75 9 - - PUNCT cana-3900 75 10 ring ring NOUN cana-3900 75 11 then then ADV cana-3900 75 12 π‘₯ π‘₯ DET cana-3900 75 13 ∈ ∈ NOUN cana-3900 75 14 π‘₯𝑁 π‘₯𝑁 NOUN cana-3900 75 15 and and CCONJ cana-3900 75 16 since since SCONJ cana-3900 75 17 π‘₯𝑁 π‘₯𝑁 NOUN cana-3900 75 18 = = NOUN cana-3900 75 19 𝑁π‘₯2 𝑁π‘₯2 NOUN cana-3900 75 20 we we PRON cana-3900 75 21 get get VERB cana-3900 75 22 π‘₯ π‘₯ PRON cana-3900 75 23 = = PUNCT cana-3900 75 24 𝑛π‘₯2 𝑛π‘₯2 VERB cana-3900 75 25 for for ADP cana-3900 75 26 some some DET cana-3900 75 27 𝑛 𝑛 PRON cana-3900 75 28 ∈ ∈ NOUN cana-3900 75 29 𝑁. 𝑁. PROPN cana-3900 75 30 thereforeπ‘₯2 thereforeπ‘₯2 NOUN cana-3900 76 1 = = SYM cana-3900 76 2 0 0 NUM cana-3900 76 3 β‡’ β‡’ NOUN cana-3900 76 4 π‘₯ π‘₯ X cana-3900 77 1 = = SYM cana-3900 77 2 0 0 X cana-3900 77 3 . . PUNCT cana-3900 78 1 n n PROPN cana-3900 78 2 has have VERB cana-3900 78 3 no no DET cana-3900 78 4 non non ADJ cana-3900 78 5 - - ADJ cana-3900 78 6 zero zero ADJ cana-3900 78 7 nilpotent nilpotent ADJ cana-3900 78 8 elements element NOUN cana-3900 78 9 , , PUNCT cana-3900 78 10 from from ADP cana-3900 78 11 k k PROPN cana-3900 78 12 ( ( PUNCT cana-3900 78 13 4 4 NUM cana-3900 78 14 ) ) PUNCT cana-3900 78 15 . . PUNCT cana-3900 79 1 corollary3.2.2 corollary3.2.2 PROPN cana-3900 79 2 . . PUNCT cana-3900 80 1 communications communication NOUN cana-3900 80 2 on on ADP cana-3900 80 3 applied apply VERB cana-3900 80 4 nonlinear nonlinear ADJ cana-3900 80 5 analysis analysis NOUN cana-3900 80 6 issn issn NOUN cana-3900 80 7 : : PUNCT cana-3900 80 8 1074 1074 NUM cana-3900 80 9 - - PUNCT cana-3900 80 10 133x 133x NUM cana-3900 80 11 vol vol NOUN cana-3900 80 12 32 32 NUM cana-3900 80 13 no no NOUN cana-3900 80 14 . . PUNCT cana-3900 81 1 9s 9s NUM cana-3900 81 2 ( ( PUNCT cana-3900 81 3 2025 2025 NUM cana-3900 81 4 ) ) PUNCT cana-3900 81 5 348 348 NUM cana-3900 81 6 https://internationalpubls.com https://internationalpubls.com X cana-3900 82 1 if if SCONJ cana-3900 82 2 n n PRON cana-3900 82 3 is be AUX cana-3900 82 4 a a DET cana-3900 82 5 𝛿1near 𝛿1near NOUN cana-3900 82 6 - - PUNCT cana-3900 82 7 ring ring NOUN cana-3900 82 8 without without ADP cana-3900 82 9 non non ADJ cana-3900 82 10 - - ADJ cana-3900 82 11 zero zero ADJ cana-3900 82 12 nilpotent nilpotent ADJ cana-3900 82 13 elements element NOUN cana-3900 82 14 , , PUNCT cana-3900 82 15 then then ADV cana-3900 82 16 from from ADP cana-3900 82 17 k k PROPN cana-3900 82 18 ( ( PUNCT cana-3900 82 19 2 2 NUM cana-3900 82 20 ) ) PUNCT cana-3900 82 21 , , PUNCT cana-3900 82 22 we we PRON cana-3900 82 23 see see VERB cana-3900 82 24 that that SCONJ cana-3900 82 25 n n PRON cana-3900 82 26 has have AUX cana-3900 82 27 ( ( PUNCT cana-3900 82 28 * * PROPN cana-3900 82 29 , , PUNCT cana-3900 82 30 ifp ifp NOUN cana-3900 82 31 ) ) PUNCT cana-3900 82 32 . . PUNCT cana-3900 83 1 it it PRON cana-3900 83 2 is be AUX cana-3900 83 3 obvious obvious ADJ cana-3900 83 4 that that SCONJ cana-3900 83 5 the the DET cana-3900 83 6 property property NOUN cana-3900 83 7 𝛿1is 𝛿1i NOUN cana-3900 83 8 preserved preserve VERB cana-3900 83 9 by by ADP cana-3900 83 10 near near ADJ cana-3900 83 11 - - PUNCT cana-3900 83 12 ring ring NOUN cana-3900 83 13 homomorphisms homomorphism NOUN cana-3900 83 14 . . PUNCT cana-3900 84 1 consequently consequently ADV cana-3900 84 2 , , PUNCT cana-3900 84 3 we we PRON cana-3900 84 4 have have VERB cana-3900 84 5 proposition proposition NOUN cana-3900 84 6 3.2.3 3.2.3 NUM cana-3900 84 7 any any DET cana-3900 84 8 homomorphism homomorphism NOUN cana-3900 84 9 image image NOUN cana-3900 84 10 of of ADP cana-3900 84 11 a a DET cana-3900 84 12 𝛿1near 𝛿1near NOUN cana-3900 84 13 - - PUNCT cana-3900 84 14 ring ring NOUN cana-3900 84 15 is be AUX cana-3900 84 16 𝛿1 𝛿1 NOUN cana-3900 84 17 - - PUNCT cana-3900 84 18 near near ADP cana-3900 84 19 - - PUNCT cana-3900 84 20 ring ring NOUN cana-3900 84 21 . . PUNCT cana-3900 85 1 proof proof NOUN cana-3900 85 2 let let VERB cana-3900 85 3 𝑓 𝑓 PRON cana-3900 85 4 : : PUNCT cana-3900 85 5 n n X cana-3900 85 6 β†’ β†’ SYM cana-3900 85 7 n n CCONJ cana-3900 85 8 β€² β€² NOUN cana-3900 85 9 be be AUX cana-3900 85 10 a a DET cana-3900 85 11 near near ADJ cana-3900 85 12 - - PUNCT cana-3900 85 13 ring ring NOUN cana-3900 85 14 epimorphism epimorphism NOUN cana-3900 85 15 . . PUNCT cana-3900 86 1 let let VERB cana-3900 86 2 π‘₯ π‘₯ PROPN cana-3900 86 3 β€² β€² VERB cana-3900 86 4 , , PUNCT cana-3900 86 5 𝑦′ 𝑦′ X cana-3900 86 6 ∈ ∈ NOUN cana-3900 86 7 𝑁 𝑁 PROPN cana-3900 86 8 β€². β€². NOUN cana-3900 86 9 then then ADV cana-3900 86 10 there there PRON cana-3900 86 11 exist exist VERB cana-3900 86 12 π‘₯ π‘₯ PRON cana-3900 86 13 , , PUNCT cana-3900 86 14 𝑦 𝑦 NOUN cana-3900 86 15 ∈ ∈ NOUN cana-3900 87 1 𝑁 𝑁 ADP cana-3900 87 2 such such ADJ cana-3900 87 3 that that SCONJ cana-3900 87 4 𝑓 𝑓 DET cana-3900 87 5 ( ( PUNCT cana-3900 87 6 π‘₯ π‘₯ NOUN cana-3900 87 7 ) ) PUNCT cana-3900 87 8 = = SYM cana-3900 88 1 π‘₯β€² π‘₯β€² PROPN cana-3900 88 2 , , PUNCT cana-3900 88 3 𝑓 𝑓 PRON cana-3900 88 4 ( ( PUNCT cana-3900 88 5 𝑦 𝑦 NOUN cana-3900 88 6 ) ) PUNCT cana-3900 88 7 = = SYM cana-3900 88 8 𝑦 𝑦 NUM cana-3900 88 9 β€². β€². NOUN cana-3900 88 10 also also ADV cana-3900 88 11 for for ADP cana-3900 88 12 𝑛′ 𝑛′ NOUN cana-3900 88 13 ∈ ∈ NOUN cana-3900 88 14 𝑁 𝑁 PROPN cana-3900 88 15 β€² β€² NOUN cana-3900 88 16 there there ADV cana-3900 88 17 exist exist VERB cana-3900 88 18 𝑛 𝑛 DET cana-3900 88 19 ∈ ∈ NOUN cana-3900 88 20 𝑁 𝑁 PROPN cana-3900 88 21 such such ADJ cana-3900 88 22 that that SCONJ cana-3900 88 23 𝑓 𝑓 DET cana-3900 88 24 ( ( PUNCT cana-3900 88 25 𝑛 𝑛 NOUN cana-3900 88 26 ) ) PUNCT cana-3900 88 27 = = SYM cana-3900 88 28 𝑛′. 𝑛′. NOUN cana-3900 88 29 since since SCONJ cana-3900 88 30 n n NUM cana-3900 88 31 is be AUX cana-3900 88 32 𝛿1 𝛿1 NOUN cana-3900 88 33 βˆ— βˆ— NOUN cana-3900 88 34 near near ADP cana-3900 88 35 - - PUNCT cana-3900 88 36 ring ring NOUN cana-3900 88 37 , , PUNCT cana-3900 88 38 π‘₯𝑁𝑦 π‘₯𝑁𝑦 NOUN cana-3900 88 39 = = PUNCT cana-3900 88 40 𝑁π‘₯2𝑦2 𝑁π‘₯2𝑦2 NOUN cana-3900 88 41 … … PUNCT cana-3900 88 42 … … PUNCT cana-3900 88 43 … … PUNCT cana-3900 88 44 … … PUNCT cana-3900 88 45 … … PUNCT cana-3900 88 46 ( ( PUNCT cana-3900 88 47 3 3 X cana-3900 88 48 ) ) PUNCT cana-3900 88 49 now now ADV cana-3900 88 50 , , PUNCT cana-3900 88 51 π‘₯ π‘₯ PROPN cana-3900 88 52 ′𝑛′𝑦 ′𝑛′𝑦 VERB cana-3900 88 53 β€² β€² NOUN cana-3900 88 54 = = SYM cana-3900 89 1 𝑓 𝑓 PROPN cana-3900 89 2 ( ( PUNCT cana-3900 89 3 π‘₯)𝑓 π‘₯)𝑓 X cana-3900 89 4 ( ( PUNCT cana-3900 89 5 𝑛)𝑓 𝑛)𝑓 X cana-3900 89 6 ( ( PUNCT cana-3900 89 7 𝑦 𝑦 NOUN cana-3900 89 8 ) ) PUNCT cana-3900 89 9 = = SYM cana-3900 89 10 𝑓 𝑓 PROPN cana-3900 89 11 ( ( PUNCT cana-3900 89 12 π‘₯𝑛𝑦 π‘₯𝑛𝑦 PROPN cana-3900 89 13 ) ) PUNCT cana-3900 90 1 [ [ X cana-3900 90 2 since since SCONJ cana-3900 90 3 𝑓 𝑓 PRON cana-3900 90 4 is be AUX cana-3900 90 5 a a DET cana-3900 90 6 homomorphism homomorphism NOUN cana-3900 90 7 ] ] PUNCT cana-3900 90 8 = = SYM cana-3900 90 9 𝑓 𝑓 X cana-3900 90 10 ( ( PUNCT cana-3900 90 11 𝑛π‘₯2𝑦2 𝑛π‘₯2𝑦2 NOUN cana-3900 90 12 ) ) PUNCT cana-3900 90 13 [ [ X cana-3900 90 14 by by ADP cana-3900 90 15 equation equation NOUN cana-3900 90 16 ( ( PUNCT cana-3900 90 17 3 3 NUM cana-3900 90 18 ) ) PUNCT cana-3900 90 19 ] ] PUNCT cana-3900 91 1 = = X cana-3900 91 2 𝑓(𝑛)𝑓(π‘₯2)𝑓(𝑦2 𝑓(𝑛)𝑓(π‘₯2)𝑓(𝑦2 PROPN cana-3900 91 3 ) ) PUNCT cana-3900 91 4 therefore therefore ADV cana-3900 91 5 , , PUNCT cana-3900 91 6 π‘₯ π‘₯ PROPN cana-3900 91 7 ′𝑛′𝑦 ′𝑛′𝑦 VERB cana-3900 91 8 β€² β€² NUM cana-3900 91 9 ∈ ∈ NOUN cana-3900 91 10 𝑁′π‘₯2′𝑦2β€². 𝑁′π‘₯2′𝑦2β€². NOUN cana-3900 91 11 consequently consequently ADV cana-3900 91 12 , , PUNCT cana-3900 91 13 π‘₯ π‘₯ PRON cana-3900 91 14 ′𝑁 ′𝑁 PROPN cana-3900 91 15 ′𝑦′ ′𝑦′ NOUN cana-3900 91 16 βŠ† βŠ† NUM cana-3900 91 17 𝑁 𝑁 PROPN cana-3900 91 18 β€²π‘₯2′𝑦2β€² β€²π‘₯2′𝑦2β€² NOUN cana-3900 91 19 … … PUNCT cana-3900 91 20 … … PUNCT cana-3900 91 21 … … PUNCT cana-3900 91 22 … … PUNCT cana-3900 91 23 … … PUNCT cana-3900 91 24 . . PUNCT cana-3900 91 25 . . PUNCT cana-3900 92 1 ( ( PUNCT cana-3900 92 2 4 4 X cana-3900 92 3 ) ) PUNCT cana-3900 92 4 similarly similarly ADV cana-3900 92 5 , , PUNCT cana-3900 92 6 𝑁′π‘₯2′𝑦2β€² 𝑁′π‘₯2′𝑦2β€² PROPN cana-3900 92 7 βŠ† βŠ† NUM cana-3900 92 8 π‘₯ π‘₯ DET cana-3900 92 9 ′𝑁′𝑦 ′𝑁′𝑦 PROPN cana-3900 92 10 β€² β€² NUM cana-3900 92 11 … … PUNCT cana-3900 92 12 …… …… NOUN cana-3900 92 13 …… …… NOUN cana-3900 92 14 … … PUNCT cana-3900 92 15 ( ( PUNCT cana-3900 92 16 5 5 X cana-3900 92 17 ) ) PUNCT cana-3900 92 18 combining combine VERB cana-3900 92 19 equations equation NOUN cana-3900 92 20 ( ( PUNCT cana-3900 92 21 4 4 NUM cana-3900 92 22 ) ) PUNCT cana-3900 92 23 and and CCONJ cana-3900 92 24 ( ( PUNCT cana-3900 92 25 5 5 NUM cana-3900 92 26 ) ) PUNCT cana-3900 92 27 , , PUNCT cana-3900 92 28 we we PRON cana-3900 92 29 get get VERB cana-3900 92 30 π‘₯ π‘₯ PRON cana-3900 92 31 ′𝑁′𝑦 ′𝑁′𝑦 NOUN cana-3900 92 32 β€² β€² NOUN cana-3900 92 33 βŠ† βŠ† NUM cana-3900 92 34 𝑁′π‘₯2′𝑦2β€² 𝑁′π‘₯2′𝑦2β€² PROPN cana-3900 92 35 hence hence ADV cana-3900 92 36 n n ADV cana-3900 92 37 β€² β€² NOUN cana-3900 92 38 is be AUX cana-3900 92 39 also also ADV cana-3900 92 40 𝛿1 𝛿1 ADJ cana-3900 92 41 near near ADP cana-3900 92 42 - - PUNCT cana-3900 92 43 ring ring NOUN cana-3900 93 1 and and CCONJ cana-3900 93 2 the the DET cana-3900 93 3 desired desire VERB cana-3900 93 4 result result NOUN cana-3900 93 5 follows follow VERB cana-3900 93 6 . . PUNCT cana-3900 94 1 as as ADP cana-3900 94 2 an an DET cana-3900 94 3 immediate immediate ADJ cana-3900 94 4 consequence consequence NOUN cana-3900 94 5 of of ADP cana-3900 94 6 proposition proposition NOUN cana-3900 94 7 3.2.3we 3.2.3we NUM cana-3900 94 8 have have VERB cana-3900 94 9 the the DET cana-3900 94 10 following follow VERB cana-3900 94 11 theorem theorem NOUN cana-3900 94 12 : : PUNCT cana-3900 94 13 theorem theorem VERB cana-3900 94 14 3.2.4 3.2.4 NUM cana-3900 94 15 every every DET cana-3900 94 16 𝛿1near 𝛿1near NOUN cana-3900 94 17 - - PUNCT cana-3900 94 18 ring ring NOUN cana-3900 94 19 n n NOUN cana-3900 94 20 is be AUX cana-3900 94 21 isomorphic isomorphic ADJ cana-3900 94 22 to to ADP cana-3900 94 23 a a DET cana-3900 94 24 subdirect subdirect NOUN cana-3900 94 25 product product NOUN cana-3900 94 26 of of ADP cana-3900 94 27 subdirectly subdirectly ADV cana-3900 94 28 irreducible irreducible ADJ cana-3900 94 29 𝛿1 𝛿1 NOUN cana-3900 94 30 near near ADP cana-3900 94 31 - - PUNCT cana-3900 94 32 ring ring NOUN cana-3900 94 33 . . PUNCT cana-3900 95 1 proof proof NOUN cana-3900 95 2 . . PUNCT cana-3900 96 1 by by ADP cana-3900 96 2 theorem theorem ADJ cana-3900 96 3 1.62 1.62 NUM cana-3900 96 4 , , PUNCT cana-3900 96 5 p.26 p.26 PROPN cana-3900 96 6 of of ADP cana-3900 96 7 pilz pilz PROPN cana-3900 96 8 [ [ X cana-3900 96 9 3 3 NUM cana-3900 96 10 ] ] PUNCT cana-3900 96 11 , , PUNCT cana-3900 96 12 n n X cana-3900 96 13 is be AUX cana-3900 96 14 isomorphic isomorphic ADJ cana-3900 96 15 to to ADP cana-3900 96 16 a a DET cana-3900 96 17 sub sub NOUN cana-3900 96 18 direct direct ADJ cana-3900 96 19 product product NOUN cana-3900 96 20 of of ADP cana-3900 96 21 sub sub NOUN cana-3900 96 22 directly directly ADV cana-3900 96 23 irreducible irreducible ADJ cana-3900 96 24 near near ADP cana-3900 96 25 - - PUNCT cana-3900 96 26 ring ring NOUN cana-3900 96 27 ni ni PROPN cana-3900 96 28 's 's PART cana-3900 96 29 , , PUNCT cana-3900 96 30 say say INTJ cana-3900 96 31 , , PUNCT cana-3900 96 32 and and CCONJ cana-3900 96 33 each each DET cana-3900 96 34 ni ni PROPN cana-3900 96 35 is be AUX cana-3900 96 36 a a DET cana-3900 96 37 homomorphic homomorphic ADJ cana-3900 96 38 image image NOUN cana-3900 96 39 of of ADP cana-3900 96 40 n n NOUN cana-3900 96 41 under under ADP cana-3900 96 42 the the DET cana-3900 96 43 projection projection NOUN cana-3900 96 44 map map NOUN cana-3900 96 45 πœ‹π‘–. πœ‹π‘–. ADP cana-3900 96 46 the the DET cana-3900 96 47 desired desire VERB cana-3900 96 48 result result NOUN cana-3900 96 49 now now ADV cana-3900 96 50 follows follow VERB cana-3900 96 51 from from ADP cana-3900 96 52 proposition proposition NOUN cana-3900 96 53 3.2.3 3.2.3 NUM cana-3900 96 54 . . PUNCT cana-3900 97 1 we we PRON cana-3900 97 2 shall shall AUX cana-3900 97 3 now now ADV cana-3900 97 4 discuss discuss VERB cana-3900 97 5 the the DET cana-3900 97 6 behavior behavior NOUN cana-3900 97 7 of of ADP cana-3900 97 8 n n NOUN cana-3900 97 9 - - PUNCT cana-3900 97 10 subgroups subgroup NOUN cana-3900 97 11 and and CCONJ cana-3900 97 12 ideals ideal NOUN cana-3900 97 13 of of ADP cana-3900 97 14 𝛿1 𝛿1 NOUN cana-3900 97 15 nearring nearre VERB cana-3900 97 16 . . PUNCT cana-3900 98 1 to to PART cana-3900 98 2 start start VERB cana-3900 98 3 with with ADP cana-3900 98 4 we we PRON cana-3900 98 5 have have VERB cana-3900 98 6 the the DET cana-3900 98 7 following following NOUN cana-3900 98 8 : : PUNCT cana-3900 98 9 proposition proposition NOUN cana-3900 98 10 3.2.5 3.2.5 NUM cana-3900 98 11 . . PUNCT cana-3900 99 1 let let VERB cana-3900 99 2 n n PRON cana-3900 99 3 be be AUX cana-3900 99 4 a a DET cana-3900 99 5 𝛿1near 𝛿1near NOUN cana-3900 99 6 - - PUNCT cana-3900 99 7 ring ring NOUN cana-3900 99 8 with with ADP cana-3900 99 9 identity identity NOUN cana-3900 99 10 , , PUNCT cana-3900 99 11 if if SCONJ cana-3900 99 12 n n NOUN cana-3900 99 13 is be AUX cana-3900 99 14 left leave VERB cana-3900 99 15 bipotent bipotent NOUN cana-3900 99 16 , , PUNCT cana-3900 99 17 and and CCONJ cana-3900 99 18 then then ADV cana-3900 99 19 every every DET cana-3900 99 20 n n NOUN cana-3900 99 21 - - PUNCT cana-3900 99 22 subgroup subgroup NOUN cana-3900 99 23 of of ADP cana-3900 99 24 n n PROPN cana-3900 99 25 is be AUX cana-3900 99 26 invariant invariant ADJ cana-3900 99 27 . . PUNCT cana-3900 100 1 proof proof NOUN cana-3900 100 2 . . PUNCT cana-3900 101 1 let let VERB cana-3900 101 2 n n PRON cana-3900 101 3 be be AUX cana-3900 101 4 a a DET cana-3900 101 5 𝛿1 𝛿1 NOUN cana-3900 101 6 near near ADP cana-3900 101 7 ring ring NOUN cana-3900 101 8 with with ADP cana-3900 101 9 identity identity NOUN cana-3900 101 10 . . PUNCT cana-3900 102 1 then then ADV cana-3900 102 2 for for SCONJ cana-3900 102 3 all all DET cana-3900 102 4 x x NOUN cana-3900 102 5 , , PUNCT cana-3900 102 6 y y PROPN cana-3900 102 7 in in ADP cana-3900 102 8 n n CCONJ cana-3900 102 9 , , PUNCT cana-3900 102 10 π‘₯𝑁 π‘₯𝑁 ADJ cana-3900 102 11 = = NOUN cana-3900 102 12 𝑁π‘₯2 𝑁π‘₯2 NOUN cana-3900 102 13 … … PUNCT cana-3900 102 14 … … PUNCT cana-3900 102 15 … … PUNCT cana-3900 102 16 … … PUNCT cana-3900 102 17 … … X cana-3900 102 18 .(6 .(6 NOUN cana-3900 102 19 ) ) PUNCT cana-3900 102 20 . . PUNCT cana-3900 103 1 let let VERB cana-3900 103 2 a a DET cana-3900 103 3 be be AUX cana-3900 103 4 any any DET cana-3900 103 5 n n NOUN cana-3900 103 6 - - PUNCT cana-3900 103 7 subgroup subgroup NOUN cana-3900 103 8 of of ADP cana-3900 103 9 n n CCONJ cana-3900 103 10 , , PUNCT cana-3900 103 11 then then ADV cana-3900 103 12 𝐴 𝐴 PROPN cana-3900 103 13 = = PUNCT cana-3900 103 14 οƒ₯ οƒ₯ X cana-3900 103 15 οƒŽax οƒŽax X cana-3900 103 16 nx nx NUM cana-3900 103 17 … … SYM cana-3900 103 18 … … PUNCT cana-3900 103 19 … … PUNCT cana-3900 103 20 …… …… X cana-3900 103 21 ( ( PUNCT cana-3900 103 22 7 7 NUM cana-3900 103 23 ) ) PUNCT cana-3900 103 24 . . PUNCT cana-3900 104 1 now now ADV cana-3900 104 2 , , PUNCT cana-3900 104 3 𝑁π‘₯𝑁 𝑁π‘₯𝑁 NOUN cana-3900 104 4 = = SYM cana-3900 104 5 𝑁𝑁π‘₯21 𝑁𝑁π‘₯21 NOUN cana-3900 104 6 = = NOUN cana-3900 104 7 𝑁π‘₯2 𝑁π‘₯2 NOUN cana-3900 105 1 [ [ X cana-3900 105 2 by by ADP cana-3900 105 3 equation equation NOUN cana-3900 105 4 ( ( PUNCT cana-3900 105 5 6 6 NUM cana-3900 105 6 ) ) PUNCT cana-3900 105 7 ] ] PUNCT cana-3900 106 1 βŠ† βŠ† X cana-3900 106 2 𝑁π‘₯. 𝑁π‘₯. PROPN cana-3900 106 3 [ [ X cana-3900 106 4 since since SCONJ cana-3900 106 5 n n PROPN cana-3900 106 6 is be AUX cana-3900 106 7 left leave VERB cana-3900 106 8 bipotent bipotent NOUN cana-3900 106 9 ] ] PUNCT cana-3900 106 10 ( ( PUNCT cana-3900 106 11 i.e. i.e. X cana-3900 106 12 ) ) PUNCT cana-3900 106 13 𝑁π‘₯𝑁 𝑁π‘₯𝑁 PROPN cana-3900 106 14 βŠ† βŠ† NUM cana-3900 106 15 𝑁π‘₯ 𝑁π‘₯ NOUN cana-3900 106 16 …… …… NOUN cana-3900 106 17 …… …… NOUN cana-3900 106 18 . . PUNCT cana-3900 107 1 ( ( PUNCT cana-3900 107 2 8) 8) NUM cana-3900 107 3 . . PUNCT cana-3900 108 1 therefore therefore ADV cana-3900 108 2 , , PUNCT cana-3900 108 3 𝐴𝑁 𝐴𝑁 PROPN cana-3900 108 4 =( =( NOUN cana-3900 108 5 οƒ₯ οƒ₯ X cana-3900 108 6 οƒŽax οƒŽax X cana-3900 108 7 nx nx NUM cana-3900 108 8 ) ) PUNCT cana-3900 108 9 𝑁[by 𝑁[by NOUN cana-3900 108 10 equation equation NOUN cana-3900 108 11 ( ( PUNCT cana-3900 108 12 7 7 NUM cana-3900 108 13 ) ) PUNCT cana-3900 108 14 ] ] PUNCT cana-3900 109 1 βŠ† βŠ† NUM cana-3900 109 2 nnx nnx NOUN cana-3900 109 3 ax ax NOUN cana-3900 109 4 οƒ₯ οƒ₯ X cana-3900 109 5 οƒŽ οƒŽ PROPN cana-3900 109 6 βŠ†οƒ₯ βŠ†οƒ₯ PROPN cana-3900 110 1 οƒŽax οƒŽax PUNCT cana-3900 110 2 nx nx X cana-3900 111 1 [ [ PUNCT cana-3900 111 2 by by ADP cana-3900 111 3 equation equation NOUN cana-3900 111 4 ( ( PUNCT cana-3900 111 5 8)]= 8)]= NUM cana-3900 111 6 𝐴 𝐴 PROPN cana-3900 111 7 ( ( PUNCT cana-3900 111 8 ie.)𝐴𝑁 ie.)𝐴𝑁 NOUN cana-3900 111 9 βŠ† βŠ† NUM cana-3900 111 10 𝐴. 𝐴. NOUN cana-3900 111 11 consequently consequently ADV cana-3900 111 12 , , PUNCT cana-3900 111 13 a a PRON cana-3900 111 14 is be AUX cana-3900 111 15 invariant invariant ADJ cana-3900 111 16 nsubgroup nsubgroup NOUN cana-3900 111 17 . . PUNCT cana-3900 112 1 communications communication NOUN cana-3900 112 2 on on ADP cana-3900 112 3 applied apply VERB cana-3900 112 4 nonlinear nonlinear ADJ cana-3900 112 5 analysis analysis NOUN cana-3900 112 6 issn issn NOUN cana-3900 112 7 : : PUNCT cana-3900 112 8 1074 1074 NUM cana-3900 112 9 - - PUNCT cana-3900 112 10 133x 133x NUM cana-3900 112 11 vol vol NOUN cana-3900 112 12 32 32 NUM cana-3900 112 13 no no NOUN cana-3900 112 14 . . PUNCT cana-3900 113 1 9s 9s NUM cana-3900 113 2 ( ( PUNCT cana-3900 113 3 2025 2025 NUM cana-3900 113 4 ) ) PUNCT cana-3900 113 5 349 349 NUM cana-3900 114 1 https://internationalpubls.com https://internationalpubls.com X cana-3900 114 2 proposition proposition NOUN cana-3900 114 3 3.2.6 3.2.6 NUM cana-3900 114 4 . . PUNCT cana-3900 115 1 let let VERB cana-3900 115 2 n n PRON cana-3900 115 3 be be AUX cana-3900 115 4 a a DET cana-3900 115 5 𝛿1near 𝛿1near NOUN cana-3900 115 6 - - PUNCT cana-3900 115 7 ring ring NOUN cana-3900 115 8 . . PUNCT cana-3900 116 1 then then ADV cana-3900 116 2 every every DET cana-3900 116 3 left leave VERB cana-3900 116 4 ideal ideal NOUN cana-3900 116 5 of of ADP cana-3900 116 6 n n PROPN cana-3900 116 7 is be AUX cana-3900 116 8 an an DET cana-3900 116 9 ideal ideal NOUN cana-3900 116 10 . . PUNCT cana-3900 117 1 proof proof NOUN cana-3900 117 2 . . PUNCT cana-3900 118 1 let let VERB cana-3900 118 2 a a DET cana-3900 118 3 be be AUX cana-3900 118 4 a a DET cana-3900 118 5 left left ADJ cana-3900 118 6 ideal ideal NOUN cana-3900 118 7 of of ADP cana-3900 118 8 n. n. NOUN cana-3900 118 9 since since SCONJ cana-3900 118 10 n n PROPN cana-3900 118 11 is be AUX cana-3900 118 12 zero zero NUM cana-3900 118 13 - - PUNCT cana-3900 118 14 symmetric symmetric ADJ cana-3900 118 15 , , PUNCT cana-3900 118 16 𝑁𝐴 𝑁𝐴 PROPN cana-3900 118 17 βŠ† βŠ† NUM cana-3900 118 18 𝐴 𝐴 PROPN cana-3900 118 19 i.e. i.e. X cana-3900 118 20 a a PRON cana-3900 118 21 is be AUX cana-3900 118 22 an an DET cana-3900 118 23 n n NOUN cana-3900 118 24 - - PUNCT cana-3900 118 25 subgroup subgroup NOUN cana-3900 118 26 of of ADP cana-3900 118 27 n. n. PROPN cana-3900 118 28 proceeding proceeding NOUN cana-3900 118 29 as as ADP cana-3900 118 30 in in ADP cana-3900 118 31 proposition proposition NOUN cana-3900 118 32 3.2.5 3.2.5 NUM cana-3900 118 33 we we PRON cana-3900 118 34 get get VERB cana-3900 118 35 𝐴𝑁 𝐴𝑁 PROPN cana-3900 118 36 βŠ† βŠ† NUM cana-3900 118 37 𝐴. 𝐴. NOUN cana-3900 118 38 hence hence ADV cana-3900 118 39 a a PRON cana-3900 118 40 becomes become VERB cana-3900 118 41 an an DET cana-3900 118 42 ideal ideal NOUN cana-3900 118 43 . . PUNCT cana-3900 119 1 it it PRON cana-3900 119 2 is be AUX cana-3900 119 3 easy easy ADJ cana-3900 119 4 to to PART cana-3900 119 5 observe observe VERB cana-3900 119 6 the the DET cana-3900 119 7 following follow VERB cana-3900 119 8 : : PUNCT cana-3900 119 9 corollary corollary ADJ cana-3900 119 10 3.2.7 3.2.7 NUM cana-3900 119 11 every every DET cana-3900 119 12 left leave VERB cana-3900 119 13 ideal ideal NOUN cana-3900 119 14 ( ( PUNCT cana-3900 119 15 and and CCONJ cana-3900 119 16 therefore therefore ADV cana-3900 119 17 every every DET cana-3900 119 18 ideal ideal NOUN cana-3900 119 19 ) ) PUNCT cana-3900 119 20 of of ADP cana-3900 119 21 a a DET cana-3900 119 22 𝛿1 𝛿1 NOUN cana-3900 119 23 near near ADP cana-3900 119 24 - - PUNCT cana-3900 119 25 ring ring NOUN cana-3900 119 26 n n NOUN cana-3900 119 27 is be AUX cana-3900 119 28 an an DET cana-3900 119 29 invariant invariant ADJ cana-3900 119 30 n n CCONJ cana-3900 119 31 - - PUNCT cana-3900 119 32 subgroup subgroup NOUN cana-3900 119 33 of of ADP cana-3900 119 34 n. n. PROPN cana-3900 119 35 proposition proposition NOUN cana-3900 120 1 3.2.8 3.2.8 NUM cana-3900 120 2 . . PUNCT cana-3900 121 1 if if SCONJ cana-3900 121 2 n n PRON cana-3900 121 3 is be AUX cana-3900 121 4 a a DET cana-3900 121 5 𝛿1near 𝛿1near NOUN cana-3900 121 6 - - PUNCT cana-3900 121 7 ring ring NOUN cana-3900 121 8 with with ADP cana-3900 121 9 identity identity NOUN cana-3900 121 10 , , PUNCT cana-3900 121 11 then then ADV cana-3900 121 12 n n PRON cana-3900 121 13 has have VERB cana-3900 121 14 strong strong ADJ cana-3900 121 15 ifp ifp PROPN cana-3900 121 16 . . PUNCT cana-3900 122 1 proof proof NOUN cana-3900 122 2 . . PUNCT cana-3900 123 1 let let VERB cana-3900 123 2 n n PRON cana-3900 123 3 be be AUX cana-3900 123 4 a a DET cana-3900 123 5 𝛿1near 𝛿1near ADJ cana-3900 123 6 ring ring NOUN cana-3900 123 7 with with ADP cana-3900 123 8 identity identity NOUN cana-3900 123 9 . . PUNCT cana-3900 124 1 then then ADV cana-3900 124 2 for for SCONJ cana-3900 124 3 all all DET cana-3900 124 4 x x NOUN cana-3900 124 5 , , PUNCT cana-3900 124 6 y y PROPN cana-3900 124 7 in in ADP cana-3900 124 8 n n CCONJ cana-3900 124 9 , , PUNCT cana-3900 124 10 π‘₯𝑁 π‘₯𝑁 ADJ cana-3900 124 11 = = NOUN cana-3900 124 12 𝑁π‘₯2 𝑁π‘₯2 NOUN cana-3900 124 13 … … PUNCT cana-3900 124 14 … … PUNCT cana-3900 124 15 … … PUNCT cana-3900 124 16 … … PUNCT cana-3900 124 17 … … PUNCT cana-3900 124 18 … … PUNCT cana-3900 124 19 .(9 .(9 NUM cana-3900 124 20 ) ) PUNCT cana-3900 124 21 in in ADP cana-3900 124 22 view view NOUN cana-3900 124 23 of of ADP cana-3900 124 24 proposition proposition NOUN cana-3900 124 25 9.2 9.2 NUM cana-3900 124 26 pilz pilz NOUN cana-3900 125 1 [ [ X cana-3900 125 2 3 3 NUM cana-3900 125 3 ] ] PUNCT cana-3900 125 4 , , PUNCT cana-3900 125 5 we we PRON cana-3900 125 6 need need VERB cana-3900 125 7 only only ADV cana-3900 125 8 to to PART cana-3900 125 9 establish establish VERB cana-3900 125 10 that that SCONJ cana-3900 125 11 for for ADP cana-3900 125 12 all all DET cana-3900 125 13 ideals ideal NOUN cana-3900 125 14 i i PRON cana-3900 125 15 of of ADP cana-3900 125 16 n n PROPN cana-3900 125 17 and and CCONJ cana-3900 125 18 for for ADP cana-3900 125 19 all all DET cana-3900 125 20 a a DET cana-3900 125 21 , , PUNCT cana-3900 125 22 b b NOUN cana-3900 125 23 , , PUNCT cana-3900 125 24 n n CCONJ cana-3900 125 25 in in ADP cana-3900 125 26 n n CCONJ cana-3900 125 27 , , PUNCT cana-3900 125 28 π‘Žπ‘ π‘Žπ‘ PROPN cana-3900 125 29 ∈ ∈ PROPN cana-3900 125 30 𝐼 𝐼 PROPN cana-3900 125 31 β‡’ β‡’ NOUN cana-3900 125 32 π‘Žπ‘›π‘ π‘Žπ‘›π‘ NOUN cana-3900 125 33 ∈ ∈ PROPN cana-3900 125 34 𝐼. 𝐼. PROPN cana-3900 125 35 since since SCONJ cana-3900 125 36 i i PRON cana-3900 125 37 is be AUX cana-3900 125 38 an an DET cana-3900 125 39 ideal ideal NOUN cana-3900 125 40 , , PUNCT cana-3900 125 41 𝐼𝑁 𝐼𝑁 PROPN cana-3900 125 42 βŠ† βŠ† NUM cana-3900 125 43 𝐼 𝐼 PROPN cana-3900 125 44 and and CCONJ cana-3900 125 45 since since SCONJ cana-3900 125 46 n n ADV cana-3900 125 47 is be AUX cana-3900 125 48 zero zero NUM cana-3900 125 49 - - PUNCT cana-3900 125 50 symmetric symmetric ADJ cana-3900 125 51 , , PUNCT cana-3900 125 52 i i PRON cana-3900 125 53 is be AUX cana-3900 125 54 an an DET cana-3900 125 55 nsubgroup nsubgroup NOUN cana-3900 125 56 of of ADP cana-3900 125 57 n. n. NOUN cana-3900 125 58 i.e. i.e. X cana-3900 125 59 𝑁𝐼 𝑁𝐼 PROPN cana-3900 125 60 βŠ† βŠ† X cana-3900 125 61 𝐼. 𝐼. PROPN cana-3900 125 62 now now ADV cana-3900 125 63 π‘Žπ‘› π‘Žπ‘› VERB cana-3900 125 64 ∈ ∈ NOUN cana-3900 125 65 π‘Žπ‘ π‘Žπ‘ VERB cana-3900 125 66 = = X cana-3900 125 67 π‘π‘Ž2 π‘π‘Ž2 NOUN cana-3900 126 1 [ [ X cana-3900 126 2 by by ADP cana-3900 126 3 equation equation NOUN cana-3900 126 4 ( ( PUNCT cana-3900 126 5 9 9 NUM cana-3900 126 6 ) ) PUNCT cana-3900 126 7 ] ] PUNCT cana-3900 126 8 β‡’ β‡’ VERB cana-3900 126 9 π‘Žπ‘› π‘Žπ‘› PROPN cana-3900 126 10 = = PUNCT cana-3900 126 11 π‘›β€²π‘Ž2 π‘›β€²π‘Ž2 NOUN cana-3900 126 12 for for ADP cana-3900 126 13 some some DET cana-3900 126 14 𝑛′ 𝑛′ NOUN cana-3900 126 15 ∈ ∈ NOUN cana-3900 126 16 𝑁 𝑁 PROPN cana-3900 126 17 β‡’ β‡’ NOUN cana-3900 126 18 π‘Žπ‘›π‘ π‘Žπ‘›π‘ NOUN cana-3900 126 19 = = SYM cana-3900 126 20 ( ( PUNCT cana-3900 126 21 π‘›β€²π‘Ž2)𝑏 π‘›β€²π‘Ž2)𝑏 X cana-3900 126 22 = = SYM cana-3900 126 23 ( ( PUNCT cana-3900 126 24 𝑛′ 𝑛′ NOUN cana-3900 126 25 π‘Ž)(π‘Žπ‘ π‘Ž)(π‘Žπ‘ NUM cana-3900 126 26 ) ) PUNCT cana-3900 126 27 ∈ ∈ PROPN cana-3900 126 28 𝑁𝐼 𝑁𝐼 PROPN cana-3900 126 29 β‡’ β‡’ PROPN cana-3900 126 30 π‘Žπ‘›π‘ π‘Žπ‘›π‘ PROPN cana-3900 126 31 ∈ ∈ PROPN cana-3900 126 32 𝐼. 𝐼. PROPN cana-3900 126 33 notation notation NOUN cana-3900 126 34 3.2.9 3.2.9 NUM cana-3900 126 35 if if SCONJ cana-3900 126 36 a a DET cana-3900 126 37 𝛿1near 𝛿1near NOUN cana-3900 126 38 - - PUNCT cana-3900 126 39 ring ring NOUN cana-3900 126 40 n n NOUN cana-3900 126 41 is be AUX cana-3900 126 42 an an DET cana-3900 126 43 s s X cana-3900 126 44 ( ( PUNCT cana-3900 126 45 or or CCONJ cana-3900 126 46 s')-near s')-near ADJ cana-3900 126 47 - - PUNCT cana-3900 126 48 ring ring NOUN cana-3900 126 49 then then ADV cana-3900 126 50 we we PRON cana-3900 126 51 write write VERB cana-3900 126 52 that that SCONJ cana-3900 126 53 n n PRON cana-3900 126 54 is be AUX cana-3900 126 55 an an DET cana-3900 126 56 s s X cana-3900 126 57 𝛿1near 𝛿1near NOUN cana-3900 126 58 - - PUNCT cana-3900 126 59 ring ring NOUN cana-3900 126 60 ( ( PUNCT cana-3900 126 61 or or CCONJ cana-3900 126 62 s'𝛿1nearring s'𝛿1nearring NOUN cana-3900 126 63 ) ) PUNCT cana-3900 126 64 . . PUNCT cana-3900 127 1 remark remark VERB cana-3900 127 2 3.2.10 3.2.10 NUM cana-3900 127 3 for for ADP cana-3900 127 4 an an DET cana-3900 127 5 s𝛿1 s𝛿1 NOUN cana-3900 127 6 near near ADP cana-3900 127 7 - - PUNCT cana-3900 127 8 ring ring NOUN cana-3900 127 9 , , PUNCT cana-3900 127 10 we we PRON cana-3900 127 11 see see VERB cana-3900 127 12 that that PRON cana-3900 127 13 for for ADP cana-3900 127 14 all all DET cana-3900 127 15 x x NOUN cana-3900 127 16 in in ADP cana-3900 127 17 n n CCONJ cana-3900 127 18 , , PUNCT cana-3900 127 19 π‘₯ π‘₯ PRON cana-3900 127 20 ∈ ∈ PROPN cana-3900 127 21 𝑁π‘₯ 𝑁π‘₯ PROPN cana-3900 127 22 = = PUNCT cana-3900 127 23 π‘₯2 π‘₯2 NOUN cana-3900 127 24 𝑁 𝑁 PROPN cana-3900 127 25 β‡’ β‡’ VERB cana-3900 127 26 π‘₯ π‘₯ X cana-3900 127 27 = = PUNCT cana-3900 127 28 π‘₯2𝑛 π‘₯2𝑛 PROPN cana-3900 127 29 for for ADP cana-3900 127 30 some some DET cana-3900 127 31 𝑛 𝑛 PRON cana-3900 127 32 ∈ ∈ NOUN cana-3900 127 33 𝑁. 𝑁. PROPN cana-3900 127 34 hence hence ADV cana-3900 127 35 π‘₯2 π‘₯2 NOUN cana-3900 127 36 = = SYM cana-3900 127 37 0 0 NUM cana-3900 127 38 β‡’ β‡’ NOUN cana-3900 127 39 π‘₯ π‘₯ PROPN cana-3900 127 40 = = SYM cana-3900 127 41 0 0 PROPN cana-3900 127 42 and and CCONJ cana-3900 127 43 k k X cana-3900 127 44 ( ( PUNCT cana-3900 127 45 4 4 X cana-3900 127 46 ) ) PUNCT cana-3900 127 47 demands demand VERB cana-3900 127 48 that that SCONJ cana-3900 127 49 𝐿 𝐿 PROPN cana-3900 127 50 = = SYM cana-3900 127 51 { { PUNCT cana-3900 127 52 0 0 NUM cana-3900 127 53 } } PUNCT cana-3900 127 54 . . PUNCT cana-3900 128 1 proposition proposition NOUN cana-3900 128 2 3.2.11 3.2.11 NUM cana-3900 128 3 . . PUNCT cana-3900 129 1 in in ADP cana-3900 129 2 a a DET cana-3900 129 3 𝛿1 𝛿1 NOUN cana-3900 129 4 near near ADP cana-3900 129 5 - - PUNCT cana-3900 129 6 ring ring NOUN cana-3900 129 7 , , PUNCT cana-3900 129 8 𝐸 𝐸 PROPN cana-3900 129 9 βŠ† βŠ† NUM cana-3900 129 10 𝐢(𝑁 𝐢(𝑁 NUM cana-3900 129 11 ) ) PUNCT cana-3900 129 12 . . PUNCT cana-3900 130 1 proof proof NOUN cana-3900 130 2 . . PUNCT cana-3900 131 1 let let VERB cana-3900 131 2 e e PRON cana-3900 131 3 ∈ ∈ PROPN cana-3900 131 4 e. e. PROPN cana-3900 131 5 since since SCONJ cana-3900 131 6 n n PROPN cana-3900 131 7 is be AUX cana-3900 131 8 𝛿1near 𝛿1near NOUN cana-3900 131 9 - - PUNCT cana-3900 131 10 ring ring NOUN cana-3900 131 11 , , PUNCT cana-3900 131 12 𝑒𝑁𝑒 𝑒𝑁𝑒 ADJ cana-3900 131 13 = = PUNCT cana-3900 131 14 𝑁𝑒2𝑒 𝑁𝑒2𝑒 NOUN cana-3900 131 15 2 2 NUM cana-3900 131 16 = = SYM cana-3900 131 17 𝑁𝑒. 𝑁𝑒. PROPN cana-3900 131 18 therefore therefore ADV cana-3900 131 19 forsome forsome NOUN cana-3900 131 20 n n PROPN cana-3900 131 21 in in ADP cana-3900 131 22 n n CCONJ cana-3900 131 23 , , PUNCT cana-3900 131 24 𝑒𝑛𝑒 𝑒𝑛𝑒 X cana-3900 131 25 = = SYM cana-3900 131 26 𝑒𝑒 𝑒𝑒 PROPN cana-3900 131 27 and and CCONJ cana-3900 131 28 𝑛𝑒 𝑛𝑒 NOUN cana-3900 131 29 = = PUNCT cana-3900 131 30 𝑒𝑣𝑒 𝑒𝑣𝑒 VERB cana-3900 131 31 for for ADP cana-3900 131 32 some some DET cana-3900 131 33 u u NOUN cana-3900 131 34 , , PUNCT cana-3900 131 35 v v NOUN cana-3900 131 36 in in ADP cana-3900 131 37 n. n. PROPN cana-3900 131 38 now,𝑒𝑛𝑒 now,𝑒𝑛𝑒 PROPN cana-3900 131 39 = = SYM cana-3900 131 40 𝑒(𝑒𝑒 𝑒(𝑒𝑒 PROPN cana-3900 131 41 ) ) PUNCT cana-3900 131 42 and and CCONJ cana-3900 131 43 𝑒(𝑛𝑒 𝑒(𝑛𝑒 NUM cana-3900 131 44 ) ) PUNCT cana-3900 131 45 = = PUNCT cana-3900 131 46 𝑒𝑣𝑒. 𝑒𝑣𝑒. PUNCT cana-3900 131 47 thus thus ADV cana-3900 131 48 𝑒𝑛𝑒 𝑒𝑛𝑒 X cana-3900 131 49 = = SYM cana-3900 131 50 𝑛𝑒 𝑛𝑒 PROPN cana-3900 131 51 for for ADP cana-3900 131 52 all all DET cana-3900 131 53 n n NOUN cana-3900 131 54 in in ADP cana-3900 131 55 n. n. NOUN cana-3900 131 56 … … PUNCT cana-3900 131 57 … … SYM cana-3900 131 58 …… …… NOUN cana-3900 131 59 .... .... PUNCT cana-3900 132 1 ( ( PUNCT cana-3900 132 2 10).also 10).also NUM cana-3900 132 3 we we PRON cana-3900 132 4 have have VERB cana-3900 132 5 ( ( PUNCT cana-3900 132 6 𝑒𝑛𝑒 𝑒𝑛𝑒 X cana-3900 132 7 – – PUNCT cana-3900 132 8 𝑒𝑛)𝑒 𝑒𝑛)𝑒 X cana-3900 132 9 = = NOUN cana-3900 132 10 0 0 PROPN cana-3900 132 11 . . PUNCT cana-3900 133 1 this this PRON cana-3900 133 2 implies imply VERB cana-3900 133 3 𝑒(𝑒𝑛𝑒 𝑒(𝑒𝑛𝑒 X cana-3900 133 4 – – PUNCT cana-3900 133 5 𝑒𝑛 𝑒𝑛 NOUN cana-3900 133 6 ) ) PUNCT cana-3900 133 7 = = SYM cana-3900 133 8 0 0 NUM cana-3900 133 9 and and CCONJ cana-3900 133 10 𝑒𝑛(𝑒𝑛𝑒 𝑒𝑛(𝑒𝑛𝑒 PROPN cana-3900 133 11 – – PUNCT cana-3900 133 12 𝑒𝑛 𝑒𝑛 NOUN cana-3900 133 13 ) ) PUNCT cana-3900 133 14 = = SYM cana-3900 133 15 0 0 NUM cana-3900 133 16 . . PUNCT cana-3900 134 1 also also ADV cana-3900 134 2 𝑒𝑛𝑒(𝑒𝑛𝑒 𝑒𝑛𝑒(𝑒𝑛𝑒 X cana-3900 134 3 – – PUNCT cana-3900 134 4 𝑒𝑛 𝑒𝑛 NOUN cana-3900 134 5 ) ) PUNCT cana-3900 134 6 = = SYM cana-3900 135 1 𝑒𝑛. 𝑒𝑛. ADP cana-3900 135 2 0 0 NUM cana-3900 136 1 = = SYM cana-3900 136 2 0 0 PUNCT cana-3900 137 1 [ [ X cana-3900 137 2 since since SCONJ cana-3900 137 3 n n PROPN cana-3900 137 4 is be AUX cana-3900 137 5 zero zero NUM cana-3900 137 6 - - PUNCT cana-3900 137 7 symmetric symmetric ADJ cana-3900 137 8 ] ] X cana-3900 137 9 . . PUNCT cana-3900 138 1 now now ADV cana-3900 138 2 , , PUNCT cana-3900 138 3 𝑒𝑛𝑒(𝑒𝑛𝑒 𝑒𝑛𝑒(𝑒𝑛𝑒 X cana-3900 138 4 – – PUNCT cana-3900 138 5 𝑒𝑛 𝑒𝑛 NOUN cana-3900 138 6 ) ) PUNCT cana-3900 138 7 βˆ’ βˆ’ PROPN cana-3900 138 8 𝑒𝑛(𝑒𝑛𝑒 𝑒𝑛(𝑒𝑛𝑒 PROPN cana-3900 138 9 – – PUNCT cana-3900 138 10 𝑒𝑛 𝑒𝑛 NOUN cana-3900 138 11 ) ) PUNCT cana-3900 138 12 = = SYM cana-3900 138 13 0 0 X cana-3900 138 14 . . PUNCT cana-3900 139 1 consequently consequently ADV cana-3900 139 2 , , PUNCT cana-3900 139 3 ( ( PUNCT cana-3900 139 4 𝑒𝑛𝑒 𝑒𝑛𝑒 PROPN cana-3900 139 5 βˆ’ βˆ’ PROPN cana-3900 139 6 𝑒𝑛)2 𝑒𝑛)2 PROPN cana-3900 139 7 = = PROPN cana-3900 139 8 0 0 PROPN cana-3900 139 9 and and CCONJ cana-3900 139 10 k k PROPN cana-3900 139 11 ( ( PUNCT cana-3900 139 12 4 4 X cana-3900 139 13 ) ) PUNCT cana-3900 139 14 guarantees guarantee VERB cana-3900 139 15 𝑒𝑛𝑒 𝑒𝑛𝑒 PRON cana-3900 139 16 βˆ’ βˆ’ PUNCT cana-3900 139 17 𝑒𝑛 𝑒𝑛 NOUN cana-3900 139 18 = = SYM cana-3900 139 19 0.therefore,𝑒𝑛𝑒 0.therefore,𝑒𝑛𝑒 NUM cana-3900 140 1 = = SYM cana-3900 140 2 𝑒𝑛 𝑒𝑛 PROPN cana-3900 140 3 for for ADP cana-3900 140 4 all all DET cana-3900 140 5 n n NOUN cana-3900 140 6 in in ADP cana-3900 140 7 n n CCONJ cana-3900 140 8 … … PUNCT cana-3900 140 9 …… …… NOUN cana-3900 140 10 …… …… NOUN cana-3900 140 11 .. .. PUNCT cana-3900 140 12 ( ( PUNCT cana-3900 140 13 11).from 11).from NUM cana-3900 140 14 equations equation NOUN cana-3900 140 15 ( ( PUNCT cana-3900 140 16 10 10 NUM cana-3900 140 17 ) ) PUNCT cana-3900 140 18 and and CCONJ cana-3900 140 19 ( ( PUNCT cana-3900 140 20 11 11 X cana-3900 140 21 ) ) PUNCT cana-3900 140 22 we we PRON cana-3900 140 23 get get VERB cana-3900 140 24 𝑒𝑛 𝑒𝑛 ADP cana-3900 140 25 = = PUNCT cana-3900 140 26 𝑛𝑒 𝑛𝑒 PROPN cana-3900 140 27 for for ADP cana-3900 140 28 all all DET cana-3900 140 29 n n NOUN cana-3900 140 30 in in ADP cana-3900 140 31 n. n. NOUN cana-3900 140 32 thus thus ADV cana-3900 140 33 𝐸 𝐸 PROPN cana-3900 140 34 βŠ† βŠ† NUM cana-3900 140 35 𝐢(𝑁 𝐢(𝑁 NUM cana-3900 140 36 ) ) PUNCT cana-3900 140 37 . . PUNCT cana-3900 141 1 remark remark PROPN cana-3900 141 2 3.2.12 3.2.12 NUM cana-3900 141 3 . . PUNCT cana-3900 142 1 communications communication NOUN cana-3900 142 2 on on ADP cana-3900 142 3 applied apply VERB cana-3900 142 4 nonlinear nonlinear ADJ cana-3900 142 5 analysis analysis NOUN cana-3900 142 6 issn issn NOUN cana-3900 142 7 : : PUNCT cana-3900 142 8 1074 1074 NUM cana-3900 142 9 - - PUNCT cana-3900 142 10 133x 133x NUM cana-3900 142 11 vol vol NOUN cana-3900 142 12 32 32 NUM cana-3900 142 13 no no NOUN cana-3900 142 14 . . PUNCT cana-3900 143 1 9s 9s NUM cana-3900 143 2 ( ( PUNCT cana-3900 143 3 2025 2025 NUM cana-3900 143 4 ) ) PUNCT cana-3900 143 5 350 350 NUM cana-3900 143 6 https://internationalpubls.com https://internationalpubls.com X cana-3900 144 1 it it PRON cana-3900 144 2 is be AUX cana-3900 144 3 worth worth ADJ cana-3900 144 4 noting note VERB cana-3900 144 5 that that SCONJ cana-3900 144 6 we we PRON cana-3900 144 7 do do AUX cana-3900 144 8 not not PART cana-3900 144 9 stipulate stipulate VERB cana-3900 144 10 that that SCONJ cana-3900 144 11 n n PRON cana-3900 144 12 admits admit VERB cana-3900 144 13 mate mate NOUN cana-3900 144 14 functions function NOUN cana-3900 144 15 for for ADP cana-3900 144 16 the the DET cana-3900 144 17 validity validity NOUN cana-3900 144 18 of of ADP cana-3900 144 19 the the DET cana-3900 144 20 above above ADJ cana-3900 144 21 results result NOUN cana-3900 144 22 . . PUNCT cana-3900 145 1 proposition proposition NOUN cana-3900 145 2 3.2.13 3.2.13 NUM cana-3900 145 3 . . PUNCT cana-3900 146 1 let let VERB cana-3900 146 2 n n PRON cana-3900 146 3 be be AUX cana-3900 146 4 a a DET cana-3900 146 5 𝛿1near 𝛿1near NOUN cana-3900 146 6 - - PUNCT cana-3900 146 7 ring ring NOUN cana-3900 146 8 with with ADP cana-3900 146 9 identity identity NOUN cana-3900 146 10 . . PUNCT cana-3900 147 1 then then ADV cana-3900 147 2 n n PRON cana-3900 147 3 has have VERB cana-3900 147 4 a a DET cana-3900 147 5 mate mate NOUN cana-3900 147 6 function function NOUN cana-3900 147 7 if if SCONJ cana-3900 147 8 and and CCONJ cana-3900 147 9 only only ADV cana-3900 147 10 if if SCONJ cana-3900 147 11 n n PRON cana-3900 147 12 is be AUX cana-3900 147 13 an an DET cana-3900 147 14 s'-near s'-near NOUN cana-3900 147 15 - - PUNCT cana-3900 147 16 ring ring NOUN cana-3900 147 17 . . PUNCT cana-3900 148 1 proof proof NOUN cana-3900 148 2 . . PUNCT cana-3900 149 1 when when SCONJ cana-3900 149 2 n n PRON cana-3900 149 3 has have VERB cana-3900 149 4 a a DET cana-3900 149 5 mate mate NOUN cana-3900 149 6 function function NOUN cana-3900 149 7 ` ` PUNCT cana-3900 149 8 m m NOUN cana-3900 149 9 ' ' PUNCT cana-3900 149 10 for for ADP cana-3900 149 11 all all DET cana-3900 149 12 π‘₯ π‘₯ DET cana-3900 149 13 ∈ ∈ PROPN cana-3900 149 14 𝑁 𝑁 PROPN cana-3900 149 15 , , PUNCT cana-3900 149 16 π‘₯ π‘₯ X cana-3900 149 17 = = PUNCT cana-3900 149 18 π‘₯𝑓(π‘₯)π‘₯ π‘₯𝑓(π‘₯)π‘₯ VERB cana-3900 149 19 ∈ ∈ NOUN cana-3900 149 20 π‘₯𝑁 π‘₯𝑁 NOUN cana-3900 149 21 and and CCONJ cana-3900 149 22 obviously obviously ADV cana-3900 149 23 n n PRON cana-3900 149 24 is be AUX cana-3900 149 25 an an DET cana-3900 149 26 s'-near s'-near NOUN cana-3900 149 27 - - PUNCT cana-3900 149 28 ring ring NOUN cana-3900 149 29 . . PUNCT cana-3900 150 1 conversely conversely ADV cana-3900 150 2 let let VERB cana-3900 150 3 n n PRON cana-3900 150 4 be be AUX cana-3900 150 5 an an DET cana-3900 150 6 s s NOUN cana-3900 150 7 ' ' PUNCT cana-3900 150 8 near near NOUN cana-3900 150 9 - - PUNCT cana-3900 150 10 ring ring NOUN cana-3900 150 11 . . PUNCT cana-3900 151 1 π‘₯ π‘₯ PRON cana-3900 151 2 ∈ ∈ PROPN cana-3900 151 3 π‘₯𝑁 π‘₯𝑁 NOUN cana-3900 151 4 = = NOUN cana-3900 151 5 𝑁π‘₯2 𝑁π‘₯2 NOUN cana-3900 151 6 β‡’ β‡’ NOUN cana-3900 151 7 π‘₯ π‘₯ X cana-3900 151 8 = = PUNCT cana-3900 151 9 𝑛π‘₯2 𝑛π‘₯2 VERB cana-3900 151 10 for for ADP cana-3900 151 11 some some DET cana-3900 151 12 𝑛 𝑛 PRON cana-3900 151 13 ∈ ∈ NOUN cana-3900 151 14 𝑁 𝑁 PROPN cana-3900 151 15 β‡’ β‡’ NOUN cana-3900 151 16 π‘₯2 π‘₯2 NOUN cana-3900 151 17 = = PUNCT cana-3900 151 18 π‘₯𝑛π‘₯2 π‘₯𝑛π‘₯2 NOUN cana-3900 151 19 β‡’ β‡’ NOUN cana-3900 151 20 ( ( PUNCT cana-3900 151 21 π‘₯ π‘₯ DET cana-3900 151 22 βˆ’ βˆ’ PROPN cana-3900 151 23 π‘₯𝑛π‘₯ π‘₯𝑛π‘₯ NOUN cana-3900 151 24 ) ) PUNCT cana-3900 151 25 π‘₯ π‘₯ PROPN cana-3900 151 26 = = SYM cana-3900 151 27 0 0 NUM cana-3900 151 28 .using .use VERB cana-3900 151 29 proposition proposition NOUN cana-3900 151 30 3.2.1(iii 3.2.1(iii NUM cana-3900 151 31 ) ) PUNCT cana-3900 151 32 and and CCONJ cana-3900 151 33 corollary corollary ADJ cana-3900 151 34 3.2.2 3.2.2 NUM cana-3900 151 35 we we PRON cana-3900 151 36 get get VERB cana-3900 151 37 π‘₯(π‘₯ π‘₯(π‘₯ NOUN cana-3900 151 38 βˆ’ βˆ’ PROPN cana-3900 151 39 π‘₯𝑛π‘₯ π‘₯𝑛π‘₯ NOUN cana-3900 151 40 ) ) PUNCT cana-3900 151 41 = = SYM cana-3900 151 42 0 0 NUM cana-3900 151 43 and and CCONJ cana-3900 151 44 π‘₯𝑛π‘₯(π‘₯ π‘₯𝑛π‘₯(π‘₯ PROPN cana-3900 151 45 βˆ’ βˆ’ PROPN cana-3900 151 46 π‘₯𝑛π‘₯ π‘₯𝑛π‘₯ NOUN cana-3900 151 47 ) ) PUNCT cana-3900 151 48 = = SYM cana-3900 151 49 0 0 NUM cana-3900 152 1 and and CCONJ cana-3900 152 2 consequently consequently ADV cana-3900 152 3 ( ( PUNCT cana-3900 152 4 π‘₯ π‘₯ NOUN cana-3900 152 5 βˆ’ βˆ’ PUNCT cana-3900 152 6 π‘₯𝑛π‘₯)2 π‘₯𝑛π‘₯)2 NOUN cana-3900 152 7 = = SYM cana-3900 152 8 0 0 NUM cana-3900 152 9 . . PUNCT cana-3900 153 1 since since SCONJ cana-3900 153 2 𝐿 𝐿 PROPN cana-3900 153 3 = = SYM cana-3900 153 4 { { PUNCT cana-3900 153 5 0 0 NUM cana-3900 153 6 } } PUNCT cana-3900 153 7 we we PRON cana-3900 153 8 get get VERB cana-3900 153 9 π‘₯ π‘₯ PRON cana-3900 153 10 βˆ’ βˆ’ NOUN cana-3900 153 11 π‘₯𝑛π‘₯ π‘₯𝑛π‘₯ NOUN cana-3900 153 12 = = NOUN cana-3900 153 13 0i.e 0i.e NUM cana-3900 153 14 . . PUNCT cana-3900 154 1 π‘₯ π‘₯ X cana-3900 155 1 = = PUNCT cana-3900 155 2 π‘₯𝑓(π‘₯)π‘₯ π‘₯𝑓(π‘₯)π‘₯ VERB cana-3900 155 3 where where SCONJ cana-3900 155 4 we we PRON cana-3900 155 5 get get VERB cana-3900 155 6 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3900 155 7 ) ) PUNCT cana-3900 155 8 = = SYM cana-3900 155 9 𝑛. 𝑛. NOUN cana-3900 155 10 this this PRON cana-3900 155 11 guarantees guarantee VERB cana-3900 155 12 that that SCONJ cana-3900 155 13 𝑓 𝑓 DET cana-3900 155 14 ∢ ∢ NOUN cana-3900 155 15 𝑁 𝑁 PROPN cana-3900 155 16 β†’ β†’ SYM cana-3900 155 17 𝑁 𝑁 PROPN cana-3900 155 18 is be AUX cana-3900 155 19 a a DET cana-3900 155 20 mate mate NOUN cana-3900 155 21 function function NOUN cana-3900 155 22 for for ADP cana-3900 155 23 n. n. NOUN cana-3900 155 24 proposition proposition PROPN cana-3900 155 25 3.2.14 3.2.14 NUM cana-3900 155 26 . . PUNCT cana-3900 156 1 let let VERB cana-3900 156 2 n n PRON cana-3900 156 3 be be AUX cana-3900 156 4 an an DET cana-3900 156 5 𝑆’-𝛿1near 𝑆’-𝛿1near NOUN cana-3900 156 6 - - PUNCT cana-3900 156 7 ring ring NOUN cana-3900 156 8 . . PUNCT cana-3900 157 1 then then ADV cana-3900 157 2 n n PRON cana-3900 157 3 has have VERB cana-3900 157 4 a a DET cana-3900 157 5 p3 p3 NOUN cana-3900 157 6 mate mate NOUN cana-3900 157 7 function function NOUN cana-3900 157 8 . . PUNCT cana-3900 158 1 proof proof NOUN cana-3900 158 2 . . PUNCT cana-3900 159 1 when when SCONJ cana-3900 159 2 n n X cana-3900 159 3 is be AUX cana-3900 159 4 an an DET cana-3900 159 5 s'𝛿1near s'𝛿1near ADJ cana-3900 159 6 - - PUNCT cana-3900 159 7 ring ring NOUN cana-3900 159 8 it it PRON cana-3900 159 9 admits admit VERB cana-3900 159 10 a a DET cana-3900 159 11 mate mate NOUN cana-3900 159 12 function function NOUN cana-3900 159 13 β€˜ ' PUNCT cana-3900 159 14 𝑓 𝑓 X cana-3900 159 15 ’ ' PUNCT cana-3900 159 16 . . PUNCT cana-3900 160 1 from from ADP cana-3900 160 2 proposition proposition NOUN cana-3900 160 3 3.2.12 3.2.12 NUM cana-3900 160 4 we we PRON cana-3900 160 5 have have VERB cana-3900 160 6 π‘₯ π‘₯ NOUN cana-3900 160 7 = = PUNCT cana-3900 160 8 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3900 160 9 ) ) PUNCT cana-3900 160 10 π‘₯2 π‘₯2 NOUN cana-3900 160 11 β‡’ β‡’ NOUN cana-3900 160 12 ( ( PUNCT cana-3900 160 13 π‘₯𝑓(π‘₯ π‘₯𝑓(π‘₯ NOUN cana-3900 160 14 ) ) PUNCT cana-3900 160 15 βˆ’ βˆ’ PROPN cana-3900 160 16 𝑓(π‘₯)π‘₯ 𝑓(π‘₯)π‘₯ NOUN cana-3900 160 17 ) ) PUNCT cana-3900 160 18 π‘₯ π‘₯ NOUN cana-3900 160 19 = = SYM cana-3900 160 20 0 0 NUM cana-3900 160 21 β‡’ β‡’ NOUN cana-3900 160 22 ( ( PUNCT cana-3900 160 23 π‘₯𝑓(π‘₯ π‘₯𝑓(π‘₯ NOUN cana-3900 160 24 ) ) PUNCT cana-3900 160 25 βˆ’ βˆ’ PROPN cana-3900 160 26 𝑓(π‘₯)π‘₯)2 𝑓(π‘₯)π‘₯)2 NOUN cana-3900 160 27 = = NOUN cana-3900 160 28 0 0 NUM cana-3900 160 29 . . PUNCT cana-3900 161 1 [ [ X cana-3900 161 2 since since SCONJ cana-3900 161 3 n n PRON cana-3900 161 4 has have VERB cana-3900 161 5 ( ( PUNCT cana-3900 161 6 * * PROPN cana-3900 161 7 , , PUNCT cana-3900 161 8 ifp ifp NOUN cana-3900 161 9 ) ) PUNCT cana-3900 161 10 ] ] PUNCT cana-3900 161 11 π‘₯𝑓(π‘₯ π‘₯𝑓(π‘₯ NOUN cana-3900 161 12 ) ) PUNCT cana-3900 161 13 βˆ’ βˆ’ PROPN cana-3900 162 1 𝑓(π‘₯)π‘₯ 𝑓(π‘₯)π‘₯ NOUN cana-3900 162 2 = = SYM cana-3900 162 3 0 0 NUM cana-3900 162 4 β‡’ β‡’ NOUN cana-3900 162 5 π‘₯𝑓(π‘₯ π‘₯𝑓(π‘₯ NOUN cana-3900 162 6 ) ) PUNCT cana-3900 162 7 = = SYM cana-3900 162 8 𝑓(π‘₯)π‘₯ 𝑓(π‘₯)π‘₯ ADJ cana-3900 162 9 i.e. i.e. X cana-3900 162 10 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3900 162 11 ) ) PUNCT cana-3900 162 12 ∈ ∈ PROPN cana-3900 162 13 𝐢(π‘₯ 𝐢(π‘₯ NOUN cana-3900 162 14 ) ) PUNCT cana-3900 162 15 i.e. i.e. X cana-3900 162 16 β€˜ ' PUNCT cana-3900 162 17 𝑓 𝑓 X cana-3900 162 18 ’ ' PUNCT cana-3900 162 19 is be AUX cana-3900 162 20 a a DET cana-3900 162 21 p3 p3 NOUN cana-3900 162 22 mate mate NOUN cana-3900 162 23 function function NOUN cana-3900 162 24 . . PUNCT cana-3900 163 1 proposition proposition NOUN cana-3900 163 2 3.2.15 3.2.15 NUM cana-3900 163 3 . . PUNCT cana-3900 164 1 if if SCONJ cana-3900 164 2 n n PRON cana-3900 164 3 has have VERB cana-3900 164 4 property property NOUN cana-3900 164 5 𝛿1and 𝛿1and ADP cana-3900 164 6 a a DET cana-3900 164 7 mate mate NOUN cana-3900 164 8 function function NOUN cana-3900 164 9 β€˜ ' PUNCT cana-3900 164 10 f f X cana-3900 164 11 ’ ' PUNCT cana-3900 164 12 then then ADV cana-3900 164 13 𝐿 𝐿 PROPN cana-3900 164 14 = = SYM cana-3900 164 15 { { PUNCT cana-3900 164 16 0 0 NUM cana-3900 164 17 } } PUNCT cana-3900 164 18 and and CCONJ cana-3900 164 19 n n PROPN cana-3900 164 20 has have VERB cana-3900 164 21 ( ( PUNCT cana-3900 164 22 * * PROPN cana-3900 164 23 , , PUNCT cana-3900 164 24 ifp ifp NOUN cana-3900 164 25 ) ) PUNCT cana-3900 164 26 . . PUNCT cana-3900 165 1 we we PRON cana-3900 165 2 now now ADV cana-3900 165 3 give give VERB cana-3900 165 4 a a DET cana-3900 165 5 complete complete ADJ cana-3900 165 6 characterization characterization NOUN cana-3900 165 7 of of ADP cana-3900 165 8 𝛿1 𝛿1 NOUN cana-3900 165 9 when when SCONJ cana-3900 165 10 they they PRON cana-3900 165 11 admit admit VERB cana-3900 165 12 mate mate NOUN cana-3900 165 13 functions function NOUN cana-3900 165 14 . . PUNCT cana-3900 166 1 theorem theorem PROPN cana-3900 166 2 3.2.16 3.2.16 NUM cana-3900 166 3 . . PUNCT cana-3900 167 1 let let VERB cana-3900 167 2 n n PRON cana-3900 167 3 be be AUX cana-3900 167 4 a a DET cana-3900 167 5 near near NOUN cana-3900 167 6 - - PUNCT cana-3900 167 7 ring ring NOUN cana-3900 167 8 with with ADP cana-3900 167 9 a a DET cana-3900 167 10 zero zero NUM cana-3900 167 11 symmetric symmetric ADJ cana-3900 167 12 mate mate NOUN cana-3900 167 13 function function NOUN cana-3900 167 14 β€˜ ' PUNCT cana-3900 167 15 𝑓 𝑓 X cana-3900 167 16 ’ ' PUNCT cana-3900 167 17 and and CCONJ cana-3900 167 18 left leave VERB cana-3900 167 19 bipotent bipotent NOUN cana-3900 167 20 . . PUNCT cana-3900 168 1 then then ADV cana-3900 168 2 the the DET cana-3900 168 3 following follow VERB cana-3900 168 4 statements statement NOUN cana-3900 168 5 are be AUX cana-3900 168 6 equivalent equivalent ADJ cana-3900 168 7 : : PUNCT cana-3900 168 8 ( ( PUNCT cana-3900 168 9 i i NOUN cana-3900 168 10 ) ) PUNCT cana-3900 168 11 n n PRON cana-3900 168 12 is be AUX cana-3900 168 13 𝛿1 𝛿1 ADJ cana-3900 168 14 ( ( PUNCT cana-3900 168 15 ii ii NOUN cana-3900 168 16 ) ) PUNCT cana-3900 168 17 𝐸 𝐸 PROPN cana-3900 168 18 βŠ† βŠ† NUM cana-3900 168 19 𝐢(𝑁 𝐢(𝑁 NUM cana-3900 168 20 ) ) PUNCT cana-3900 168 21 proof proof NOUN cana-3900 168 22 . . PUNCT cana-3900 169 1 ( ( PUNCT cana-3900 169 2 ii ii NOUN cana-3900 169 3 ) ) PUNCT cana-3900 169 4 β‡’ β‡’ NOUN cana-3900 169 5 ( ( PUNCT cana-3900 169 6 i i NOUN cana-3900 169 7 ) ) PUNCT cana-3900 169 8 let let VERB cana-3900 169 9 𝐸 𝐸 PRON cana-3900 169 10 βŠ† βŠ† NUM cana-3900 169 11 𝐢(𝑁 𝐢(𝑁 NUM cana-3900 169 12 ) ) PUNCT cana-3900 169 13 . . PUNCT cana-3900 170 1 now now ADV cana-3900 170 2 𝑁π‘₯2𝑦2 𝑁π‘₯2𝑦2 NOUN cana-3900 170 3 = = SYM cana-3900 170 4 𝑁π‘₯𝑦2 𝑁π‘₯𝑦2 NOUN cana-3900 171 1 [ [ X cana-3900 171 2 n n NOUN cana-3900 171 3 is be AUX cana-3900 171 4 left leave VERB cana-3900 171 5 bipotent bipotent NOUN cana-3900 171 6 ] ] X cana-3900 172 1 = = X cana-3900 172 2 ( ( PUNCT cana-3900 172 3 𝑁𝑓(π‘₯)π‘₯)𝑦2 𝑁𝑓(π‘₯)π‘₯)𝑦2 X cana-3900 172 4 [ [ PUNCT cana-3900 172 5 by by ADP cana-3900 172 6 k k PROPN cana-3900 172 7 ( ( PUNCT cana-3900 172 8 1)]= 1)]= X cana-3900 172 9 ( ( PUNCT cana-3900 172 10 𝑓(π‘₯)π‘₯𝑁)𝑦2[since 𝑓(π‘₯)π‘₯𝑁)𝑦2[since NOUN cana-3900 172 11 e e NOUN cana-3900 172 12 βŠ† βŠ† NUM cana-3900 172 13 c c PROPN cana-3900 172 14 ( ( PUNCT cana-3900 172 15 n n X cana-3900 172 16 ) ) PUNCT cana-3900 172 17 ] ] PUNCT cana-3900 173 1 = = PUNCT cana-3900 173 2 π‘₯𝑓(π‘₯)𝑁𝑦 π‘₯𝑓(π‘₯)𝑁𝑦 NOUN cana-3900 174 1 [ [ X cana-3900 174 2 n n NOUN cana-3900 174 3 is be AUX cana-3900 174 4 left leave VERB cana-3900 174 5 bipotent]= bipotent]= PROPN cana-3900 174 6 π‘₯𝑁𝑦 π‘₯𝑁𝑦 NOUN cana-3900 174 7 [ [ PUNCT cana-3900 174 8 by by ADP cana-3900 174 9 k k PROPN cana-3900 174 10 ( ( PUNCT cana-3900 174 11 1)].i.e 1)].i.e NUM cana-3900 174 12 . . PUNCT cana-3900 175 1 𝑁π‘₯2𝑦2 𝑁π‘₯2𝑦2 NOUN cana-3900 176 1 = = SYM cana-3900 176 2 π‘₯𝑁𝑦 π‘₯𝑁𝑦 NOUN cana-3900 176 3 proof proof NOUN cana-3900 176 4 of of ADP cana-3900 176 5 β€˜ ' PUNCT cana-3900 176 6 ( ( PUNCT cana-3900 176 7 i i NOUN cana-3900 176 8 ) ) PUNCT cana-3900 176 9 β‡’(ii β‡’(ii PROPN cana-3900 176 10 ) ) PUNCT cana-3900 176 11 ’ ' PUNCT cana-3900 176 12 is be AUX cana-3900 176 13 similar similar ADJ cana-3900 176 14 . . PUNCT cana-3900 177 1 remark remark NOUN cana-3900 177 2 3.2.17 3.2.17 NUM cana-3900 177 3 let let VERB cana-3900 177 4 n n PRON cana-3900 177 5 admit admit VERB cana-3900 177 6 a a DET cana-3900 177 7 mate mate NOUN cana-3900 177 8 function function NOUN cana-3900 177 9 β€˜ ' PUNCT cana-3900 177 10 𝑓(π‘₯ 𝑓(π‘₯ NOUN cana-3900 177 11 ) ) PUNCT cana-3900 177 12 ’ ' PUNCT cana-3900 177 13 and and CCONJ cana-3900 177 14 let let VERB cana-3900 177 15 e e NOUN cana-3900 177 16 βŠ† βŠ† NUM cana-3900 177 17 c c X cana-3900 177 18 ( ( PUNCT cana-3900 177 19 n n CCONJ cana-3900 177 20 ) ) PUNCT cana-3900 177 21 . . PUNCT cana-3900 178 1 it it PRON cana-3900 178 2 is be AUX cana-3900 178 3 easy easy ADJ cana-3900 178 4 to to PART cana-3900 178 5 observe observe VERB cana-3900 178 6 that that SCONJ cana-3900 178 7 for for ADP cana-3900 178 8 every every DET cana-3900 178 9 x x NOUN cana-3900 178 10 in in ADP cana-3900 178 11 n,π‘₯ n,π‘₯ NOUN cana-3900 178 12 = = SYM cana-3900 178 13 π‘₯ π‘₯ PRON cana-3900 178 14 𝑓(π‘₯)π‘₯ 𝑓(π‘₯)π‘₯ NOUN cana-3900 178 15 β‡’ β‡’ VERB cana-3900 178 16 π‘₯ π‘₯ X cana-3900 178 17 = = PUNCT cana-3900 178 18 𝑓(π‘₯)π‘₯2 𝑓(π‘₯)π‘₯2 PROPN cana-3900 178 19 . . PUNCT cana-3900 179 1 consequently consequently ADV cana-3900 179 2 proposition proposition VERB cana-3900 179 3 3.2.16 3.2.16 NUM cana-3900 179 4 guarantees guarantee VERB cana-3900 179 5 that that SCONJ cana-3900 179 6 m m NOUN cana-3900 179 7 is be AUX cana-3900 179 8 a a DET cana-3900 179 9 p3 p3 NOUN cana-3900 179 10 mate mate NOUN cana-3900 179 11 function function NOUN cana-3900 179 12 . . PUNCT cana-3900 180 1 theorem theorem VERB cana-3900 180 2 3.2.18 3.2.18 NUM cana-3900 180 3 every every DET cana-3900 180 4 n n NOUN cana-3900 180 5 - - PUNCT cana-3900 180 6 subgroup subgroup NOUN cana-3900 180 7 of of ADP cana-3900 180 8 n n PROPN cana-3900 180 9 is be AUX cana-3900 180 10 an an DET cana-3900 180 11 ideal ideal NOUN cana-3900 180 12 in in ADP cana-3900 180 13 an an DET cana-3900 180 14 s'𝛿1near s'𝛿1near ADJ cana-3900 180 15 - - PUNCT cana-3900 180 16 ring ring NOUN cana-3900 180 17 . . PUNCT cana-3900 181 1 proof proof NOUN cana-3900 181 2 : : PUNCT cana-3900 181 3 since since SCONJ cana-3900 181 4 n n PRON cana-3900 181 5 is be AUX cana-3900 181 6 an an DET cana-3900 181 7 s'-𝛿1near s'-𝛿1near NOUN cana-3900 181 8 - - PUNCT cana-3900 181 9 ring ring NOUN cana-3900 181 10 , , PUNCT cana-3900 181 11 it it PRON cana-3900 181 12 admits admit VERB cana-3900 181 13 a a DET cana-3900 181 14 mate mate NOUN cana-3900 181 15 function function NOUN cana-3900 181 16 β€˜ ' PUNCT cana-3900 181 17 f(x f(x PROPN cana-3900 181 18 ) ) PUNCT cana-3900 181 19 ’ ' PUNCT cana-3900 182 1 [ [ X cana-3900 182 2 from from ADP cana-3900 182 3 proposition proposition NOUN cana-3900 182 4 3.2.13 3.2.13 NUM cana-3900 182 5 ] ] PUNCT cana-3900 182 6 and and CCONJ cana-3900 182 7 l l NOUN cana-3900 182 8 { { PUNCT cana-3900 182 9 0 0 NUM cana-3900 182 10 } } PUNCT cana-3900 182 11 [ [ X cana-3900 182 12 from from ADP cana-3900 182 13 proposition proposition NOUN cana-3900 182 14 3.2.3(iii 3.2.3(iii NUM cana-3900 182 15 ) ) PUNCT cana-3900 182 16 ] ] PUNCT cana-3900 182 17 . . PUNCT cana-3900 183 1 it it PRON cana-3900 183 2 is be AUX cana-3900 183 3 clear clear ADJ cana-3900 183 4 from from ADP cana-3900 183 5 k k PROPN cana-3900 183 6 ( ( PUNCT cana-3900 183 7 2 2 NUM cana-3900 183 8 ) ) PUNCT cana-3900 183 9 that that PRON cana-3900 183 10 n n PRON cana-3900 183 11 has have AUX cana-3900 183 12 ( ( PUNCT cana-3900 183 13 * * PROPN cana-3900 183 14 , , PUNCT cana-3900 183 15 ifp ifp NOUN cana-3900 183 16 ) ) PUNCT cana-3900 183 17 . . PUNCT cana-3900 184 1 again again ADV cana-3900 184 2 for for ADP cana-3900 184 3 any any DET cana-3900 184 4 non non ADJ cana-3900 184 5 - - ADJ cana-3900 184 6 empty empty ADJ cana-3900 184 7 𝑆 𝑆 PROPN cana-3900 184 8 βŠ† βŠ† NUM cana-3900 184 9 𝑁,(0 𝑁,(0 NOUN cana-3900 184 10 : : PUNCT cana-3900 184 11 s s X cana-3900 184 12 ) ) PUNCT cana-3900 184 13 is be AUX cana-3900 184 14 an an DET cana-3900 184 15 ideal ideal NOUN cana-3900 184 16 of of ADP cana-3900 184 17 n n PRON cana-3900 184 18 [ [ PUNCT cana-3900 184 19 by by ADP cana-3900 184 20 k(3 k(3 PROPN cana-3900 184 21 ) ) PUNCT cana-3900 184 22 ] ] PUNCT cana-3900 184 23 . . PUNCT cana-3900 185 1 if if SCONJ cana-3900 185 2 m m NOUN cana-3900 185 3 is be AUX cana-3900 185 4 any any DET cana-3900 185 5 n n NOUN cana-3900 185 6 - - PUNCT cana-3900 185 7 subgroup subgroup NOUN cana-3900 185 8 of of ADP cana-3900 185 9 n n CCONJ cana-3900 185 10 , , PUNCT cana-3900 185 11 then then ADV cana-3900 185 12 = = SYM cana-3900 185 13 βˆ‘ βˆ‘ PROPN cana-3900 185 14 𝑁π‘₯π‘₯βˆˆπ‘€ 𝑁π‘₯π‘₯βˆˆπ‘€ PROPN cana-3900 185 15 . . PUNCT cana-3900 186 1 we we PRON cana-3900 186 2 first first ADV cana-3900 186 3 show show VERB cana-3900 186 4 that that SCONJ cana-3900 186 5 each each DET cana-3900 186 6 𝑁π‘₯ 𝑁π‘₯ PROPN cana-3900 186 7 is be AUX cana-3900 186 8 an an DET cana-3900 186 9 ideal ideal NOUN cana-3900 186 10 . . PUNCT cana-3900 187 1 let let VERB cana-3900 187 2 𝑆 𝑆 PROPN cana-3900 187 3 = = SYM cana-3900 187 4 ( ( PUNCT cana-3900 187 5 0 0 NUM cana-3900 187 6 ∢ ∢ NOUN cana-3900 187 7 𝑁π‘₯ 𝑁π‘₯ PROPN cana-3900 187 8 ) ) PUNCT cana-3900 187 9 . . PUNCT cana-3900 188 1 we we PRON cana-3900 188 2 claim claim VERB cana-3900 188 3 that that SCONJ cana-3900 188 4 𝑁π‘₯ 𝑁π‘₯ PROPN cana-3900 188 5 = = SYM cana-3900 188 6 ( ( PUNCT cana-3900 188 7 0 0 NUM cana-3900 188 8 ∢ ∢ PROPN cana-3900 188 9 𝑆 𝑆 PROPN cana-3900 188 10 ) ) PUNCT cana-3900 188 11 . . PUNCT cana-3900 189 1 clearly𝑁π‘₯ clearly𝑁π‘₯ PUNCT cana-3900 190 1 βŠ‚ βŠ‚ PROPN cana-3900 190 2 ( ( PUNCT cana-3900 190 3 0 0 NUM cana-3900 190 4 ∢ ∢ NOUN cana-3900 190 5 𝑆 𝑆 PROPN cana-3900 190 6 ) ) PUNCT cana-3900 190 7 … … PUNCT cana-3900 190 8 … … SYM cana-3900 190 9 …… …… NOUN cana-3900 190 10 .. .. PUNCT cana-3900 190 11 ( ( PUNCT cana-3900 190 12 12 12 NUM cana-3900 190 13 ) ) PUNCT cana-3900 190 14 now now ADV cana-3900 190 15 if if SCONJ cana-3900 190 16 𝑦 𝑦 PRON cana-3900 190 17 ∈ ∈ NOUN cana-3900 190 18 ( ( PUNCT cana-3900 190 19 0 0 NUM cana-3900 190 20 ∢ ∢ PROPN cana-3900 190 21 𝑆 𝑆 PROPN cana-3900 190 22 ) ) PUNCT cana-3900 190 23 then then ADV cana-3900 190 24 𝑦𝑆 𝑦𝑆 VERB cana-3900 190 25 = = SYM cana-3900 190 26 { { PUNCT cana-3900 190 27 0 0 NUM cana-3900 190 28 } } PUNCT cana-3900 190 29 . . PUNCT cana-3900 191 1 also also ADV cana-3900 191 2 ( ( PUNCT cana-3900 191 3 𝑦 𝑦 NOUN cana-3900 191 4 βˆ’ βˆ’ NOUN cana-3900 191 5 𝑦𝑓(π‘₯)π‘₯)𝑓(π‘₯)π‘₯ 𝑦𝑓(π‘₯)π‘₯)𝑓(π‘₯)π‘₯ NOUN cana-3900 191 6 = = SYM cana-3900 191 7 0 0 NUM cana-3900 191 8 … … PUNCT cana-3900 191 9 … … PUNCT cana-3900 191 10 … … PUNCT cana-3900 191 11 .. .. PUNCT cana-3900 191 12 ( ( PUNCT cana-3900 191 13 13 13 NUM cana-3900 191 14 ) ) PUNCT cana-3900 191 15 β‡’ β‡’ NOUN cana-3900 191 16 ( ( PUNCT cana-3900 191 17 𝑦 𝑦 NOUN cana-3900 191 18 βˆ’ βˆ’ NOUN cana-3900 191 19 𝑦𝑓(π‘₯)π‘₯)𝑁𝑓(π‘₯)π‘₯ 𝑦𝑓(π‘₯)π‘₯)𝑁𝑓(π‘₯)π‘₯ PROPN cana-3900 191 20 = = SYM cana-3900 191 21 { { PUNCT cana-3900 191 22 0 0 NUM cana-3900 191 23 } } PUNCT cana-3900 191 24 β‡’ β‡’ NOUN cana-3900 191 25 ( ( PUNCT cana-3900 191 26 𝑦 𝑦 NOUN cana-3900 191 27 βˆ’ βˆ’ PROPN cana-3900 191 28 𝑦𝑓(π‘₯)π‘₯)𝑁π‘₯ 𝑦𝑓(π‘₯)π‘₯)𝑁π‘₯ PROPN cana-3900 191 29 = = SYM cana-3900 191 30 { { PUNCT cana-3900 191 31 0 0 NUM cana-3900 191 32 } } PUNCT cana-3900 192 1 [ [ X cana-3900 192 2 using use VERB cana-3900 192 3 k(1 k(1 NOUN cana-3900 192 4 ) ) PUNCT cana-3900 192 5 ] ] PUNCT cana-3900 192 6 β‡’ β‡’ NOUN cana-3900 192 7 ( ( PUNCT cana-3900 192 8 𝑦 𝑦 NOUN cana-3900 192 9 βˆ’ βˆ’ PROPN cana-3900 192 10 𝑦𝑓(π‘₯)π‘₯ 𝑦𝑓(π‘₯)π‘₯ NOUN cana-3900 192 11 ) ) PUNCT cana-3900 192 12 ∈ ∈ PROPN cana-3900 192 13 ( ( PUNCT cana-3900 192 14 0 0 NUM cana-3900 192 15 ∢ ∢ NOUN cana-3900 192 16 𝑁π‘₯ 𝑁π‘₯ PROPN cana-3900 192 17 ) ) PUNCT cana-3900 192 18 = = SYM cana-3900 192 19 𝑆. 𝑆. PROPN cana-3900 192 20 since since SCONJ cana-3900 192 21 𝑦𝑆 𝑦𝑆 NOUN cana-3900 192 22 = = PUNCT cana-3900 192 23 { { PUNCT cana-3900 192 24 0 0 NUM cana-3900 192 25 } } PUNCT cana-3900 192 26 , , PUNCT cana-3900 192 27 𝑦 𝑦 X cana-3900 192 28 ( ( PUNCT cana-3900 192 29 𝑦 𝑦 NOUN cana-3900 192 30 βˆ’ βˆ’ PROPN cana-3900 192 31 𝑦𝑓(π‘₯)π‘₯ 𝑦𝑓(π‘₯)π‘₯ PROPN cana-3900 192 32 ) ) PUNCT cana-3900 192 33 = = SYM cana-3900 192 34 0 0 NUM cana-3900 192 35 … … SYM cana-3900 192 36 …… …… NOUN cana-3900 192 37 .... .... PUNCT cana-3900 192 38 ( ( PUNCT cana-3900 192 39 14 14 NUM cana-3900 192 40 ) ) PUNCT cana-3900 192 41 .using .use VERB cana-3900 192 42 the the DET cana-3900 192 43 fact fact NOUN cana-3900 192 44 that that SCONJ cana-3900 192 45 n n PRON cana-3900 192 46 has have AUX cana-3900 192 47 ( ( PUNCT cana-3900 192 48 * * PROPN cana-3900 192 49 , , PUNCT cana-3900 192 50 ifp ifp NOUN cana-3900 192 51 ) ) PUNCT cana-3900 192 52 , , PUNCT cana-3900 192 53 it it PRON cana-3900 192 54 is be AUX cana-3900 192 55 easy easy ADJ cana-3900 192 56 to to SCONJ cana-3900 192 57 communications communication NOUN cana-3900 192 58 on on ADP cana-3900 192 59 applied apply VERB cana-3900 192 60 nonlinear nonlinear ADJ cana-3900 192 61 analysis analysis NOUN cana-3900 192 62 issn issn NOUN cana-3900 192 63 : : PUNCT cana-3900 192 64 1074 1074 NUM cana-3900 192 65 - - PUNCT cana-3900 192 66 133x 133x NUM cana-3900 192 67 vol vol NOUN cana-3900 192 68 32 32 NUM cana-3900 192 69 no no NOUN cana-3900 192 70 . . PUNCT cana-3900 193 1 9s 9s NUM cana-3900 193 2 ( ( PUNCT cana-3900 193 3 2025 2025 NUM cana-3900 193 4 ) ) PUNCT cana-3900 193 5 351 351 NUM cana-3900 193 6 https://internationalpubls.com https://internationalpubls.com X cana-3900 193 7 get get VERB cana-3900 193 8 from from ADP cana-3900 193 9 equations equation NOUN cana-3900 193 10 ( ( PUNCT cana-3900 193 11 13 13 NUM cana-3900 193 12 ) ) PUNCT cana-3900 193 13 and and CCONJ cana-3900 193 14 ( ( PUNCT cana-3900 193 15 14 14 NUM cana-3900 193 16 ) ) PUNCT cana-3900 193 17 , , PUNCT cana-3900 193 18 ( ( PUNCT cana-3900 193 19 𝑦 𝑦 NOUN cana-3900 193 20 βˆ’ βˆ’ PROPN cana-3900 193 21 𝑦𝑓(π‘₯)π‘₯)2 𝑦𝑓(π‘₯)π‘₯)2 PROPN cana-3900 193 22 = = SYM cana-3900 194 1 0 0 X cana-3900 194 2 . . PUNCT cana-3900 195 1 since since SCONJ cana-3900 195 2 𝐿 𝐿 PROPN cana-3900 195 3 = = SYM cana-3900 195 4 { { PUNCT cana-3900 195 5 0 0 NUM cana-3900 195 6 } } PUNCT cana-3900 195 7 we we PRON cana-3900 195 8 get get VERB cana-3900 195 9 ( ( PUNCT cana-3900 195 10 𝑦 𝑦 NOUN cana-3900 195 11 βˆ’ βˆ’ NOUN cana-3900 195 12 𝑦𝑓(π‘₯)π‘₯ 𝑦𝑓(π‘₯)π‘₯ PROPN cana-3900 195 13 ) ) PUNCT cana-3900 195 14 = = SYM cana-3900 195 15 0 0 NUM cana-3900 196 1 β‡’ β‡’ NOUN cana-3900 196 2 𝑦 𝑦 NOUN cana-3900 196 3 = = SYM cana-3900 196 4 𝑦𝑓(π‘₯)π‘₯ 𝑦𝑓(π‘₯)π‘₯ PROPN cana-3900 196 5 β‡’ β‡’ VERB cana-3900 196 6 𝑦 𝑦 SYM cana-3900 196 7 ∈ ∈ PROPN cana-3900 196 8 𝑁π‘₯ 𝑁π‘₯ PROPN cana-3900 196 9 . . PUNCT cana-3900 197 1 therefore therefore ADV cana-3900 197 2 ( ( PUNCT cana-3900 197 3 0 0 NUM cana-3900 197 4 ∢ ∢ PROPN cana-3900 197 5 𝑆 𝑆 PROPN cana-3900 197 6 ) ) PUNCT cana-3900 197 7 βŠ‚ βŠ‚ PROPN cana-3900 197 8 𝑁π‘₯ 𝑁π‘₯ NOUN cana-3900 197 9 …… …… NOUN cana-3900 197 10 …… …… NOUN cana-3900 197 11 . . PUNCT cana-3900 198 1 ( ( PUNCT cana-3900 198 2 15 15 NUM cana-3900 198 3 ) ) PUNCT cana-3900 198 4 from from ADP cana-3900 198 5 equations equation NOUN cana-3900 198 6 ( ( PUNCT cana-3900 198 7 12 12 NUM cana-3900 198 8 ) ) PUNCT cana-3900 198 9 and and CCONJ cana-3900 198 10 ( ( PUNCT cana-3900 198 11 15 15 X cana-3900 198 12 ) ) PUNCT cana-3900 198 13 we we PRON cana-3900 198 14 get get VERB cana-3900 198 15 𝑁π‘₯ 𝑁π‘₯ NOUN cana-3900 198 16 = = SYM cana-3900 198 17 ( ( PUNCT cana-3900 198 18 0 0 NUM cana-3900 198 19 ∢ ∢ PROPN cana-3900 198 20 𝑆 𝑆 PROPN cana-3900 198 21 ) ) PUNCT cana-3900 198 22 and and CCONJ cana-3900 198 23 hence hence ADV cana-3900 198 24 nx nx PROPN cana-3900 198 25 is be AUX cana-3900 198 26 an an DET cana-3900 198 27 ideal ideal NOUN cana-3900 198 28 . . PUNCT cana-3900 199 1 the the DET cana-3900 199 2 desired desire VERB cana-3900 199 3 result result NOUN cana-3900 199 4 now now ADV cana-3900 199 5 follows follow VERB cana-3900 199 6 . . PUNCT cana-3900 200 1 remarks remark VERB cana-3900 200 2 3.2.19 3.2.19 NUM cana-3900 200 3 . . PUNCT cana-3900 201 1 ( ( PUNCT cana-3900 201 2 a a X cana-3900 201 3 ) ) PUNCT cana-3900 201 4 it it PRON cana-3900 201 5 is be AUX cana-3900 201 6 worth worth ADJ cana-3900 201 7 noting note VERB cana-3900 201 8 that that SCONJ cana-3900 201 9 in in ADP cana-3900 201 10 a a DET cana-3900 201 11 𝛿1near 𝛿1near NOUN cana-3900 201 12 - - PUNCT cana-3900 201 13 ring ring NOUN cana-3900 201 14 with with ADP cana-3900 201 15 mate mate NOUN cana-3900 201 16 functions function NOUN cana-3900 201 17 the the DET cana-3900 201 18 concepts concept NOUN cana-3900 201 19 of of ADP cana-3900 201 20 n n NOUN cana-3900 201 21 - - PUNCT cana-3900 201 22 subgroups subgroup NOUN cana-3900 201 23 , , PUNCT cana-3900 201 24 left leave VERB cana-3900 201 25 ideals ideal NOUN cana-3900 201 26 , , PUNCT cana-3900 201 27 right right ADJ cana-3900 201 28 ideals ideal NOUN cana-3900 201 29 and and CCONJ cana-3900 201 30 ideals ideal NOUN cana-3900 201 31 are be AUX cana-3900 201 32 equivalent equivalent ADJ cana-3900 201 33 . . PUNCT cana-3900 202 1 ( ( PUNCT cana-3900 202 2 b b X cana-3900 202 3 ) ) PUNCT cana-3900 202 4 recall recall NOUN cana-3900 202 5 that that SCONJ cana-3900 202 6 the the DET cana-3900 202 7 nil nil ADJ cana-3900 202 8 radical radical NOUN cana-3900 202 9 of of ADP cana-3900 202 10 n n PROPN cana-3900 202 11 is be AUX cana-3900 202 12 the the DET cana-3900 202 13 greatest great ADJ cana-3900 202 14 nil nil NOUN cana-3900 202 15 ideal ideal NOUN cana-3900 202 16 of of ADP cana-3900 202 17 n. n. NOUN cana-3900 202 18 since since SCONJ cana-3900 202 19 l l NOUN cana-3900 202 20 = = PUNCT cana-3900 202 21 { { PUNCT cana-3900 202 22 0 0 NUM cana-3900 202 23 } } PUNCT cana-3900 202 24 , , PUNCT cana-3900 202 25 for for ADP cana-3900 202 26 an an DET cana-3900 202 27 s'-𝛿1near s'-𝛿1near NOUN cana-3900 202 28 - - PUNCT cana-3900 202 29 ring ring NOUN cana-3900 202 30 n n CCONJ cana-3900 202 31 , , PUNCT cana-3900 202 32 it it PRON cana-3900 202 33 follows follow VERB cana-3900 202 34 that that SCONJ cana-3900 202 35 the the DET cana-3900 202 36 nil nil ADJ cana-3900 202 37 radical radical NOUN cana-3900 202 38 of of ADP cana-3900 202 39 𝑁 𝑁 PROPN cana-3900 202 40 = = SYM cana-3900 202 41 { { PUNCT cana-3900 202 42 0 0 NUM cana-3900 202 43 } } PUNCT cana-3900 202 44 . . PUNCT cana-3900 203 1 proposition proposition NOUN cana-3900 203 2 3.2.20 3.2.20 NUM cana-3900 203 3 let let VERB cana-3900 203 4 n n PRON cana-3900 203 5 be be AUX cana-3900 203 6 an an DET cana-3900 203 7 s'-𝛿1near s'-𝛿1near NOUN cana-3900 203 8 - - PUNCT cana-3900 203 9 ring ring NOUN cana-3900 203 10 . . PUNCT cana-3900 204 1 then then ADV cana-3900 204 2 any any DET cana-3900 204 3 n n NOUN cana-3900 204 4 - - PUNCT cana-3900 204 5 subgroup subgroup NOUN cana-3900 204 6 of of ADP cana-3900 204 7 n n PROPN cana-3900 204 8 is be AUX cana-3900 204 9 a a DET cana-3900 204 10 completely completely ADV cana-3900 204 11 semi semi ADJ cana-3900 204 12 prime prime ADJ cana-3900 204 13 ideal ideal NOUN cana-3900 204 14 . . PUNCT cana-3900 205 1 proof proof NOUN cana-3900 205 2 . . PUNCT cana-3900 206 1 suppose suppose VERB cana-3900 206 2 i i PRON cana-3900 206 3 is be AUX cana-3900 206 4 an an DET cana-3900 206 5 n n NOUN cana-3900 206 6 - - PUNCT cana-3900 206 7 subgroup subgroup NOUN cana-3900 206 8 of of ADP cana-3900 206 9 n. n. PROPN cana-3900 206 10 from from ADP cana-3900 206 11 theorem theorem ADJ cana-3900 206 12 3.2.18 3.2.18 NUM cana-3900 206 13 it it PRON cana-3900 206 14 follows follow VERB cana-3900 206 15 that that SCONJ cana-3900 206 16 i i PRON cana-3900 206 17 is be AUX cana-3900 206 18 an an DET cana-3900 206 19 ideal ideal NOUN cana-3900 206 20 . . PUNCT cana-3900 207 1 let let VERB cana-3900 207 2 π‘₯2 π‘₯2 PROPN cana-3900 207 3 ∈ ∈ PROPN cana-3900 207 4 𝐼. 𝐼. PROPN cana-3900 207 5 since since SCONJ cana-3900 207 6 n n NUM cana-3900 207 7 has have VERB cana-3900 207 8 strong strong ADJ cana-3900 207 9 ifp ifp NOUN cana-3900 207 10 , , PUNCT cana-3900 207 11 π‘₯𝑓(π‘₯)π‘₯ π‘₯𝑓(π‘₯)π‘₯ VERB cana-3900 207 12 ∈ ∈ NOUN cana-3900 207 13 𝐼 𝐼 PROPN cana-3900 207 14 i.e. i.e. X cana-3900 207 15 π‘₯ π‘₯ X cana-3900 207 16 ∈i ∈i NOUN cana-3900 207 17 . . PUNCT cana-3900 208 1 hence hence ADV cana-3900 208 2 i i PRON cana-3900 208 3 is be AUX cana-3900 208 4 a a DET cana-3900 208 5 completely completely ADV cana-3900 208 6 semi semi ADJ cana-3900 208 7 prime prime ADJ cana-3900 208 8 ideal ideal NOUN cana-3900 208 9 . . PUNCT cana-3900 209 1 proposition proposition NOUN cana-3900 209 2 3.2.21 3.2.21 NUM cana-3900 209 3 an an DET cana-3900 209 4 s'-𝛿1near s'-𝛿1near NOUN cana-3900 209 5 - - PUNCT cana-3900 209 6 ring ring NOUN cana-3900 209 7 has have VERB cana-3900 209 8 property property NOUN cana-3900 209 9 p4 p4 ADJ cana-3900 209 10 proof proof NOUN cana-3900 209 11 . . PUNCT cana-3900 210 1 let let VERB cana-3900 210 2 i i PRON cana-3900 210 3 be be AUX cana-3900 210 4 an an DET cana-3900 210 5 ideal ideal NOUN cana-3900 210 6 of of ADP cana-3900 210 7 n n NUM cana-3900 210 8 and and CCONJ cana-3900 210 9 let let VERB cana-3900 210 10 π‘₯𝑦 π‘₯𝑦 NOUN cana-3900 210 11 ∈ ∈ NOUN cana-3900 210 12 𝐼. 𝐼. NOUN cana-3900 210 13 now now ADV cana-3900 210 14 ( ( PUNCT cana-3900 210 15 𝑦π‘₯)2 𝑦π‘₯)2 PROPN cana-3900 210 16 = = SYM cana-3900 210 17 ( ( PUNCT cana-3900 210 18 𝑦π‘₯)(𝑦π‘₯ 𝑦π‘₯)(𝑦π‘₯ PROPN cana-3900 210 19 ) ) PUNCT cana-3900 210 20 = = SYM cana-3900 211 1 𝑦(π‘₯𝑦)π‘₯ 𝑦(π‘₯𝑦)π‘₯ PROPN cana-3900 211 2 ∈ ∈ PROPN cana-3900 211 3 𝑁𝐼𝑁 𝑁𝐼𝑁 PROPN cana-3900 211 4 βŠ‚ βŠ‚ X cana-3900 211 5 𝐼 𝐼 ADP cana-3900 212 1 [ [ X cana-3900 212 2 using use VERB cana-3900 212 3 remark remark NOUN cana-3900 212 4 3.2.19 3.2.19 NUM cana-3900 212 5 ( ( PUNCT cana-3900 212 6 a a NOUN cana-3900 212 7 ) ) PUNCT cana-3900 212 8 ] ] PUNCT cana-3900 212 9 β‡’ β‡’ NOUN cana-3900 212 10 ( ( PUNCT cana-3900 212 11 𝑦π‘₯)2 𝑦π‘₯)2 PROPN cana-3900 212 12 ∈ ∈ NOUN cana-3900 212 13 𝐼 𝐼 ADP cana-3900 212 14 .using .use VERB cana-3900 212 15 proposition proposition NOUN cana-3900 212 16 3.2.20 3.2.20 NUM cana-3900 212 17 we we PRON cana-3900 212 18 get get VERB cana-3900 212 19 𝑦π‘₯ 𝑦π‘₯ NOUN cana-3900 212 20 ∈ ∈ NOUN cana-3900 212 21 𝐼. 𝐼. NOUN cana-3900 212 22 i.e. i.e. X cana-3900 212 23 ) ) PUNCT cana-3900 212 24 . . PUNCT cana-3900 213 1 n n PROPN cana-3900 213 2 has have VERB cana-3900 213 3 property property NOUN cana-3900 213 4 p4 p4 ADJ cana-3900 213 5 . . PUNCT cana-3900 214 1 3.3 3.3 NUM cana-3900 214 2 in in ADP cana-3900 214 3 this this DET cana-3900 214 4 section section NOUN cana-3900 214 5 we we PRON cana-3900 214 6 obtain obtain VERB cana-3900 214 7 a a DET cana-3900 214 8 structure structure NOUN cana-3900 214 9 theorem theorem VERB cana-3900 214 10 for for ADP cana-3900 214 11 𝜹𝟏 𝜹𝟏 NOUN cana-3900 214 12 near near ADP cana-3900 214 13 - - PUNCT cana-3900 214 14 ring ring NOUN cana-3900 214 15 . . PUNCT cana-3900 215 1 throughout throughout ADP cana-3900 215 2 this this DET cana-3900 215 3 section section NOUN cana-3900 215 4 n n PRON cana-3900 215 5 denotes denote VERB cana-3900 215 6 an an DET cana-3900 215 7 s'-𝛿1near s'-𝛿1near NOUN cana-3900 215 8 - - PUNCT cana-3900 215 9 ring ring NOUN cana-3900 215 10 and and CCONJ cana-3900 215 11 m m NOUN cana-3900 215 12 is be AUX cana-3900 215 13 a a DET cana-3900 215 14 mate mate NOUN cana-3900 215 15 function function NOUN cana-3900 215 16 for for ADP cana-3900 215 17 n. n. NOUN cana-3900 215 18 theorem theorem NOUN cana-3900 215 19 3.3.1 3.3.1 NUM cana-3900 215 20 . . PUNCT cana-3900 216 1 n n PRON cana-3900 216 2 is be AUX cana-3900 216 3 sub sub NOUN cana-3900 216 4 directly directly ADV cana-3900 216 5 irreducible irreducible ADJ cana-3900 216 6 if if SCONJ cana-3900 216 7 and and CCONJ cana-3900 216 8 only only ADV cana-3900 216 9 if if SCONJ cana-3900 216 10 n n PRON cana-3900 216 11 is be AUX cana-3900 216 12 a a DET cana-3900 216 13 near near ADJ cana-3900 216 14 - - PUNCT cana-3900 216 15 field field NOUN cana-3900 216 16 . . PUNCT cana-3900 217 1 proof proof NOUN cana-3900 217 2 . . PUNCT cana-3900 218 1 suppose suppose VERB cana-3900 218 2 n n PRON cana-3900 218 3 is be AUX cana-3900 218 4 sub sub NOUN cana-3900 218 5 directly directly ADV cana-3900 218 6 irreducible irreducible ADJ cana-3900 218 7 . . PUNCT cana-3900 219 1 first first ADV cana-3900 219 2 we we PRON cana-3900 219 3 claim claim VERB cana-3900 219 4 that that SCONJ cana-3900 219 5 no no DET cana-3900 219 6 non non ADJ cana-3900 219 7 - - ADJ cana-3900 219 8 zero zero ADJ cana-3900 219 9 idempotent idempotent NOUN cana-3900 219 10 of of ADP cana-3900 219 11 n n PROPN cana-3900 219 12 is be AUX cana-3900 219 13 a a DET cana-3900 219 14 zerodivisor zerodivisor NOUN cana-3900 219 15 . . PUNCT cana-3900 220 1 let let VERB cana-3900 220 2 j j PROPN cana-3900 220 3 be be AUX cana-3900 220 4 the the DET cana-3900 220 5 set set NOUN cana-3900 220 6 of of ADP cana-3900 220 7 all all DET cana-3900 220 8 non non ADJ cana-3900 220 9 - - ADJ cana-3900 220 10 zero zero NUM cana-3900 220 11 idempotent idempotent NOUN cana-3900 220 12 which which PRON cana-3900 220 13 are be AUX cana-3900 220 14 zero zero NUM cana-3900 220 15 - - PUNCT cana-3900 220 16 divisors divisor NOUN cana-3900 220 17 and and CCONJ cana-3900 220 18 let let VERB cana-3900 220 19 β‰  β‰  PROPN cana-3900 220 20 βˆ… βˆ… NOUN cana-3900 220 21 . . PUNCT cana-3900 221 1 let let VERB cana-3900 221 2 𝐼 𝐼 PROPN cana-3900 221 3 = = SYM cana-3900 221 4 β‹‚ β‹‚ PROPN cana-3900 221 5 ( ( PUNCT cana-3900 221 6 0 0 NUM cana-3900 221 7 ∢ ∢ NOUN cana-3900 221 8 𝑒)π‘’βˆˆπ½ 𝑒)π‘’βˆˆπ½ NOUN cana-3900 221 9 . . PUNCT cana-3900 222 1 since since SCONJ cana-3900 222 2 n n PRON cana-3900 222 3 is be AUX cana-3900 222 4 subdirectly subdirectly ADV cana-3900 222 5 irreducible irreducible ADJ cana-3900 222 6 , , PUNCT cana-3900 222 7 𝐼 𝐼 PROPN cana-3900 222 8 β‰  β‰  PROPN cana-3900 222 9 βˆ…. βˆ…. AUX cana-3900 222 10 let let VERB cana-3900 222 11 π‘Ž π‘Ž PRON cana-3900 222 12 ∈ ∈ NOUN cana-3900 222 13 𝐼 𝐼 ADP cana-3900 222 14 βˆ’ βˆ’ PROPN cana-3900 222 15 { { PUNCT cana-3900 222 16 0 0 NUM cana-3900 222 17 } } PUNCT cana-3900 222 18 . . PUNCT cana-3900 223 1 thus thus ADV cana-3900 223 2 π‘Žπ‘’ π‘Žπ‘’ X cana-3900 223 3 = = SYM cana-3900 223 4 0 0 NUM cana-3900 223 5 for for ADP cana-3900 223 6 all all DET cana-3900 223 7 e e NOUN cana-3900 223 8 in in ADP cana-3900 223 9 j j PROPN cana-3900 223 10 … … PUNCT cana-3900 223 11 … … PUNCT cana-3900 223 12 … … SYM cana-3900 223 13 . . PUNCT cana-3900 223 14 … … PUNCT cana-3900 223 15 . . PUNCT cana-3900 224 1 ( ( PUNCT cana-3900 224 2 16 16 NUM cana-3900 224 3 ) ) PUNCT cana-3900 224 4 this this DET cana-3900 224 5 β‡’ β‡’ NOUN cana-3900 224 6 𝑓(π‘Ž)π‘Žπ‘’ 𝑓(π‘Ž)π‘Žπ‘’ PROPN cana-3900 224 7 = = SYM cana-3900 224 8 0 0 NUM cana-3900 224 9 β‡’ β‡’ NOUN cana-3900 224 10 𝑒𝑓(π‘Ž)π‘Ž 𝑒𝑓(π‘Ž)π‘Ž PUNCT cana-3900 225 1 = = SYM cana-3900 225 2 0 0 PUNCT cana-3900 226 1 [ [ X cana-3900 226 2 using use VERB cana-3900 226 3 k k PROPN cana-3900 226 4 ( ( PUNCT cana-3900 226 5 2 2 NUM cana-3900 226 6 ) ) PUNCT cana-3900 226 7 ] ] PUNCT cana-3900 226 8 β‡’ β‡’ PROPN cana-3900 226 9 𝑓(π‘Ž 𝑓(π‘Ž NOUN cana-3900 226 10 ) ) PUNCT cana-3900 226 11 π‘Ž π‘Ž PRON cana-3900 226 12 ∈ ∈ PROPN cana-3900 226 13 𝐽 𝐽 NOUN cana-3900 226 14 . . PUNCT cana-3900 227 1 from from ADP cana-3900 227 2 equation equation NOUN cana-3900 227 3 ( ( PUNCT cana-3900 227 4 16 16 NUM cana-3900 227 5 ) ) PUNCT cana-3900 227 6 we we PRON cana-3900 227 7 get get VERB cana-3900 227 8 π‘Žπ‘“(π‘Ž)π‘Ž π‘Žπ‘“(π‘Ž)π‘Ž PROPN cana-3900 227 9 = = SYM cana-3900 227 10 0 0 NUM cana-3900 227 11 β‡’ β‡’ NOUN cana-3900 227 12 π‘Ž π‘Ž PROPN cana-3900 227 13 = = SYM cana-3900 227 14 0 0 NUM cana-3900 227 15 . . NUM cana-3900 227 16 … … PUNCT cana-3900 227 17 … … PUNCT cana-3900 227 18 … … PUNCT cana-3900 227 19 … … PUNCT cana-3900 227 20 . . PUNCT cana-3900 227 21 . . PUNCT cana-3900 227 22 . . PUNCT cana-3900 228 1 ( ( PUNCT cana-3900 228 2 17 17 NUM cana-3900 228 3 ) ) PUNCT cana-3900 228 4 . . PUNCT cana-3900 229 1 this this DET cana-3900 229 2 contradiction contradiction NOUN cana-3900 229 3 implies imply VERB cana-3900 229 4 that that SCONJ cana-3900 229 5 no no DET cana-3900 229 6 non non ADJ cana-3900 229 7 - - ADJ cana-3900 229 8 zero zero ADJ cana-3900 229 9 idempotent idempotent NOUN cana-3900 229 10 of of ADP cana-3900 229 11 n n PROPN cana-3900 229 12 is be AUX cana-3900 229 13 a a DET cana-3900 229 14 zero zero NUM cana-3900 229 15 - - PUNCT cana-3900 229 16 divisor divisor NOUN cana-3900 229 17 . . PUNCT cana-3900 230 1 we we PRON cana-3900 230 2 shall shall AUX cana-3900 230 3 now now ADV cana-3900 230 4 prove prove VERB cana-3900 230 5 that that SCONJ cana-3900 230 6 n n PRON cana-3900 230 7 has have VERB cana-3900 230 8 no no DET cana-3900 230 9 non non ADJ cana-3900 230 10 - - ADJ cana-3900 230 11 trivial trivial ADJ cana-3900 230 12 n n CCONJ cana-3900 230 13 - - PUNCT cana-3900 230 14 subgroups subgroup NOUN cana-3900 230 15 . . PUNCT cana-3900 231 1 let let VERB cana-3900 231 2 m m PRON cana-3900 231 3 be be AUX cana-3900 231 4 any any DET cana-3900 231 5 n n NOUN cana-3900 231 6 - - PUNCT cana-3900 231 7 subgroup subgroup NOUN cana-3900 231 8 of of ADP cana-3900 231 9 n n PRON cana-3900 231 10 such such ADJ cana-3900 231 11 that that SCONJ cana-3900 231 12 𝑀 𝑀 PROPN cana-3900 231 13 β‰  β‰  PROPN cana-3900 231 14 { { PUNCT cana-3900 231 15 0 0 NUM cana-3900 231 16 } } PUNCT cana-3900 231 17 and and CCONJ cana-3900 231 18 let let VERB cana-3900 231 19 π‘₯(β‰  π‘₯(β‰  PROPN cana-3900 231 20 0 0 NUM cana-3900 231 21 ) ) PUNCT cana-3900 231 22 ∈ ∈ NOUN cana-3900 231 23 𝑀. 𝑀. PROPN cana-3900 231 24 let let VERB cana-3900 231 25 n n PRON cana-3900 231 26 be be AUX cana-3900 231 27 a a DET cana-3900 231 28 𝛿1near 𝛿1near ADJ cana-3900 231 29 ring ring NOUN cana-3900 231 30 . . PUNCT cana-3900 232 1 then then ADV cana-3900 232 2 for for ADP cana-3900 232 3 all all DET cana-3900 232 4 x x NOUN cana-3900 232 5 , , PUNCT cana-3900 232 6 y y PROPN cana-3900 232 7 in in ADP cana-3900 232 8 n n CCONJ cana-3900 232 9 , , PUNCT cana-3900 232 10 π‘₯𝑁𝑦 π‘₯𝑁𝑦 NOUN cana-3900 232 11 = = PUNCT cana-3900 232 12 𝑁π‘₯2𝑦2 𝑁π‘₯2𝑦2 NOUN cana-3900 232 13 .putting .putte VERB cana-3900 232 14 π‘₯ π‘₯ NOUN cana-3900 232 15 = = SYM cana-3900 232 16 1 1 NUM cana-3900 232 17 , , PUNCT cana-3900 232 18 we we PRON cana-3900 232 19 get get VERB cana-3900 232 20 π‘₯𝑁. π‘₯𝑁. ADP cana-3900 232 21 1 1 NUM cana-3900 232 22 = = NOUN cana-3900 232 23 𝑁π‘₯2 𝑁π‘₯2 ADJ cana-3900 232 24 . . PUNCT cana-3900 233 1 1 1 NUM cana-3900 233 2 for for ADP cana-3900 233 3 all all DET cana-3900 233 4 x x NOUN cana-3900 233 5 in in ADP cana-3900 233 6 n. n. NOUN cana-3900 233 7 β‡’ β‡’ NOUN cana-3900 234 1 𝑁𝑦 𝑁𝑦 PROPN cana-3900 234 2 = = PUNCT cana-3900 234 3 𝑁𝑦2 𝑁𝑦2 NOUN cana-3900 234 4 for for ADP cana-3900 234 5 all all DET cana-3900 234 6 y y PROPN cana-3900 234 7 in in ADP cana-3900 234 8 n. n. NOUN cana-3900 234 9 for for ADP cana-3900 234 10 any any DET cana-3900 234 11 𝑛 𝑛 PRON cana-3900 234 12 ∈ ∈ NOUN cana-3900 234 13 𝑁 𝑁 PROPN cana-3900 234 14 , , PUNCT cana-3900 234 15 there there PRON cana-3900 234 16 exists exist VERB cana-3900 234 17 𝑛1 𝑛1 ADJ cana-3900 234 18 in in ADP cana-3900 234 19 n n CCONJ cana-3900 234 20 such such ADJ cana-3900 234 21 that that SCONJ cana-3900 234 22 𝑛𝑦 𝑛𝑦 NOUN cana-3900 234 23 = = SYM cana-3900 234 24 𝑛1𝑦2 𝑛1𝑦2 NOUN cana-3900 234 25 β‡’ β‡’ NOUN cana-3900 234 26 ( ( PUNCT cana-3900 234 27 𝑛 𝑛 PRON cana-3900 234 28 βˆ’ βˆ’ PROPN cana-3900 234 29 𝑛1𝑦)𝑦 𝑛1𝑦)𝑦 NOUN cana-3900 234 30 = = SYM cana-3900 234 31 0 0 NUM cana-3900 234 32 β‡’ β‡’ NOUN cana-3900 234 33 ( ( PUNCT cana-3900 234 34 𝑛 𝑛 DET cana-3900 234 35 βˆ’ βˆ’ PROPN cana-3900 234 36 𝑛1𝑦)𝑓(𝑦)𝑦 𝑛1𝑦)𝑓(𝑦)𝑦 NOUN cana-3900 234 37 = = SYM cana-3900 234 38 0 0 NUM cana-3900 234 39 β‡’ β‡’ NOUN cana-3900 234 40 𝑛 𝑛 PRON cana-3900 234 41 βˆ’ βˆ’ PROPN cana-3900 234 42 𝑛1𝑦 𝑛1𝑦 PROPN cana-3900 234 43 = = SYM cana-3900 234 44 0[by 0[by NUM cana-3900 234 45 equation equation NOUN cana-3900 234 46 17]β‡’ 17]β‡’ NOUN cana-3900 234 47 𝑛 𝑛 NOUN cana-3900 234 48 = = SYM cana-3900 234 49 𝑛1𝑦 𝑛1𝑦 PROPN cana-3900 234 50 ∈ ∈ PROPN cana-3900 234 51 𝑁𝑀 𝑁𝑀 PROPN cana-3900 234 52 βŠ† βŠ† NUM cana-3900 234 53 𝑀. 𝑀. NOUN cana-3900 234 54 therefore therefore ADV cana-3900 234 55 𝑁 𝑁 PROPN cana-3900 234 56 βŠ† βŠ† NUM cana-3900 234 57 𝑀 𝑀 PROPN cana-3900 234 58 i.e. i.e. X cana-3900 234 59 𝑀 𝑀 PROPN cana-3900 234 60 = = SYM cana-3900 234 61 𝑁. 𝑁. PROPN cana-3900 234 62 thus thus ADV cana-3900 234 63 n n PRON cana-3900 234 64 has have VERB cana-3900 234 65 no no DET cana-3900 234 66 non non ADJ cana-3900 234 67 - - ADJ cana-3900 234 68 trivial trivial ADJ cana-3900 234 69 n n CCONJ cana-3900 234 70 - - PUNCT cana-3900 234 71 subgroups subgroup NOUN cana-3900 234 72 . . PUNCT cana-3900 235 1 clearly clearly ADV cana-3900 235 2 for for ADP cana-3900 235 3 𝑛 𝑛 PRON cana-3900 235 4 ∈ ∈ PROPN cana-3900 235 5 𝑁 𝑁 PROPN cana-3900 235 6 βˆ’ βˆ’ PROPN cana-3900 235 7 { { PUNCT cana-3900 235 8 0 0 NUM cana-3900 235 9 } } PUNCT cana-3900 235 10 , , PUNCT cana-3900 235 11 nn nn X cana-3900 235 12 is be AUX cana-3900 235 13 an an DET cana-3900 235 14 n n NOUN cana-3900 235 15 - - PUNCT cana-3900 235 16 subgroup subgroup NOUN cana-3900 235 17 of of ADP cana-3900 235 18 n. n. NOUN cana-3900 235 19 consequently consequently ADV cana-3900 235 20 𝑁𝑛 𝑁𝑛 PROPN cana-3900 235 21 = = SYM cana-3900 235 22 𝑁 𝑁 PROPN cana-3900 235 23 for for ADP cana-3900 235 24 all all DET cana-3900 235 25 𝑛 𝑛 DET cana-3900 235 26 ∈ ∈ NOUN cana-3900 236 1 𝑁 𝑁 PROPN cana-3900 236 2 βˆ’ βˆ’ PROPN cana-3900 236 3 { { PUNCT cana-3900 236 4 0 0 NUM cana-3900 236 5 } } PUNCT cana-3900 236 6 …… …… NOUN cana-3900 236 7 .. .. PUNCT cana-3900 236 8 … … PUNCT cana-3900 236 9 . . PUNCT cana-3900 237 1 ( ( PUNCT cana-3900 237 2 18 18 NUM cana-3900 237 3 ) ) PUNCT cana-3900 237 4 .also .also PUNCT cana-3900 237 5 , , PUNCT cana-3900 237 6 it it PRON cana-3900 237 7 is be AUX cana-3900 237 8 clear clear ADJ cana-3900 237 9 that that SCONJ cana-3900 237 10 𝑁𝑑 𝑁𝑑 ADV cana-3900 237 11 β‰  β‰  PROPN cana-3900 237 12 { { PUNCT cana-3900 237 13 0 0 NUM cana-3900 237 14 } } PUNCT cana-3900 237 15 [ [ X cana-3900 237 16 since since SCONJ cana-3900 237 17 e e PROPN cana-3900 237 18 βŠ† βŠ† NUM cana-3900 237 19 c(n)]βŠ† c(n)]βŠ† NOUN cana-3900 237 20 𝑁𝑑 𝑁𝑑 NOUN cana-3900 237 21 ] ] X cana-3900 237 22 . . PUNCT cana-3900 238 1 this this PRON cana-3900 238 2 and and CCONJ cana-3900 238 3 equation equation NOUN cana-3900 238 4 ( ( PUNCT cana-3900 238 5 18 18 NUM cana-3900 238 6 ) ) PUNCT cana-3900 238 7 guarantee guarantee VERB cana-3900 238 8 that that SCONJ cana-3900 238 9 n n PRON cana-3900 238 10 is be AUX cana-3900 238 11 a a DET cana-3900 238 12 near near ADJ cana-3900 238 13 - - PUNCT cana-3900 238 14 field field NOUN cana-3900 238 15 . . PUNCT cana-3900 239 1 [ [ X cana-3900 239 2 theorem theorem ADJ cana-3900 239 3 8.3 8.3 NUM cana-3900 239 4 , , PUNCT cana-3900 239 5 pilz pilz PROPN cana-3900 240 1 [ [ X cana-3900 240 2 3 3 NUM cana-3900 240 3 ] ] PUNCT cana-3900 240 4 ) ) PUNCT cana-3900 240 5 ] ] PUNCT cana-3900 240 6 converse converse NOUN cana-3900 240 7 is be AUX cana-3900 240 8 obvious obvious ADJ cana-3900 240 9 . . PUNCT cana-3900 241 1 as as ADP cana-3900 241 2 an an DET cana-3900 241 3 immediate immediate ADJ cana-3900 241 4 consequence consequence NOUN cana-3900 241 5 of of ADP cana-3900 241 6 theorem theorem NOUN cana-3900 241 7 3.1 3.1 NUM cana-3900 241 8 , , PUNCT cana-3900 241 9 we we PRON cana-3900 241 10 have have VERB cana-3900 241 11 the the DET cana-3900 241 12 following follow VERB cana-3900 241 13 : : PUNCT cana-3900 241 14 corollary corollary ADJ cana-3900 241 15 3.3.2 3.3.2 PROPN cana-3900 241 16 . . PUNCT cana-3900 242 1 n n PRON cana-3900 242 2 has have VERB cana-3900 242 3 no no DET cana-3900 242 4 non non ADJ cana-3900 242 5 - - ADJ cana-3900 242 6 zero zero NUM cana-3900 242 7 zero zero NUM cana-3900 242 8 - - PUNCT cana-3900 242 9 divisors divisor NOUN cana-3900 242 10 if if SCONJ cana-3900 242 11 and and CCONJ cana-3900 242 12 only only ADV cana-3900 242 13 if if SCONJ cana-3900 242 14 n n PRON cana-3900 242 15 is be AUX cana-3900 242 16 a a DET cana-3900 242 17 near near ADJ cana-3900 242 18 - - PUNCT cana-3900 242 19 field field NOUN cana-3900 242 20 . . PUNCT cana-3900 243 1 we we PRON cana-3900 243 2 are be AUX cana-3900 243 3 now now ADV cana-3900 243 4 in in ADP cana-3900 243 5 a a DET cana-3900 243 6 position position NOUN cana-3900 243 7 to to PART cana-3900 243 8 give give VERB cana-3900 243 9 a a DET cana-3900 243 10 structure structure NOUN cana-3900 243 11 theorem theorem VERB cana-3900 243 12 for for ADP cana-3900 243 13 n. n. NOUN cana-3900 243 14 communications communication NOUN cana-3900 243 15 on on ADP cana-3900 243 16 applied apply VERB cana-3900 243 17 nonlinear nonlinear ADJ cana-3900 243 18 analysis analysis NOUN cana-3900 243 19 issn issn NOUN cana-3900 243 20 : : PUNCT cana-3900 243 21 1074 1074 NUM cana-3900 243 22 - - PUNCT cana-3900 243 23 133x 133x NUM cana-3900 243 24 vol vol NOUN cana-3900 243 25 32 32 NUM cana-3900 244 1 no no NOUN cana-3900 244 2 . . PUNCT cana-3900 245 1 9s 9s NUM cana-3900 245 2 ( ( PUNCT cana-3900 245 3 2025 2025 NUM cana-3900 245 4 ) ) PUNCT cana-3900 245 5 352 352 NUM cana-3900 246 1 https://internationalpubls.com https://internationalpubls.com X cana-3900 246 2 theorem theorem VERB cana-3900 246 3 3.3.3 3.3.3 PROPN cana-3900 246 4 . . PUNCT cana-3900 247 1 n n PRON cana-3900 247 2 is be AUX cana-3900 247 3 isomorphic isomorphic ADJ cana-3900 247 4 to to ADP cana-3900 247 5 a a DET cana-3900 247 6 sub sub NOUN cana-3900 247 7 direct direct ADJ cana-3900 247 8 product product NOUN cana-3900 247 9 of of ADP cana-3900 247 10 near near ADJ cana-3900 247 11 - - PUNCT cana-3900 247 12 fields field NOUN cana-3900 247 13 . . PUNCT cana-3900 248 1 proof proof NOUN cana-3900 248 2 . . PUNCT cana-3900 249 1 from from ADP cana-3900 249 2 theorem theorem ADJ cana-3900 249 3 3.2.4 3.2.4 NUM cana-3900 249 4 , , PUNCT cana-3900 249 5 n n X cana-3900 249 6 is be AUX cana-3900 249 7 isomorphic isomorphic ADJ cana-3900 249 8 to to ADP cana-3900 249 9 a a DET cana-3900 249 10 sub sub NOUN cana-3900 249 11 direct direct ADJ cana-3900 249 12 product product NOUN cana-3900 249 13 of of ADP cana-3900 249 14 sub sub NOUN cana-3900 249 15 directly directly ADV cana-3900 249 16 irreducible irreducible ADJ cana-3900 249 17 𝛿1nearring 𝛿1nearring NOUN cana-3900 249 18 , , PUNCT cana-3900 249 19 ni ni PROPN cana-3900 249 20 's 's PART cana-3900 249 21 , , PUNCT cana-3900 249 22 say say INTJ cana-3900 249 23 . . PUNCT cana-3900 250 1 obviously obviously ADV cana-3900 250 2 the the DET cana-3900 250 3 existence existence NOUN cana-3900 250 4 of of ADP cana-3900 250 5 a a DET cana-3900 250 6 mate mate NOUN cana-3900 250 7 function function NOUN cana-3900 250 8 is be AUX cana-3900 250 9 preserved preserve VERB cana-3900 250 10 under under ADP cana-3900 250 11 homomorphisms homomorphism NOUN cana-3900 250 12 . . PUNCT cana-3900 251 1 hence hence ADV cana-3900 251 2 each each DET cana-3900 251 3 ni ni PROPN cana-3900 251 4 admits admit VERB cana-3900 251 5 a a DET cana-3900 251 6 mate mate NOUN cana-3900 251 7 function function NOUN cana-3900 251 8 . . PUNCT cana-3900 252 1 appealing appeal VERB cana-3900 252 2 to to PART cana-3900 252 3 theorem theorem VERB cana-3900 252 4 3.3.1 3.3.1 NUM cana-3900 252 5 we we PRON cana-3900 252 6 get get VERB cana-3900 252 7 n n PRON cana-3900 252 8 is be AUX cana-3900 252 9 isomorphic isomorphic ADJ cana-3900 252 10 to to ADP cana-3900 252 11 a a DET cana-3900 252 12 sub sub NOUN cana-3900 252 13 direct direct ADJ cana-3900 252 14 product product NOUN cana-3900 252 15 of of ADP cana-3900 252 16 near near ADJ cana-3900 252 17 - - PUNCT cana-3900 252 18 fields field NOUN cana-3900 252 19 . . PUNCT cana-3900 253 1 remark remark PROPN cana-3900 253 2 3.3.4 3.3.4 NUM cana-3900 253 3 . . PUNCT cana-3900 254 1 from from ADP cana-3900 254 2 8.11 8.11 NUM cana-3900 254 3 of of ADP cana-3900 254 4 [ [ X cana-3900 254 5 3 3 NUM cana-3900 254 6 ] ] PUNCT cana-3900 254 7 , , PUNCT cana-3900 254 8 the the DET cana-3900 254 9 additive additive ADJ cana-3900 254 10 group group NOUN cana-3900 254 11 of of ADP cana-3900 254 12 a a DET cana-3900 254 13 near near ADJ cana-3900 254 14 - - PUNCT cana-3900 254 15 field field NOUN cana-3900 254 16 is be AUX cana-3900 254 17 abelian abelian ADJ cana-3900 254 18 . . PUNCT cana-3900 255 1 it it PRON cana-3900 255 2 follows follow VERB cana-3900 255 3 that that SCONJ cana-3900 255 4 for for ADP cana-3900 255 5 any any DET cana-3900 255 6 𝛿1 𝛿1 NOUN cana-3900 255 7 near near ADP cana-3900 255 8 - - PUNCT cana-3900 255 9 ring ring NOUN cana-3900 255 10 n n NOUN cana-3900 255 11 with with ADP cana-3900 255 12 mate mate NOUN cana-3900 255 13 functions function NOUN cana-3900 255 14 , , PUNCT cana-3900 255 15 ( ( PUNCT cana-3900 255 16 n n CCONJ cana-3900 255 17 , , PUNCT cana-3900 255 18 + + X cana-3900 255 19 ) ) PUNCT cana-3900 255 20 is be AUX cana-3900 255 21 abelian abelian ADJ cana-3900 255 22 . . PUNCT cana-3900 256 1 proposition proposition NOUN cana-3900 256 2 3.3.5 3.3.5 X cana-3900 256 3 . . PUNCT cana-3900 257 1 let let VERB cana-3900 257 2 n n PRON cana-3900 257 3 be be AUX cana-3900 257 4 a a DET cana-3900 257 5 boolean boolean ADJ cana-3900 257 6 near near ADP cana-3900 257 7 - - PUNCT cana-3900 257 8 ring ring NOUN cana-3900 257 9 . . PUNCT cana-3900 258 1 then then ADV cana-3900 258 2 n n PROPN cana-3900 258 3 is be AUX cana-3900 258 4 𝛿1if 𝛿1if PUNCT cana-3900 258 5 and and CCONJ cana-3900 258 6 only only ADV cana-3900 258 7 if if SCONJ cana-3900 258 8 it it PRON cana-3900 258 9 is be AUX cana-3900 258 10 a a DET cana-3900 258 11 commutative commutative ADJ cana-3900 258 12 ring ring NOUN cana-3900 258 13 . . PUNCT cana-3900 259 1 proof proof NOUN cana-3900 259 2 . . PUNCT cana-3900 260 1 we we PRON cana-3900 260 2 observe observe VERB cana-3900 260 3 that that SCONJ cana-3900 260 4 identity identity NOUN cana-3900 260 5 function function NOUN cana-3900 260 6 is be AUX cana-3900 260 7 a a DET cana-3900 260 8 mate mate NOUN cana-3900 260 9 function function NOUN cana-3900 260 10 for for ADP cana-3900 260 11 n. n. NOUN cana-3900 260 12 appealing appeal VERB cana-3900 260 13 to to PART cana-3900 260 14 theorem theorem VERB cana-3900 260 15 3.2.16 3.2.16 NUM cana-3900 260 16 and and CCONJ cana-3900 260 17 remark remark NOUN cana-3900 260 18 3.3.4 3.3.4 NUM cana-3900 260 19 we we PRON cana-3900 260 20 see see VERB cana-3900 260 21 that that SCONJ cana-3900 260 22 when when SCONJ cana-3900 260 23 n n PRON cana-3900 260 24 is be AUX cana-3900 260 25 a a DET cana-3900 260 26 𝛿1near 𝛿1near NOUN cana-3900 260 27 - - PUNCT cana-3900 260 28 ring ring NOUN cana-3900 260 29 , , PUNCT cana-3900 260 30 𝑁 𝑁 PROPN cana-3900 260 31 = = SYM cana-3900 260 32 𝐸 𝐸 PROPN cana-3900 260 33 βŠ‚ βŠ‚ PROPN cana-3900 260 34 𝐢(𝑁 𝐢(𝑁 NUM cana-3900 260 35 ) ) PUNCT cana-3900 260 36 and and CCONJ cana-3900 260 37 ( ( PUNCT cana-3900 260 38 𝑁 𝑁 PROPN cana-3900 260 39 , , PUNCT cana-3900 260 40 + + NOUN cana-3900 260 41 ) ) PUNCT cana-3900 260 42 is be AUX cana-3900 260 43 abelian abelian ADJ cana-3900 260 44 and and CCONJ cana-3900 260 45 hence hence ADV cana-3900 260 46 n n PRON cana-3900 260 47 is be AUX cana-3900 260 48 a a DET cana-3900 260 49 commutative commutative ADJ cana-3900 260 50 ring ring NOUN cana-3900 260 51 . . PUNCT cana-3900 261 1 conversely conversely ADV cana-3900 261 2 , , PUNCT cana-3900 261 3 n n PRON cana-3900 261 4 is be AUX cana-3900 261 5 boolean boolean ADJ cana-3900 261 6 and and CCONJ cana-3900 261 7 a a DET cana-3900 261 8 commutative commutative ADJ cana-3900 261 9 ring ring NOUN cana-3900 261 10 . . PUNCT cana-3900 262 1 then then ADV cana-3900 262 2 for for ADP cana-3900 262 3 all all DET cana-3900 262 4 x x NOUN cana-3900 262 5 in in ADP cana-3900 262 6 n,π‘₯𝑛 n,π‘₯𝑛 NUM cana-3900 262 7 = = SYM cana-3900 262 8 𝑛π‘₯ 𝑛π‘₯ PROPN cana-3900 262 9 for for ADP cana-3900 262 10 all all DET cana-3900 262 11 n n NOUN cana-3900 262 12 in in ADP cana-3900 262 13 nβ‡’ nβ‡’ NOUN cana-3900 262 14 π‘₯𝑛𝑦 π‘₯𝑛𝑦 ADP cana-3900 262 15 = = SYM cana-3900 262 16 𝑛π‘₯𝑦 𝑛π‘₯𝑦 X cana-3900 262 17 for for ADP cana-3900 262 18 all all DET cana-3900 262 19 y y PROPN cana-3900 262 20 in in ADP cana-3900 262 21 n,β‡’ n,β‡’ PROPN cana-3900 262 22 π‘₯𝑁𝑦 π‘₯𝑁𝑦 NOUN cana-3900 262 23 = = NOUN cana-3900 262 24 𝑁π‘₯2𝑦2hence 𝑁π‘₯2𝑦2hence NOUN cana-3900 262 25 the the DET cana-3900 262 26 result result NOUN cana-3900 262 27 . . PUNCT cana-3900 263 1 proposition proposition NOUN cana-3900 263 2 3.3.6 3.3.6 NUM cana-3900 263 3 . . PUNCT cana-3900 264 1 if if SCONJ cana-3900 264 2 n n PRON cana-3900 264 3 is be AUX cana-3900 264 4 distributively distributively ADV cana-3900 264 5 generated generate VERB cana-3900 264 6 and and CCONJ cana-3900 264 7 has have VERB cana-3900 264 8 no no DET cana-3900 264 9 non non ADJ cana-3900 264 10 - - ADJ cana-3900 264 11 zero zero NUM cana-3900 264 12 zero zero NUM cana-3900 264 13 - - PUNCT cana-3900 264 14 divisors divisor NOUN cana-3900 264 15 then then ADV cana-3900 264 16 n n PRON cana-3900 264 17 is be AUX cana-3900 264 18 a a DET cana-3900 264 19 division division NOUN cana-3900 264 20 ring ring NOUN cana-3900 264 21 . . PUNCT cana-3900 265 1 proof proof NOUN cana-3900 265 2 . . PUNCT cana-3900 266 1 corollary corollary ADJ cana-3900 266 2 3.3.2 3.3.2 NUM cana-3900 266 3 guarantees guarantee NOUN cana-3900 266 4 that that PRON cana-3900 266 5 n n X cana-3900 266 6 is be AUX cana-3900 266 7 a a DET cana-3900 266 8 near near ADJ cana-3900 266 9 - - PUNCT cana-3900 266 10 field field NOUN cana-3900 266 11 . . PUNCT cana-3900 267 1 also also ADV cana-3900 267 2 ( ( PUNCT cana-3900 267 3 n n X cana-3900 267 4 , , PUNCT cana-3900 267 5 + + PUNCT cana-3900 267 6 ) ) PUNCT cana-3900 267 7 is be AUX cana-3900 267 8 abelian abelian ADJ cana-3900 267 9 [ [ X cana-3900 267 10 by by ADP cana-3900 267 11 remark remark NOUN cana-3900 267 12 3.3.4 3.3.4 NUM cana-3900 267 13 ) ) PUNCT cana-3900 267 14 ] ] PUNCT cana-3900 267 15 since since SCONJ cana-3900 267 16 n n PRON cana-3900 267 17 is be AUX cana-3900 267 18 distributively distributively ADV cana-3900 267 19 generated generate VERB cana-3900 267 20 , , PUNCT cana-3900 267 21 we we PRON cana-3900 267 22 see see VERB cana-3900 267 23 that that SCONJ cana-3900 267 24 n n PRON cana-3900 267 25 is be AUX cana-3900 267 26 a a DET cana-3900 267 27 ring ring NOUN cana-3900 267 28 [ [ X cana-3900 267 29 from from ADP cana-3900 267 30 theorem theorem ADJ cana-3900 267 31 6.6(c 6.6(c NUM cana-3900 267 32 ) ) PUNCT cana-3900 267 33 of of ADP cana-3900 267 34 pilz pilz PROPN cana-3900 267 35 [ [ X cana-3900 267 36 3 3 NUM cana-3900 267 37 ] ] X cana-3900 267 38 ] ] PUNCT cana-3900 267 39 and and CCONJ cana-3900 267 40 hence hence ADV cana-3900 267 41 the the DET cana-3900 267 42 result result NOUN cana-3900 267 43 . . PUNCT cana-3900 268 1 references reference NOUN cana-3900 268 2 [ [ X cana-3900 268 3 1 1 NUM cana-3900 268 4 ] ] X cana-3900 268 5 j.r j.r PROPN cana-3900 268 6 . . PROPN cana-3900 268 7 clay clay NOUN cana-3900 268 8 , , PUNCT cana-3900 268 9 the the DET cana-3900 268 10 near near NOUN cana-3900 268 11 - - PUNCT cana-3900 268 12 ring ring NOUN cana-3900 268 13 on on ADP cana-3900 268 14 groups group NOUN cana-3900 268 15 of of ADP cana-3900 268 16 low low ADJ cana-3900 268 17 order order NOUN cana-3900 268 18 , , PUNCT cana-3900 268 19 math math NOUN cana-3900 268 20 . . PUNCT cana-3900 269 1 z. z. PROPN cana-3900 269 2 104 104 NUM cana-3900 269 3 ( ( PUNCT cana-3900 269 4 1968 1968 NUM cana-3900 269 5 ) ) PUNCT cana-3900 269 6 , , PUNCT cana-3900 269 7 364 364 NUM cana-3900 269 8 - - SYM cana-3900 269 9 371 371 NUM cana-3900 269 10 . . PUNCT cana-3900 270 1 [ [ X cana-3900 270 2 2 2 NUM cana-3900 270 3 ] ] X cana-3900 270 4 n.h n.h PROPN cana-3900 270 5 . . PROPN cana-3900 270 6 mccoy mccoy PROPN cana-3900 270 7 , , PUNCT cana-3900 270 8 the the DET cana-3900 270 9 theory theory NOUN cana-3900 270 10 of of ADP cana-3900 270 11 ring ring PROPN cana-3900 270 12 , , PUNCT cana-3900 270 13 macmillan macmillan PROPN cana-3900 270 14 & & CCONJ cana-3900 270 15 co. co. PROPN cana-3900 270 16 , , PUNCT cana-3900 270 17 1970 1970 NUM cana-3900 270 18 . . PUNCT cana-3900 271 1 [ [ X cana-3900 271 2 3 3 X cana-3900 271 3 ] ] X cana-3900 271 4 g. g. PROPN cana-3900 271 5 pilz pilz PROPN cana-3900 271 6 , , PUNCT cana-3900 271 7 near near ADP cana-3900 271 8 - - PUNCT cana-3900 271 9 ring ring NOUN cana-3900 271 10 , , PUNCT cana-3900 271 11 north north PROPN cana-3900 271 12 holland holland PROPN cana-3900 271 13 / / SYM cana-3900 271 14 american american PROPN cana-3900 271 15 elsevier elsevier PROPN cana-3900 271 16 , , PUNCT cana-3900 271 17 amsterdam amsterdam PROPN cana-3900 271 18 , , PUNCT cana-3900 271 19 1983 1983 NUM cana-3900 271 20 . . PUNCT cana-3900 272 1 [ [ X cana-3900 272 2 4 4 X cana-3900 272 3 ] ] PUNCT cana-3900 272 4 s. s. PROPN cana-3900 272 5 suryanarayanan suryanarayanan PROPN cana-3900 272 6 and and CCONJ cana-3900 272 7 n. n. PROPN cana-3900 272 8 ganesan ganesan PROPN cana-3900 272 9 , , PUNCT cana-3900 272 10 stable stable ADJ cana-3900 272 11 and and CCONJ cana-3900 272 12 pseudo pseudo NOUN cana-3900 272 13 stable stable ADJ cana-3900 272 14 near near ADP cana-3900 272 15 - - PUNCT cana-3900 272 16 ring ring NOUN cana-3900 272 17 , , PUNCT cana-3900 272 18 indian indian ADJ cana-3900 272 19 j. j. PROPN cana-3900 272 20 pure pure PROPN cana-3900 272 21 and and CCONJ cana-3900 272 22 appl appl PROPN cana-3900 272 23 . . PUNCT cana-3900 273 1 math math NOUN cana-3900 273 2 19 19 NUM cana-3900 273 3 ( ( PUNCT cana-3900 273 4 12 12 NUM cana-3900 273 5 ) ) PUNCT cana-3900 273 6 ( ( PUNCT cana-3900 273 7 december december PROPN cana-3900 273 8 , , PUNCT cana-3900 273 9 1988 1988 NUM cana-3900 273 10 ) ) PUNCT cana-3900 273 11 , , PUNCT cana-3900 273 12 1206 1206 NUM cana-3900 273 13 - - SYM cana-3900 273 14 1216 1216 NUM cana-3900 273 15 . . PUNCT cana-3900 274 1 [ [ X cana-3900 274 2 5 5 NUM cana-3900 274 3 ] ] PUNCT cana-3900 274 4 s. s. PROPN cana-3900 274 5 suryanarayanan suryanarayanan PROPN cana-3900 274 6 , , PUNCT cana-3900 274 7 near near ADP cana-3900 274 8 - - PUNCT cana-3900 274 9 ring ring NOUN cana-3900 274 10 with with ADP cana-3900 274 11 p3 p3 NOUN cana-3900 274 12 - - PUNCT cana-3900 274 13 mate mate NOUN cana-3900 274 14 functions function NOUN cana-3900 274 15 , , PUNCT cana-3900 274 16 bull bull NOUN cana-3900 274 17 . . PUNCT cana-3900 275 1 malaysian malaysian ADJ cana-3900 275 2 math math PROPN cana-3900 275 3 . . PUNCT cana-3900 276 1 soc soc PROPN cana-3900 276 2 . . PUNCT cana-3900 277 1 ( ( PUNCT cana-3900 277 2 second second ADJ cana-3900 277 3 series series NOUN cana-3900 277 4 ) ) PUNCT cana-3900 277 5 19 19 NUM cana-3900 277 6 ( ( PUNCT cana-3900 277 7 1996 1996 NUM cana-3900 277 8 ) ) PUNCT cana-3900 277 9 , , PUNCT cana-3900 277 10 17 17 NUM cana-3900 277 11 - - SYM cana-3900 277 12 24 24 NUM cana-3900 277 13 . . PUNCT