Ne plutati z faktorialnim kilcem V abstraktnij algebri faktor kilce kilce klasiv ekvivalentnosti sho buduyetsya z deyakogo kilcya R displaystyle R za dopomogoyu deyakogo jogo idealu I displaystyle I Poznachayetsya R I displaystyle R I Zmist 1 Viznachennya 2 Prikladi 3 Vlastivosti 4 Div takozh 5 Posilannya 6 DzherelaViznachennya RedaguvatiNehaj R displaystyle R nbsp kilce a I displaystyle I nbsp deyakij jogo dvostoronnij ideal Na R displaystyle R nbsp mozhna zadati vidnoshennya ekvivalentnosti displaystyle sim nbsp a b displaystyle a sim b nbsp todi i tilki todi koli b a I displaystyle b a in I nbsp Oskilki za oznachennyam ideal ye pidgrupoyu aditivnoyi grupi kilcya Todi a a 0 I displaystyle a a 0 in I nbsp tobto a a displaystyle a sim a nbsp Yaksho b a I displaystyle b a in I nbsp to takozh a b b a I displaystyle a b b a in I nbsp tobto z a b displaystyle a sim b nbsp viplivaye b a displaystyle b sim a nbsp Yaksho b a I displaystyle b a in I nbsp ta c b I displaystyle c b in I nbsp to takozh c a c b b a I displaystyle c a c b b a in I nbsp tobto z a b displaystyle a sim b nbsp ta b c displaystyle b sim c nbsp viplivaye a c displaystyle a sim c nbsp Otzhe vidnoshennya a b displaystyle a sim b nbsp ye refleksivnim simetrichnim i tranzitivnim otzhe ye vidnoshennyam ekvivalentnosti Nehaj a a I a r r I displaystyle a a I a r r in I nbsp poznachaye klas ekvivalentnosti elementa a displaystyle a nbsp Mnozhina klasiv ekvivalentnosti vvedenogo vidnoshennya poznachayetsya R I displaystyle R I nbsp Na danij mnozhini mozhna vvesti operaciyi dodavannya i mnozhennya a b a I b I a b I a b displaystyle a b a I b I a b I a b nbsp a b a I b I a b I a b displaystyle a cdot b a I cdot b I a cdot b I a cdot b nbsp Dani viznachennya ye nesuperechlivimi tobto ne zalezhat vid viboru predstavnikiv klasu Dijsno nehaj a I a 1 I displaystyle a I a 1 I nbsp ta b I b 1 I displaystyle b I b 1 I nbsp Todi a a 1 i I displaystyle a a 1 i in I nbsp ta b b 1 j I displaystyle b b 1 j in I nbsp Zvidsi a b a 1 b 1 i j displaystyle a b a 1 b 1 i j nbsp ta a b a 1 i b 1 j a 1 b 1 i b 1 a 1 j i j displaystyle ab a 1 i b 1 j a 1 b 1 ib 1 a 1 j ij nbsp Oskilki i j i b 1 a 1 j i j I displaystyle i j ib 1 a 1 j ij in I nbsp oderzhuyetsya a b I a 1 b 1 i j I displaystyle a b I a 1 b 1 i j I nbsp ta a b I a 1 b 1 I displaystyle ab I a 1 b 1 I nbsp sho dovodit nesuperechlivist viznachennya Mnozhina viznachenih klasiv ekvivalentnosti z viznachenimi operaciyami mnozhennya i dodavannya nazivayetsya faktor kilcem kilcya R displaystyle R nbsp za idealom I displaystyle I nbsp Prikladi RedaguvatiNajprostishi prikladi faktor kilec oderzhuyutsya za dopomogoyu idealiv 0 displaystyle 0 nbsp i samogo kilcya R displaystyle R nbsp R 0 displaystyle R 0 nbsp ye izomorfnim do R displaystyle R nbsp a R R displaystyle R R nbsp ye trivialnim kilcem 0 displaystyle 0 nbsp Nehaj Z displaystyle mathbb Z nbsp kilce cilih chisel a 2 Z displaystyle 2 mathbb Z nbsp kilce parnih chisel Todi faktor kilce Z 2 Z displaystyle mathbb Z 2 mathbb Z nbsp maye lishe dva elementi sho vidpovidayut mnozhinam parnih i neparnih chisel Dane faktor kilce ye izomorfnim polyu z dvoma elementami F 2 displaystyle mathbb F 2 nbsp Bilsh zagalno mozhna rozglyanuti faktor kilce Z n Z displaystyle mathbb Z n mathbb Z nbsp sho ye izomorfnim kilcyu lishkiv za modulem n displaystyle n nbsp Nehaj R x displaystyle mathbb R x nbsp kilce mnogochleniv vid zminnoyi X displaystyle X nbsp z dijsnimi koeficiyentami i ideal I X 2 1 displaystyle I X 2 1 nbsp skladayetsya z usih dobutkiv mnogochlena X 2 1 displaystyle X 2 1 nbsp na inshi mnogochleni Faktor kilce R x X 2 1 displaystyle mathbb R x X 2 1 nbsp ye izomorfnim polyu kompleksnih chisel C displaystyle mathbb C nbsp i klas ekvivalentnosti X displaystyle X nbsp vidpovidaye uyavnij odinici i displaystyle i nbsp Uzagalnyuyuchi poperednij priklad faktor kilce mozhna vikoristati dlya pobudovi rozshirennya polya Nehaj K displaystyle K nbsp deyake pole i f displaystyle f nbsp nezvidnij mnogochlen v K X displaystyle K X nbsp Todi L K X f displaystyle L K X f nbsp ye polem sho mistit K displaystyle K nbsp Vlastivosti RedaguvatiYaksho R displaystyle R nbsp komutativne kilce to kilce R I displaystyle R I nbsp tezh ye komutativnim Obernene tverdzhennya nevirne Teorema pro gomomorfizm kilec Yaksho f displaystyle f nbsp epimorfizm syur yektivnij gomomorfizm kilcya K displaystyle mathrm K nbsp na kilce R displaystyle mathrm R nbsp to yadro ker f displaystyle ker f nbsp ye idealom kilcya K displaystyle mathrm K nbsp prichomu kilce R displaystyle mathrm R nbsp izomorfne faktor kilcyu K ker f displaystyle mathrm K ker f nbsp Navpaki yaksho J displaystyle mathrm J nbsp ideal kilcya K displaystyle mathrm K nbsp to vidobrazhennya f K K J displaystyle f mathrm K to mathrm K J nbsp viznachene umovoyu f a a J a K displaystyle f a a mathrm J forall a in mathrm K nbsp ye gomomorfizmom kilcya J displaystyle mathrm J nbsp na K J displaystyle mathrm K J nbsp z yadrom J displaystyle mathrm J nbsp Ideal J displaystyle mathrm J nbsp kilcya K displaystyle mathrm K nbsp ye prostim maksimalnim v tomu i lishe u tomu vipadku koli faktor kilce K J displaystyle mathrm K J nbsp ye oblastyu cilisnosti polem Mizh idealami kilec R displaystyle mathrm R nbsp i R I displaystyle R I nbsp isnuye tisnij zv yazok A same ideali R I displaystyle R I nbsp znahodyatsya u vzayemno odnoznachnij vidpovidnosti iz idealami kilcya R displaystyle mathrm R nbsp sho mistyat ideal I displaystyle I nbsp yak pidmnozhinu Yaksho I J displaystyle I subset J nbsp takij ideal kilcya R displaystyle mathrm R nbsp jomu stavitsya u vidpovidnist ideal J I displaystyle J I nbsp kilcya R I displaystyle R I nbsp Do togo zh faktor kilcya R J displaystyle R J nbsp i R I J I displaystyle R I J I nbsp ye izomorfnimi cherez prirodnij gomomorfizm h R J R I J I displaystyle h R J to R I J I nbsp dlya yakogo h a J a I J I displaystyle h a J a I J I nbsp Div takozh RedaguvatiFaktor mnozhina Faktor grupa Faktor prostirPosilannya RedaguvatiFaktor kilce Arhivovano 14 travnya 2011 u Wayback Machine na sajti PlanetMath Dzherela RedaguvatiVan der Varden B L Algebra Moskva Nauka 1975 623 s ISBN 5 8114 0552 9 ros Vinberg E B Kurs algebri 4 e izd Moskva MCNMO 2011 592 s ISBN 978 5 94057 685 3 ros Leng S Algebra Moskva Mir 1968 564 s ISBN 5458320840 ros Bondarenko Ye V 2012 Teoriya kilec navchalnij posibnik Kiyiv RVC Kiyivskij universitet s 64 ukr Otrimano z https uk wikipedia org w index php title Faktor kilce amp oldid 38824298
