U abstraktnij algebri faktor modulem nazivayetsya novij modul yakij mozhna viznachiti dlya dovilnogo modulya nad kilcem i jogo pidmodulya Pobudova faktor modulya ye analogom pobudovi faktorm nozhini faktor grupi faktor kilcya i faktor prostoru Zmist 1 Oznachennya 2 Prikladi 3 Vlastivosti 4 LiteraturaOznachennya RedaguvatiNehaj dano livij modul M displaystyle M nbsp nad kilcem R displaystyle R nbsp i jogo pidmodul N displaystyle N nbsp Na M displaystyle M nbsp mozhna vvesti vidnoshennya ekvivalentnosti m n displaystyle m sim n nbsp yaksho i tilki yaksho m n N displaystyle m n in N nbsp dlya bud yakih m n M displaystyle m n in M nbsp Elementami mnozhini M N displaystyle M N nbsp ye klasi ekvivalentnosti m m n n N displaystyle m m n colon n in N nbsp Suma dvoh klasiv ekvivalentnosti u M N displaystyle M N nbsp ye klasom ekvivalentnosti ekvivalentnosti sumi predstavnikiv dvoh klasiv v shozhij sposib mozhna vvesti mnozhennya na elementi R displaystyle R nbsp Konkretno m n m n displaystyle m n m n nbsp i r m r m displaystyle r m rm nbsp dlya bud yakih m n M displaystyle m n in M nbsp i r R displaystyle r in R nbsp Takim chinom M N displaystyle M N nbsp otrimuye strukturu modulya nad R displaystyle R nbsp Cej modul nazivayetsya faktor modulem modulya M displaystyle M nbsp po pidmodulyu N displaystyle N nbsp Prikladi RedaguvatiM M ye trivialnim modulem 0 M 0 ye izomorfnim M Nehaj R displaystyle mathbb R nbsp kilce dijsnih chisel i A R X displaystyle A mathbb R X nbsp kilce mnogochleniv z dijsnimi koeficiyentami sho ochevidno ye R displaystyle mathbb R nbsp modulem Rozglyanemo pidmodulB X 2 1 R X displaystyle B X 2 1 mathbb R X nbsp modulya A displaystyle A nbsp tobto pidmodul vsih mnogochleniv sho dilyatsya na X 2 1 displaystyle X 2 1 nbsp Vidnoshennya ekvivalentnosti dlya cih moduliv zadayetsya yak P X Q X displaystyle P X sim Q X nbsp yaksho i tilki yaksho zalishki vid dilennya P X displaystyle P X nbsp i Q X displaystyle Q X nbsp na X 2 1 displaystyle X 2 1 nbsp ye odnakovimi Zokrema u faktor moduli A B displaystyle A B nbsp mnogochlen X 2 1 displaystyle X 2 1 nbsp perehodit u toj zhe klas sho i 0 displaystyle 0 nbsp i faktor modul mozhna rozglyadati yak pohidnij vid R X displaystyle mathbb R X nbsp pri ototozhnenni X 2 1 0 displaystyle X 2 1 0 nbsp Faktor modul A B displaystyle A B nbsp ye izomorfnim iz dijsnim vektornim prostorom kompleksnih chisel Vlastivosti RedaguvatiFaktor modul M N displaystyle M N nbsp ye gomomorfnim obrazom modulya M displaystyle M nbsp dlya gomomorfizma yadro yakogo ye rivnim N displaystyle N nbsp i yake mozhna zapisati yakp M M N m m N displaystyle pi M longrightarrow M N m mapsto m N nbsp Vidobrazhennya p displaystyle pi nbsp nazivayetsya proyekciyeyu modulya M displaystyle M nbsp na faktor modul M N displaystyle M N nbsp Teoremi pro izomorfizmi dlya dvoh pidmoduliv M N displaystyle M N nbsp modulya Q displaystyle Q nbsp M M N M N N displaystyle M M cap N simeq M N N nbsp dlya pidmodulya N Q P displaystyle N subseteq Q subseteq P nbsp vikonuyetsya P N Q N P Q displaystyle P N Q N simeq P Q nbsp dd Kozhen gomomorfizm R moduliv f M L displaystyle f M rightarrow L nbsp yadro yakogo mistit N u yedinij sposib rozkladayetsya cherez M N tobto isnuye yedinij gomomorfizm R moduliv f M N L displaystyle tilde f M N to L nbsp dlya yakogo f p f displaystyle tilde f circ pi f nbsp Navpaki nehaj isnuye R modul P i syur yektivnij gomomorfizm p M P displaystyle p M longrightarrow P nbsp Yaksho dlya kozhnogo gomomorfizma R moduliv f M L displaystyle f M rightarrow L nbsp yadro yakogo mistit N isnuye yedinij gomomorfizm f P L displaystyle tilde f P to L nbsp dlya yakogo f p f displaystyle tilde f circ p f nbsp to M N P displaystyle M N simeq P nbsp Takim chinom dana vlastivist povnistyu harakterizuye faktor modul Vona nazivayetsya universalnoyu harakteristikoyu modulya Faktor modulyami skinchennoporodzhenih moduliv i moduliv skinchennoyi dovzhini ye skinchennoporodzheni moduli i moduli skinchennoyi dovzhini Nehaj M P displaystyle M P nbsp dva moduli nad komutativnim kilcem R displaystyle R nbsp i N M T P displaystyle N subset M T subset P nbsp yih pidmoduli Todi dlya tenzornogo dobutku vikonuyetsya vlastivist M N R P T M R P Q displaystyle M N otimes R P T simeq M otimes R P Q nbsp de Q displaystyle Q nbsp pidmodul u M R P displaystyle M otimes R P nbsp porodzhenij elementami vidu m t displaystyle m otimes t nbsp i n p displaystyle n otimes p nbsp dlya dovilnih m M t T n N p P displaystyle m in M t in T n in N p in P nbsp Yaksho S displaystyle S nbsp multiplikativna mnozhina u komutativnomu kilci R displaystyle R nbsp to dlya lokalizaciyi S 1 M N S 1 M S 1 N displaystyle S 1 M N simeq S 1 M S 1 N nbsp Yaksho B displaystyle B nbsp ye A displaystyle A nbsp algebroyu asociativnoyu z odiniceyu to B A M N B A M U displaystyle B otimes A M N simeq B otimes A M U nbsp de U displaystyle U nbsp ye obrazom B A N displaystyle B otimes A N nbsp u B A M displaystyle B otimes A M nbsp Yaksho I displaystyle I nbsp ye dvostoronnim idealom u A displaystyle A nbsp to faktor modul A I displaystyle A I nbsp ye faktor kilcem A I displaystyle A I nbsp Literatura RedaguvatiDauns John 1994 Modules and rings Cambridge University Press ISBN 9780521462587 Dummit David S Foote Richard M 2004 Abstract Algebra vid 3rd John Wiley amp Sons ISBN 0 471 43334 9 Otrimano z https uk wikipedia org w index php title Faktor modul amp oldid 38824300
