Piv txwv: Lub nplhaib Z ntawm Gaussian integers yog ib qho Z-module uas tsim los kawg, thiab Z yog Noetherian. Los ntawm yav dhau los Theorem, Z yog lub nplhaib Noetherian. Theorem: Rings ntawm feem ntawm Noetherian rings yog Noetherian.
Z X puas yog Noetherian nplhaib?
Lub nplhaib Z[X, 1 /X] yog Noetherian txij li nws yog isomorphic rau Z[X, Y]/(XY − 1).
Vim li cas Z Noetherian?
Tab sis tsuas muaj ntau lub tswv yim nyob hauv Z uas muaj I1 txij thaum lawv sib raug rau cov hom phiaj ntawm lub nplhaib finite Z/(a) los ntawm Lemma 1.21. Yog li cov saw tsis tuaj yeem ntev ntev, thiab yog li Z yog Noetherian.
Dab tsi yog Noetherian domain?
Txhua lub nplhaib zoo tagnrho, xws li cov lej, yog Noetherian vim txhua qhov zoo tagnrho yog tsim los ntawm ib lub ntsiab Qhov no suav nrog cov thawj coj zoo tshaj plaws thiab Euclidean domains. A Dedekind domain (piv txwv li, rings of integers) yog Noetherian domain uas txhua qhov zoo tshaj plaws yog tsim los ntawm ob lub ntsiab lus.
Koj ua pov thawj li cas lub nplhaib yog Noetherian?
Theorem Ib lub nplhaib R yog Noetherian yog tias thiab tsuas yog yog tias txhua qhov tsis yog qhov khoob ntawm cov hom phiaj ntawm R muaj cov ntsiab lus siab tshaj plaws Pov thawj ⇐=Cia I1 ⊆ I2 ⊆···· ib tug ascending chain of ideals of R. Put S={I1, I2, …}. Yog tias txhua qhov tsis muaj qhov khoob ntawm cov tswv yim muaj cov ntsiab lus siab tshaj plaws ces S muaj cov ntsiab lus siab tshaj plaws, hais IN.