Some Properties Of The Flat Module And Multiplication Module And Their Generalization

In this paper, we study some properties of the ?at module andthe multiplication module, characterize some properties of the ?at module,generalize the properties of the ?at module and the multiplication module.Firstly, we study the property (P) of the ring R, and conclude the condition ofthe ring R with the property (P), that is let R be a ring, N is a finite generated?at R-module, if for every finite generated submodule M of N, N/M is ?atR-module, then R have the property (P); prove the relation of ?at module andthe property (P) of the ring R, that is let R be a ring, N is a finite generated?at R-module, if for every maximum ideal m of R, Rm is self-associated, then forevery finite generated submodule M of N, N/M is a ?at R-module; characterizethe relation of the property (P) of the ring R and the ?at R-module, andconclude the equivalent relation by some methods of dimension, that is let Rbe a ring, R have the property (P) if and only if for every finite generated ?atR-module N and finite generated ?at submodule M of N, N/M is ?at moduleif and only if fPD(R) = 0 if and only if fFD(R) = 0. So the property (P) can becharacterized by the ?at module. Secondly, we generalize the properties of the?at module, and characterize some properties of the G-?at module, for examplethe local-global property and so on. So the G-?at module become more andmore concretely. At last, we study the properties of the multiplication moduleand v-operation. We definite the G-Dedekind module, and characterize the G-Dedekind module, and prove the equivalent condition of the G-Dedekind moduleand the G-Dedekind domain, that is let R be a domain, M is a multiplicationR-module, then M is G-Dedekind module if and only if R is G-Dedekind domain.