.g/h- Graded R-module Category R ~ (h) - Module Category Equivalent

The theory of rings is a great math branch, it has a long history. It seems that it couldn't be completely discovered. And the theory of the graded rings has been widely studied. Let G be an arbitrary group, a ring R is G-graded if R = g⊕∈R_g where each R gis an additive subgroup of R and R_g R_h(?)R_gh,for all g , h∈G. C. stasescu, S.Raianu and others had studied the theory of the G-graded rings in the text [2,4]. Cohen,M and Montgomery,S gave the notion of smash product R# G. Professor Shaoxue Liu gave the notion of smash product R#G/H for G-set G/H and graded ring R, and proved that the G/H-graded R-module category and the R#G/H-module category are isomorphic with the representation of matrix rings in the text [3].At the base of these, let R be a strong G-graded ring, by the property of strong G-graded rings, we discuss the equivalence between G / H-graded module category (G /H,R)-grand module category R^(H)-Mod,and the equivalence between module category R # G/H-Mod and module category R^(H)-Mod.We follow the step below. In second part, we give the definition of G-graded rings, smash product R#G/H, etc. In third part, we obtain a necessary and sufficient condition for strong G-graded rings (lemma3.1), and prove that G / H-module category (G /H,R)-grand module category R^(H)-Modare equivalent.Theorem3.2 Let R be a strong ring, then functors R(?)-and (-)_eH show that G / H-module category (G /H,R)-grand module category R^(H)-Mod are equivalent.As a special case, the following corollary is drawn, it's one of the important conclusions in the text[4]. Corollary3.3 Let R be a strong G-graded ring, then functors R(?)-and (-)_e show that G-graded module category (G ,R)?grand module category R_e-Mod are equivalent.In forth part, we discuss the equivalence between module category R # G/H-Mod and R^(H)-Mod, we prove that R # G/H-Mod and R^(H)-Modare equivalent on strong graded rings.Theorem4.1 Let R be a strong graded ring, then functors R(?)-:R # G/H-Mod→R^(H)-Modand R(?)-: R^(H)-Mod→R # G/H-Mod are a couple of mutually inverse equivalent functors, in other words, module category R # G/H-Mod and R^(H)-Modare equivalent.Also as a special case, we can conclude the following corollary.Corollary4.2:Let R be a strong G-graded ring,then functors R(?)-and R(?)- show that R_e-Mod and R # G-Mod are equivalent.We take G as a finite Group in Corollary 4.2, it is one of the important conclusions in the text [1].At last, in section 5, by the theorems above, we give some important conclusions about G / H-graded Noether-module and Artin-module (theorem5.1 to corollary 5.7).