分次环与局部化

Graded Rings and Localizations

设Ｇ是有限群，Ｒ是有单位元的Ｇ－型分次环，Ｓ是包含在Ｒ的所有齐次元素组成的集合内的乘法封闭子集，Ｓ＝ｘ∈Ｇａｅ（ｇｘ，ｘ）ａ∈Ｓ，Ｄｅｇ（ａ）＝ｇ∈Ｇ｛｝，Ｓ＝＝ｘ∈Ｇａｅ（ｇｘ，ｘｈ）ａ∈Ｓ，Ｄｅｇ（ａ）＝ｇ∈Ｇ，ｈ∈Ｇ｛｝，ＭＧ（Ｒ）表示以Ｇ的元作为行列标的｜Ｇ｜阶矩阵环．本文证明了Ｒ关于Ｓ满足左Ｏｒｅ条件当且仅当Ｒ＃Ｇ关于Ｓ满足左Ｏｒｅ条件当且仅当ＭＧ（Ｒ）关于Ｓ＝满足左Ｏｒｅ条件，而且，Ｓ－１（Ｒ＃Ｇ）≌（Ｓ－１Ｒ）＃Ｇ和Ｓ＝，－１（ＭＧ（Ｒ））≌ＭＧ（Ｓ－１Ｒ）．

Suppose G is a finite group, R is a graded ring of type G with identity and S is a multiplicatively closed subset of the set consisting of all homogeneous elements of R. Let =x∈Gae(gx,x)a∈S, Deg (a)=g∈G] and S==x∈ G ae (gx,xh)a ∈S, Deg (a)=g∈ G, h∈ G. M G(R) denotes the matrices over R with the rows columns indexed by elements of G. In this paper, we prove that R satisfies the left Ore condition with S if and only if R# G satisfies the left Ore condition with  if and only if M G(R) satisfies the left Ore condition with S=. Inaddition, we obtained the following results:  -1 (R# G) is isomorphic to (S -1 R)# G and S=, -1 (M G(R)) is isomorphic to M G(S -1 R) as rings.