分次Morita对偶，Morita对偶与Smash积

设Ｃ和ｒ都是群，是Ｇ－型分次环，是Γ－型分次环．是双分次模，Ｒ＃Ｇ是Ｒ的Ｓｍａｓｈ积，Ａ＃Γ是Ａ的Ｓｍａｓｈ积。令Ｗ＝（＿ｇＵ＿（σ－１））＿（ｇ，σ）即（ｇ，σ）位置取＿ｇＵ＿（σ－１）的元素的｜Ｇ｜×｜Γ｜矩阵的全体组成的集合，且每个矩阵的每行和每列的非零元只有有限个，按矩阵运算，Ｗ构成（Ｒ＃６，Ａ＃Γ）双模。则＿ＲＵ＿Ａ定义了一个分次Ｍｏｒｉｔａ对偶当且仅当＿（Ｒ＃Ｇ）Ｗ＿（Ａ＃Γ）定义了一个Ｍｏｒｉｔａ对偶。

Graded Morita Dualities,Morita Kualities and Smash Products

Let G and Γ be groups,a graded ring of type G with an identity, a graded ring of type Γ with an identity,R#G and A#Γsmash products of R and A, respectively. Let W=(_gU_(σ-1))_(g,σ)i.e. a set of all|G|》×|Γ| matrices whose element in the(g,σ)-position belongs to gU_(σ-1). Suppose each matrix in W only has finitely non zero elements.Then W is an(R#G,A#Γ)-bimodule with the matrix addition and multiplication and _RU_Adefines a graded Morita duality iff _(R#G)W_(A#Γ) defines a Morita duality.