Semi-commutativity of graded rings and graded modules

A ring R is said to be semi-commutative if whenever a, b ∈ R such that ab = 0, then aRb = 0. In this article, we introduce the concepts of g−semi-commutative rings and g−N−semi-commutative rings and we introduce several results concerning these two concepts. Let R be a G-graded ring and g ∈ supp(R, G). Then R is said to be a g−semi-commutative if whenever a, b ∈ R with ab = 0, then aRgb = 0. Also, R is said to be a g − N−semi-commutative if for any a ∈ R and b ∈ N(R) ⋂ Ann(a), bRg ⊆ Ann(a). We introduce an example of a G-graded ring R which is g − N-semi-commutative for some g ∈ supp(R, G) but R itself is not semi-commutative. Clearly, if R is a g−semi-commutative ring, then R is a g − N−semi-commutative ring, however, we introduce an example showing that the converse is not true in general. Several results and examples are investigated. Also, we introduce the concept of g − NE−semi-commutative rings and we introduce several results concerning g−NE−semi-commutative rings. Let R be a G-graded ring and g ∈ supp(R, G). Then R is said to be a g−NE− semi-commutative ring if whenever a ∈ N(R) and b ∈ E(R) such that ab = 0, then aRgb = 0. Clearly, g−semi-commutative rings are g −NE−semi-commutative, however, we introduce an example ...