On graded $$I_{e}$$-primary submodules of graded modules over graded commutative rings

Abstract

Let G be a group with identity e. Let R be a G-graded commutative ring with identity and M a graded R-module. In this paper, we introduce the concept of graded $$J_{e}$$ -primary submodule as a generalization of a graded primary submodule for $$\ J=\oplus _{g\in G}J_{g}$$ a fixed graded ideal of R. We give a number of results concerning of these classes of graded submodules and their homogeneous components. A proper graded submodule C of M is said to be a graded $$J_{e}$$ -primary submodule of M if whenever $$r_{h}\in h(R)$$ and $$m_{\lambda }\in h(M)$$ with $$r_{h}m_{\lambda }\in C\backslash J_{e}C$$ , implies either $$m_{\lambda }\in C$$ or $$r_{h}\in Gr((C:_{R}M)).$$