On graded 2-absorbing $I_{e}$-prime submodules of graded modules over graded commutative rings

Abstract

Let $G$ be an abelian 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 2-absorbing $I_{e}$-prime submodule as a generalization of a graded 2-absorbing prime submodule for $\ I=\oplus _{g\in G}I_{g}$ a fixed graded ideal of $R$. We give a number of results concerning these classes of graded submodules and their homogeneous components. A proper graded submodule $N$ of $M$ is said to be a graded 2-absorbing $I_{e}$-prime submodule of $M$ if whenever $% r_{h},s_{\lambda }\in h(R)$ and $m_{\alpha }\in h(M)$ with $r_{h}s_{\lambda }m_{\alpha }\in N\backslash I_{e}N$, implies either $r_{h}s_{\lambda }\in (N:_{R}M)$ or $r_{h}m_{\alpha }\in N$ or $s_{\lambda }m_{\alpha }\in N.$

Keywords

• graded 2-absorbing $i_{e}$-prime submodules
• graded $i_{e}$-prime submodules
• graded 2-absorbing submodules
• graded prime submodules
• graded 2-absorbing $i_{e}$-prime ideals