On Graded $I_e$-prime Submodules of Graded Modules Over Graded Commutative Rings

Shatha Alghueiri, Khaldoun Al-Zoubi

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity and $M$ a graded $R$-module. We introduce the concept of graded $I_e$-prime submodule as a generalization of a graded prime submodule for $I=\bigoplus_{g\in G}I_g$ a fixed graded ideal of $R$. We give a number of results concerning this class of graded submodules and their homogeneous components. A proper graded submodule $N$ of $M$ is said to be a graded $I_e$-prime submodule of $M$ if whenever $r_g\in h(R)$ and $m_h\in h(M)$ with $r_gm_h\in N-I_eN$, then either $r_g\in(N:_RM)$ or $m_h\in N$.