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

TL;DR

This paper introduces and studies graded $I_{e}$-prime submodules in graded modules over graded commutative rings, generalizing the concept of graded prime submodules and exploring their properties.

Contribution

The paper defines graded $I_{e}$-prime submodules and investigates their properties and relationships with homogeneous components in graded modules.

Findings

Characterization of graded $I_{e}$-prime submodules

Conditions for submodules to be graded $I_{e}$-prime

Relations between graded $I_{e}$-prime submodules and homogeneous components

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 $I_{e}$-prime submodule as a generalization of a graded prime submodule for $I=\oplus_{g\in G}I_{g}$ 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 $N$ 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 $r_{g}m_{h}\in N-I_{e}N,$ then either $r_{g}\in (N:_{R}M)$ or $m_{h}\in N.$