$EM-$Graded Rings

TL;DR

This paper introduces $EM-G$-graded rings, extending $EM$-rings to graded rings with a focus on homogeneous zero divisors and their annihilating contents, providing new examples and theoretical results.

Contribution

It defines $EM-G$-graded rings, extends the concept of $EM$-rings to graded rings, and offers new examples and theoretical insights into their properties.

Findings

Examples of $EM-G$-graded rings that are not $EM$-rings.

Theoretical results relating to homogeneous zero divisors.

Extension of $EM$-ring concepts to graded ring structures.

Abstract

The main goal of this article is to introduce the concept of $EM-G-$graded rings. This concept is an extension of the notion of $EM-$rings. Let $G$ be a group and $R$ be a $G-$graded commutative ring. The $G-$gradation of $R$ can be extended to $R[x]$ by taking the components $(R[x])_{\sigma}=R_{\sigma}[x]$. We define $R$ to be $EM-G-$graded ring if every homogeneous zero divisor polynomial has an annihilating content. We provide examples of $EM-G-$graded rings that are not $EM-$rings and we prove some interesting results regarding these rings.