Essential ideals represented by mod-annihilators of modules

TL;DR

This paper explores properties of mod-annihilators in modules over commutative rings, focusing on their essentiality, intersections, and relationships with ring and module structures, including annihilating graphs.

Contribution

It introduces new characterizations of mod-annihilators, their essentiality, and their connections to ring and module properties, including intersections and graph representations.

Findings

Intersection of essential mod-annihilator ideals is essential.

Mod-annihilator is injective iff the ring is non-singular and radical is zero.

Essential socle implies mod-annihilator is an intersection of maximal ideals.

Abstract

Let $R$ be a commutative ring with unity, $M$ be a unitary $R$-module and $G$ a finite abelian group (viewed as a $\mathbb{Z}$-module). The main objective of this paper is to study properties of mod-annihilators of $M$. For $x \in M$, we study the ideals $[x : M] =\{r\in R | rM\subseteq Rx\}$ of $R$ corresponding to mod-annihilator of $M$. We investigate that when $[x : M]$ is an essential ideal of $R$. We prove that arbitrary intersection of essential ideals represented by mod-annihilators is an essential ideal. We observe that $[x : M]$ is injective if and only if $R$ is non-singular and the radical of $R/[x : M]$ is zero. Moreover, if essential socle of $M$ is non-zero, then we show that $[x : M]$ is the intersection of maximal ideals and $[x : M]^2 = [x : M]$. Finally, we discuss the correspondence of essential ideals of $R$ and vertices of the annihilating graphs realized by $M$ over $R$.