Essential ideals represented by mod-annihilators of modules

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