Essential ideals represented by mod-annihilators of modules

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 as when [x : M] is an essential ideal of R. We prove that the 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.