Essential ideals represented by mod-annihilators of modules

Let R be a commutative ring with unity, M be a unitary R -module and G a finite abelian group (viewed as a Z -module). The main objective of this paper is to study properties of mod-annihilators of M . For x ∈ M , we study the ideals [ x : M ] = { r ∈ R | r M ⊆ R x } 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 .