On Annihilating Graphs Associated with Modules over Commutative Rings

Abstract

Let [Formula: see text] be a commutative ring with unity, [Formula: see text] be a unitary [Formula: see text]-module and [Formula: see text] be a simple connected graph. We examine different equivalence relations on subsets [Formula: see text], [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is the set of full-annihilators, [Formula: see text] is the set of semi-annihilators and [Formula: see text] is the set of star-annihilators in [Formula: see text]. We prove that elements [Formula: see text] are neighborhood similar in the annihilating graph [Formula: see text] if and only if the submodules [Formula: see text] and [Formula: see text] of [Formula: see text] are equal. We study the isomorphism of annihilating graphs arising from [Formula: see text] and the tensor product [Formula: see text], where [Formula: see text], [Formula: see text].