On the Roman domination number of the zero-divisor graph of a commutative ring

The rings considered in this paper are commutative with identity that admit at least one nonzero zero-divisor. Let [Formula: see text] be a ring and let [Formula: see text] denote the set of all zero-divisors of [Formula: see text]. Let us denote [Formula: see text] by [Formula: see text]. The zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is an undirected graph such that its vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text]. A Roman dominating function on a finite graph [Formula: see text] is a function [Formula: see text] that satisfies the following property: any [Formula: see text] for which [Formula: see text] must be adjacent in [Formula: see text] to at least one vertex [Formula: see text] for which [Formula: see text]. The weight of a Roman dominating function [Formula: see text] is the value [Formula: see text]. The minimum weight of a Roman dominating function on [Formula: see text] is called the Roman domination number of [Formula: see text], denoted by [Formula: see text]. As [Formula: see text] need not be finite, we restrict ourselves to Roman dominating functions [Formula: see text] on [Formula: see text] such that there are only finitely many [Formula: see text] with [Formula: see text] or [Formula: see text]. This paper aims to discuss some results on the Roman domination number of [Formula: see text] with the assumption that the domination number of [Formula: see text] is finite.