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.