The McCoy condition on skew monoid rings

Abstract

Let [Formula: see text] be an associative ring with identity, [Formula: see text] a monoid and [Formula: see text] a monoid homomorphism. When [Formula: see text] is a u.p.-monoid and [Formula: see text] is a reversible [Formula: see text]-compatible ring, then we observe that [Formula: see text] satisfies a McCoy-type property, in the context of skew monoid ring [Formula: see text]. We introduce and study the [Formula: see text]-McCoy condition on [Formula: see text], a generalization of the standard McCoy condition from polynomial rings to skew monoid rings. Several examples of reversible [Formula: see text]-compatible rings and also various examples of [Formula: see text]-McCoy rings are provided. As an application of [Formula: see text]-McCoy rings, we investigate the interplay between the ring-theoretical properties of a general skew monoid ring [Formula: see text] and the graph-theoretical properties of its zero-divisor graph [Formula: see text].