LAUSR

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