Skew Generalized Power Series Rings and the McCoy Property

Abstract

Given a ring $R$, a strictly totally ordered monoid $(S,\preceq)$ and a monoid homomorphism $\omega \colon S \to \operatorname{End}(R)$, one can construct the skew generalized power series ring $R[[S,\omega,\preceq]]$, consisting all of the functions from a monoid $S$ to a coefficient ring $R$ whose support is artinian and narrow, where the addition is pointwise, and the multiplication is given by convolution twisted by an action $\omega$ of the monoid $S$ on the ring $R$. In this paper, we consider the problem of determining some annihilator and zero-divisor properties of the skew generalized power series ring $R[[S,\omega,\preceq]]$ over an associative non-commutative ring $R$. Providing many examples, we investigate relations between McCoy property of skew generalized power series ring, namely $(S,\omega)$-McCoy property, and other standard ring-theoretic properties. We show that if $R$ is a local ring such that its Jacobson radical $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy. Also if $R$ is a semicommutative semiregular ring such that $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy ring.