Skew Generalized Power Series Rings With the McCoy Property
Peter Danchev, M. Zahiri, S. Zahiri
TL;DR
The paper addresses the problem of extending the McCoy property to skew generalized power series rings $R[[S,\omega,\preceq]]$ by introducing the $(S,\omega)$-McCoy condition. It proves a central result: if $R$ is $S$-compatible, abelian, semi-regular with $J(R)$ nilpotent, then $R$ is $(S,\omega)$-McCoy, meaning that for nonzero $f,g$ with $fg=0$ in $R[[S,\omega,\preceq]]$ there exists a nonzero $c\in R$ with $f c=0$ (equivalently $f(s)\omega_s(c)=0$ for all $s$). This theorem yields a broad family of corollaries showing that several twisted series rings, including skew Malcev-Neumann rings, skew Laurent series, and generalized power series rings, satisfy the McCoy-type property, with additional results on stability under matrix rings $S_n(R)$. The article concludes with an open question about extending the main theorem to the more general 2-primal setting. Overall, the work significantly broadens McCoy-type annihilation phenomena to a wide class of noncommutative twisted series constructions, linking abelian/semi-regular structure and nilpotent Jacobson radical to zero-divisor behavior.
Abstract
Let $R$ be a ring, $(S,\preceq)$ a strictly totally ordered monoid and suppose also $ω:S\rightarrow \text{End}(R)$ is a monoid homomorphism. A skew generalized power series ring $R[[S,ω,\preceq]]$ consists of all functions from a monoid $S$ to a coefficient ring $R$ whose support contains neither infinite descending chains nor infinite anti-chains, equipped with point-wise addition and with multiplication given by convolution twisted by an action $ω$ of the monoid $S$ on the ring $R$. Special cases of the skew generalized power series ring construction are the skew polynomial rings, skew Laurent polynomial rings, skew power series rings, skew Laurent series rings, skew monoid rings, skew group rings, skew Malcev-Neumann series rings and generalized power series rings as well as the untwisted versions of all of these objects. In the present article, we study the so-termed $(S,ω)$-McCoy condition on $R$, that is a generalization of the standard McCoy condition from polynomials to skew generalized power series, thus generalizing some of the existing results in the literature relevant to the subject.