Skew generalized power series rings with the McCoy property

Abstract Let R be a ring, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mo>⪯</m:mo> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(S,\preceq)} a strictly totally ordered monoid and suppose also <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ω</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>S</m:mi> <m:mo>→</m:mo> <m:mrow> <m:mi>End</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {\omega:S\rightarrow\mathrm{End}(R)} is a monoid homomorphism. A skew generalized power series ring <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>R</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>ω</m:mi> <m:mo>,</m:mo> <m:mo>⪯</m:mo> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {R[[S,\omega,\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-called <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(S,\omega)} -McCoy condition on R that is a generalization of the standard McCoy condition from polynomials to skew generalized power series, thereby generalizing some of the existing results in the literature relevant to the subject.