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Skew generalized power series rings and the McCoy property. (English) Zbl 1406.16026

Summary: Given a ring \(R\), a strictly totally ordered monoid \((S,\preceq)\) and a monoid homomorphism \(\omega \colon S \rightarrow \operatorname{End}(R)\), 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.

MSC:

16S35 Twisted and skew group rings, crossed products
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
06F05 Ordered semigroups and monoids
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