## 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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### References:

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