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A new class of unique product monoids with applications to ring theory. (English) Zbl 1177.16030
Let $$(S,\leq)$$ be an ordered monoid. This means that $$\leq$$ is a partial order on $$S$$ which is compatible with the operation on $$S$$. The notion of an Artinian narrow unique product monoid is introduced. Namely, $$(S,\leq)$$ is such a monoid if for every two Artinian and narrow subsets $$A,B$$ of $$S$$ there exists an element in the product $$AB$$ which is uniquely presented in the form $$ab$$, where $$a\in A$$ and $$b\in B$$. Such monoids are unique product monoids in the standard sense, that originated from the zero divisor problems for group rings and semigroup rings. On the other hand, if $$\leq$$ is the trivial order on a monoid $$S$$, then $$S$$ is a unique product monoid if $$(S,\leq)$$ is an Artinian narrow unique product monoid.
Examples explaining relations between several possible definitions of unique product monoids are given. Conditions for a skew generalized power series ring $$R[\![S,\omega]\!]$$, where $$S$$ is an Artinian narrow unique product monoid and $$\omega\colon S\to\text{End}(R)$$ is a homomorphism, to be a domain or a reduced ring are also obtained. Here skew generalized power series rings are considered in the sense introduced by the second and the third author [in Commun. Algebra 36, No. 5, 1855-1868 (2008; Zbl 1159.16032)].

##### MSC:
 16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras) 06F05 Ordered semigroups and monoids 16S36 Ordinary and skew polynomial rings and semigroup rings 20M25 Semigroup rings, multiplicative semigroups of rings
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