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Nilpotent elements and nil-Armendariz property of skew generalized power series rings. (English) Zbl 1383.16029

Summary: Let \(R\) be a ring, \((S,\leq)\) a strictly ordered monoid, and \(\omega:S \to \mathrm{End}(R)\) a monoid homomorphism. The skew generalized power series ring \(R[[S,\omega]]\) is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev-Neumann Laurent series rings. In this paper, we introduce and study the \((S,\omega)\)-nil-Armendariz condition on \(R\), a generalization of the standard nil-Armendariz condition from polynomials to skew generalized power series. We resolve the structure of \((S,\omega)\)-nil-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be \((S,\omega)\)-nil-Armendariz. The \((S,\omega)\)-nil-Armendariz condition is connected to the question of whether or not a skew generalized power series ring \(R[[S,\omega]]\) over a nil ring \(R\) is nil, which is related to a question of A. S. Amitsur [Proc. Am. Math. Soc. 7, 35–48 (1956; Zbl 0070.03004)]. As particular cases of our general results we obtain several new theorems on the nil-Armendariz condition. Our results extend and unify many existing results.

MSC:

16S36 Ordinary and skew polynomial rings and semigroup rings
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
06F05 Ordered semigroups and monoids
16U80 Generalizations of commutativity (associative rings and algebras)

Citations:

Zbl 0070.03004
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References:

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