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Some characterizations of 2-primal skew generalized power series rings. (English) Zbl 1446.16030

The skew generalized power series ring \(R[\![S,\omega]\!]\) is the ring consisting of all functions from a strictly ordered monoid S to a ring R whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by a convolution twisted by the action \(\omega\) of the monoid \(S\) on the ring \(R\).
The prime radical of a ring \(T\) and the set of all nilpotent elements in \(T\) are denoted by \(P(T)\) and \(\operatorname{nil}(T)\), respectively. Recall that \(P(T)\) is the set of all strongly nilpotent elements of \(T\). The ring \(T\) is called 2-primal if \(P(T)=\operatorname{nil}(T)\).
The goal of this paper is to provide necessary and sufficient conditions for \(R[\![S,\omega]\!]\) to be 2-primal (Theorem 3.6 and 3.16). In particular, the authors show that when \(S\) is a strictly ordered artinian narrow unique product monoid, \(\omega : S \rightarrow \operatorname{End}(R)\) is a monoid homomorphism, \(R\) is \(S\)-compatible and \(P(R)\) is a nilpotent ideal, the ring \(R[\![S,\omega]\!]\) is 2-primal if and only if \(R\) is.

MSC:

16S99 Associative rings and algebras arising under various constructions
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16S36 Ordinary and skew polynomial rings and semigroup rings
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
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