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Duo, Bézout, and distributive rings of skew power series. (English) Zbl 1176.16034

Summary: We give necessary and sufficient conditions on a ring \(R\) and an endomorphism \(\sigma\) of \(R\) for the skew power series ring \(R[\![x;\sigma]\!]\) to be right duo right Bézout. In particular, we prove that \(R[\![x;\sigma]\!]\) is right duo right Bézout if and only if \(R[\![x;\sigma]\!]\) is reduced right distributive if and only if \(R[\![x;\sigma]\!]\) is right duo of weak dimension less than or equal to \(1\) if and only if \(R\) is \(\aleph_0\)-injective strongly regular and \(\sigma\) is bijective and idempotent-stabilizing, extending to skew power series rings the Brewer-Rutter-Watkins characterization of commutative Bézout power series rings.

MSC:

16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
16D25 Ideals in associative algebras
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D50 Injective modules, self-injective associative rings
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References:

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