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.


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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