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A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS

Part of: Semigroups Conditions on elements Rings and algebras with additional structure Chain conditions, growth conditions, and other forms of finiteness Rings and algebras arising under various constructions

Published online by Cambridge University Press:  23 February 2010

GREG MARKS ,
RYSZARD MAZUREK  and
MICHAŁ ZIEMBOWSKI
GREG MARKS*
Affiliation:
Department of Mathematics and Computer Science, St. Louis University, St. Louis, MO 63103, USA (email: marks@slu.edu)
RYSZARD MAZUREK
Affiliation:
Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15–351 Bialystok, Poland (email: mazurek@pb.bialystok.pl)
MICHAŁ ZIEMBOWSKI
Affiliation:
Maxwell Institute of Sciences, School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK (email: m.ziembowski@wp.pl)
*
✉For correspondence; e-mail: marks@slu.edu
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Abstract

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Let R be a ring, S a strictly ordered monoid, and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] 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. We study the (S,ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S,ω)-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S,ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

Keywords

skew generalized power series ring(S,ω)-Armendarizsemicommutative2-primalreversiblereduceduniserial

MSC classification

Secondary: 16S99: None of the above, but in this section 16W60: Valuations, completions, formal power series and related constructions 16P60: Chain conditions on annihilators and summands 16S36: Ordinary and skew polynomial rings and semigroup rings 16U80: Generalizations of commutativity 20M25: Semigroup rings, multiplicative semigroups of rings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 3 , June 2010 , pp. 361 - 397
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The second author was supported by Bialystok University of Technology grant W/WI/7/08, MNiSW grant N N201 268435, and KBN grant 1 P03A 032 27.

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