Research Article

A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS

GREG MARKS^a1 c1, RYSZARD MAZUREK^a2 and MICHAŁ ZIEMBOWSKI^a3

^a1 Department of Mathematics and Computer Science, St. Louis University, St. Louis, MO 63103, USA (email: marks@slu.edu)

^a2 Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15–351 Bialystok, Poland (email: mazurek@pb.bialystok.pl)

^a3 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)

Abstract

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.

(Received February 05 2009)

2000 Mathematics subject classification

• primary 16S99;
• 16W60; secondary 06F05;
• 16P60;
• 16S36;
• 16U80;
• 20M25

Keywords and phrases

• skew generalized power series ring;
• (S,ω)-Armendariz;
• semicommutative;
• 2-primal;
• reversible;
• reduced;
• uniserial

Correspondence:

^c1 For correspondence; e-mail: marks@slu.edu

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.