a1 Department of Mathematics and Computer Science, St. Louis University, St. Louis, MO 63103, USA (email: firstname.lastname@example.org)
a2 Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15–351 Bialystok, Poland (email: email@example.com)
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: firstname.lastname@example.org)
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
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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.