A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS

Abstract

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.