Nil-Armendariz Condition on Skew Generalized Power Series Rings

Abstract

Let R be a ring, S a strictly ordered monoid, and $$\omega :S \to {\text{End}}(R)$$ 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 Malcev–Neumann Laurent series rings. In the current work, we study the (S, ω)-nil Armendariz condition on R, a generalization of the standard nil-Armendariz condition from polynomials and power series to skew generalized power series. We resolve the structure of (S, ω)-nil Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S, ω)-nil Armendariz, unifying and generalizing a number of known nil Armendariz-like conditions in the aforementioned special cases. For example, we show that left uniserial nilpotent semicommutative rings are nil Armendariz. Moreover, we study on the relationship between the zip and weak zip properties of a ring R and these of the ring R[[S, ω]].