Radicals of Skew Inverse Laurent Series Rings

Abstract

In this note, we continue to study zero-divisor properties of skew inverse Laurent series rings R((x −1; σ, δ)), where R is an associative ring equipped with an automorphism σ and a σ-derivation δ. We first introduce (σ, δ)-SILS Armendariz rings, a generalization of the standard Armendariz condition from ordinary polynomial ring R[x] to skew inverse Laurent series ring R((x −1; σ, δ)). We study the ring-theoretical properties of (σ, δ)-SILS Armendariz rings and using the properties of these rings, we characterize radicals of the skew inverse Laurent series ring R((x −1; σ, δ)), in terms of a (σ, δ)-SILS Armendariz ring R. We also prove that several properties transfer between R and R((x −1; σ, δ)), in case R is an σ-compatible (σ, δ)-SILS Armendariz ring.