Radicals of skew generalized power series rings

Abstract

Let R be a ring, (S, ≤) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. In this note for a (S, ω)-Armendariz ring R we study some properties of skew generalized power series ring R[[S, ω]]. In particular, among other results, we show that for a S-compatible (S, ω)-Armendariz ring R, α(R[[S, ω]]) = α(R)[[S, ω]] = Niℓ*(R)[[S, ω]], where α is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZCn, zip and 2-primal property, transfer between R and the skew generalized power series ring R[[S, ω]], in case R is S-compatible (S, ω)-Armendariz. © 2013 World Scientific Publishing Company.