Nilpotent Property of Skew Generalized Power Series Rings

Abstract

Let R be an NI ring with nil(R) nilpotent, (S, ≤) a strictly totally ordered monoid satisfying the condition that 0 ≤ s for all s ∈ S ,a ndω : S → End(R) be a compatible monoid homomorphism. Then the ring ((R S,≤ ,ω )) of the skew generalized power series with coefficients in R and exponents in S is a nilpotent p.p.-ring if and only if R is a nilpotent p.p.-ring. Furthermore, if R satisfies the descending chain condition on weak annihilators, then ((R S,≤ ,ω )) is a weak APP-ring if and only if R is a weak APP-ring. Consequently, some properties of the base ring R can be profitably generalized to the skew generalized power series ring ((R S,≤ ,ω )).