On Annihilator Ideals of Pseudo-differential Operator Rings

Abstract

Let R be a ring with a derivation δ and R((x-1; δ)) denote the pseudo-differential operator ring over R. We study the relations between the set of annihilators in R and the set of annihilators in R((x-1; δ)). Among applications, it is shown that for an Armendariz ring R of pseudo-differential operator type, the ring R((x-1; δ)) is Baer (resp., quasi-Baer, PP, right zip) if and only if R is a Baer (resp., quasi-Baer, PP, right zip) ring. For a δ-weakly rigid ring R, R((x-1; δ)) is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of left semicentral idempotents of R has a generalized countable join in R.