Polynomial extensions of Baer and quasi-Baer rings

Abstract

A ring R is called (quasi-) Baer if the right annihilator of every (ideal) nonempty subset of R is generated, as a right ideal, by an idempotent of R. Armendariz has shown that for a reduced ring R (i.e., R has no nonzero nilpotent elements), R is Baer if and only if R[ x] is Baer. In this paper, we show that for many polynomial extensions (including formal power series, Laurent polynomials, and Laurent series), a ring R is quasi-Baer if and only if the polynomial extension over R is quasi-Baer. As a consequence, we obtain a generalization of Armendariz's result for several types of polynomial extensions over reduced rings.