Commutative rings whose matrix rings are Baer rings

Abstract

A ring R with unit element is a Baer ring if every left annihilator in R has the form Re, where e is an idempotent element. K. G. Wolfson has proven [3, Corollary 15], that if R is a Priifer ring (a commutative integral domain in which every finitely generated ideal is invertible) then the ring of endomorphisms of a finitely generated free module over R is a Baer ring. In this note we view these endomorphism rings as the matrix rings Rn with which they are isomorphic, and show that under the rather modest assumption that R have descending chain condition on annihilators, the converse of this result holds. It is also shown that in this case all matrix rings over R are Baer rings if any one of them is.