Matrix Algebras over Strongly Non-Singular Rings

Abstract

We consider some existing results regarding rings for which the classes of torsion-free and non-singular right modules coincide. Here, a right R-module M is non-singular if xI is nonzero for every nonzero x in M and every essential right ideal I of R, and a right R-module M is torsion-free if Tor1(M,R/Rr)=0 for every r in R. In particular, we consider a ring R for which the classes of torsion-free and non-singular right S-modules coincide for every ring S Morita-equivalent to R. We make use of these results, as well as the existence of a Morita-equivalence between a ring R and the n × n matrix ring Matn(R) to characterize rings whose n × n matrix ring is a Baer-ring. A ring is Baer if every right (or left) annihilator is generated by an idempotent. Semi-hereditary, strongly non-singular, and Utumi rings will play an important role, and we explore these concepts and relevant results as well.