Torsion-freeness and non-singularity over right p.p.-rings

A right R-module M is non-singular if xI not equal 0 for all non-zero x is an element of M and all essential right ideals I of R. The module M is torsion-free if Tor(1)(R)(M, R/Rr) = 0 for all r is an element of R. This paper shows that, for a ring R, the classes of torsion-free and non-singular right R-modules coincide if and only if R is a right Utumi-p.p.-ring with no infinite set of orthogonal idempotents. Several examples and applications of this result are presented. Special emphasis is given to the case where the maximal right ring of quotients of R is a perfect left localization of R. (C) 2004 Elsevier Inc. All rights reserved.