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

A right R-module M is non-singular if xI≠0 for all non-zero x∈M and all essential right ideals I of R. The module M is torsion-free if Tor1R(M,R/Rr)=0 for all r∈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.