QF-3 rings and torsion theories

Abstract

In this paper, the rings which have a torsion theory T with associated torsion radical t such that R/t(R) has a minimal r-torsionfree cogenerator are studied. When T is the trivial torsion theory these are precisely the left QF-3 rings. For T = TL, the Lambek torsion theory, this class of rings is wider but, with an additional hypothesis on TL it is shown that if R has this property with respect to the Lambek torsion theory on both sides, then R is a (left and right) QF-3 ring. The results obtained are applied to get new characterizations of QF-3 rings with the ascending chain condition on left annihilators. A ring R is called left QF-3 if it has a minimal faithful left i?-module and left QF-3' if the injective envelope E(RR) is a torsionless module. These rings have been the object of extensive study (for example in [3, 9, 11, 12, 13, 16]) and, recently, Baccella has obtained in [2] structure results for the class of nonsingular, finite-dimensional QF-3 rings. In the present paper we consider a (hereditary) torsion theory r in i?-mod with associated torsion radical t and the property that R/t{R) has a minimal r-torsionfree cogenerator X, in the sense that X is a r-torsionfree module which cogenerates R/t(R) and is a direct summand of every r-torsionfree cogenerator of R/t(R). When r is the trivial torsion theory in which all i2-modules are r-torsionfree, this property defines left QF-3 rings. The class of rings which have this property with respect to the Lambek torsion theory TL is wider than the class of left QF-3 rings but when TL is strongly semiprime and R has this property for the Lambek torsion theory on both sides, This work was partially supported by the CAICYT (0784-84).