A characterization of QF− 3 rings

QF-3 rings and torsion theories

“…Our property behaves nicely when R is, furthermore, supposed to be r-artinian and this case is studied in Theorem 7. For trivial r the conditions of this theorem reduce to known characterizations of left artinian left QF-3 rings (in particular, we get as a consequence the main result of [12]) but, taking r equal to the Lambek torsion theory we obtain new characterizations of left QF-3' rings with the descending chain condition on left annihilators, QF-3 rings with ACC on left annihilators and, in the nonsingular case, of finite-dimensional left QF-3' rings.…”

“…Left artinian left QF-3 rings can be characterized by the property that every finitely generated submodule of E{RR) is torsionless (this was proved by Rutter in [12]), and also as the left noetherian rings such that E(RR) is projective. We are going to give a torsion-theoretic generalization of these facts in such a way that, corresponding to the trivial torsion theory we recover Rutter's result and for the Lambek torsion theory we get characterizations of the much larger class of left QF-3' rings with ACC on left annihilators.…”

“…REMARK. If r is the trivial torsion theory, then condition (i) of Theorem 7 is clearly equivalent to R being a left artinian left QF-3 ring, so that the equivalence of (i) and (ii) gives [12,Corollary 3].…”

“…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).…”