A characterization of QF− 3 rings

Abstract

Let R be a ring with minimum condition on left or right ideals. It is shown that R is a QF-Z ring if and only if each finitely generated submodule of the injective hull of R, regarded as a left ϋί-module, is torsionless. The same approach yields a simplified proof that R is quasi-Frobenius if and only if every finitely generated left i?-module is torsionless. A ring with identity is called a left QF-Z ring if it has a (unique) minimal faithful left module, and a QF-Z ring means a ring which is both left and right QF-Z. This class of rings originated with Thrall [9] as a generalization of quasi-Frobeni us or QF algebras and has been studied extensively in recent years. Quasi-Frobenius rings have many interesting characterizations and in most instances there exists an analogous characterization of QF-Z rings at least in the case of rings with minimum condition and often for a much larger class of rings. It is well known that a ring with minimum condition on left or right ideals is a left QF-Z ring if and only if the injective hull E{BR) of the ring R regarded as a left iϋ-module is protective. Moreover, in this case R is a QF-Z ring (cf. [6] and [8]). For semi-primary or perfect rings; however, the situation is somewhat different. Namely, a perfect ring is a left QF-Z ring if and only if E(RR) is torsionless. A module is called torsionless if it can be embedded in a direct product of copies of the ring regarded as a module over itself. In this case E(RR) need not be protective and R need not be right QF-Z (cf. [3] and [8]). However, a perfect ring is QF-Z if and only if both E(RR) and E(RR) are protective (see [8]). In this note, it is shown that if R is left perfect ring, E(RR) is protective if and only if each finitely generated submodule of E{RR) can be embedded in a free i?-module. For a ring with minimum condition on left or right ideals this latter condition is equivalent to each finitely generated submodule of E{BR) being torsionless. Thus in that case QF-Z rings may be characterized by this weaker condition. The technique of proof also yields a much simplified proof of a characterization of QF rings given by the present author in [7]. Namely, a ring with minimum condition on left or right ideals is QF if and only if each finitely generated left module is torsionless. Indeed, the characterization of QF-Z rings given here may be regarded as the analog of that result.