Faithful Left Module Research Articles

A characterization of QF− 3 rings

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.

Cotorsion radicals and projective modules

We study the notion (for categories of modules) dual to that of torsion radical and its connections with projective modules.Torsion radicals in categories of modules have been studied extensively in connection with quotient categories and rings of quotients. (See [8], [12] and [13].) In this paper we consider the dual notion, which we have called a cotorsion radical. We show that the cotorsion radicals of the category RM correspond to the idempotent ideals of R. Thus they also correspond to TTF classes in the sense of Jans [9].It is well-known that the trace ideal of a projective module is idempotent. We show that this is in fact a consequence of the natural way in which every projective module determines a cotorsion radical. As an application of these techniques we study a question raised by Endo [7], to characterize rings with the property that every finitely generated, projective and faithful left module is completely faithful. We prove that for a left perfect ring this is equivalent to being an S-ring in the sense of Kasch. This extends the similar result of Morita [15] for artinian rings.

A generalization of quasi-Frobenius rings

As a generalization of quasi-Frobenius rings, G. Azumaya [2] has investigated a ring with the property that every faithful left module is a generator' in the category of left mnodules, and he has proved that such a ring is left self-injective and a direct sum of indecomposable left ideals having minimal left ideals. In his proof, however, the existence of the faithful injective module plays an important role, while the injective module is not necessarily finitely generated,2 even if the ring is a left Artinian ring. Therefore, there naturally arises a problem: Is an Artinian ring quasi-Frobenius if every finitely generated faithful module is a generator? The purpose of this paper is to give an affirmative answer to this problem as a direct consequence from a more general result which is rather similar to that of Azumaya stated above and related to perfect rings introduced by H. Bass [3]. Throughout this paper we shall assume that the ring R has identity element 1 and all modules over it are unital. For a subset A of R we shall denote

SEMI-PERFECT QF-3 AND PP-RINGS

Mochizuki [7] studied the double centralizer of a minimal faithful leftmodule for a hereditary QF-3 algebra of finite rank over a field. In [4] Haradaapplied his theory of generalized triangular matrix rings to extend Mochizuki'sresults to left QF-3 and semi-primary partially PP-rings. The purpose of thisnote is to give a direct proof of Harada's results which extends them to semi-perfect rings.