A note on generalizations of quasi-Frobenius rings

A ring [Formula: see text] is called quasi-Frobenius, briefly QF, if [Formula: see text] is right (or left) Artinian and right (or left) self-injective. A ring [Formula: see text] is called right co-Harada if every noncosmall right [Formula: see text]-module contains a nonzero projective direct summand and [Formula: see text] satisfies the ACC on right annihilators. The class of co-Harada rings is one of the most interesting generalizations of QF rings. When considering relation between these ring classes, [K. Oshiro, Lifting modules, extending modules and their applications to QF-ring, Hokkaido Math. J. 13 (1984) 310–338, Theorem 4.3] showed that a ring [Formula: see text] is QF if and only if it is right co-Harada ring with [Formula: see text]. In this note, we show that a ring [Formula: see text] is QF if and only if it is right co-Harada ring and satisfies either [Formula: see text] or [Formula: see text].

On General Injective Rings with Chain Conditions

Kupisch proved that if R is a left and right artinian QF-2 ring and Sr = Sl, then R is QF. A weaker condition for a ring to be a QF ring was obtained by Dan and Thuyet. They proved that if R is a right artinian QF-2 ring and Sr ≤ Sl, then R is QF. In this paper, we prove that if R is a QF-2 ring satisfying ACC on right annihilators in which Sl ≤ eRR (e.g., Sr ≤ Sl with Sr ≤e RR), then R is QF. It is also proved that R is QF if and only if R is a left ef-extending, right continuous ring with ACC on right annihilators.

Weakly Projective and Weakly Injective Modules

A module M is said to be weakly N-projective if it has a projective cover π: P(M) ↠M and for each homomorphism : P(M) → N there exists an epimorphism σ:P(M) ↠M such that (kerσ) = 0, equivalently there exists a homomorphism :M ↠N such that σ= . A module M is said to be weakly projective if it is weakly N-projective for all finitely generated modules N. Weakly N-injective and weakly injective modules are defined dually. In this paper we study rings over which every weakly injective right R-module is weakly projective. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a ring R the following conditions are equivalent:(1) R is a left perfect and every weakly projective right R-module is weakly injective.(2) R is a direct sum of matrix rings over local QF-rings.(3) R is a QF-ring such that for any indecomposable projective right module eR and for any right ideal I, soc(eR/eI) = (eR/eJ)n for some positive integer n.(4) R is right artinian ring and every weakly injective right R-module is weakly projective.(5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly projective.

QF-3 endomorphism rings of Σ-quasi-projective modules

A ring R is called left QF-3 if it has a minimal faithful left R-module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a Σ-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and σ[M] denotes the category of all modules isomorphic to submodules of modules generated by M, then we show that End(RM) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of σ[M] canonically defined by M) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R-module over a left QF-3 ring, extending some of the results of [7].

Quotient Rings of a Class of Lattice-Ordered-Rings

An f-ring R with zero right annihilator is called a qf-ring if its Utumi maximal left quotient ring Q = Q(R) can be made into and f-ring extension of R. F. W. Anderson [2, Theorem 3.1] has characterized unital qf-rings with the following conditions: For each q ∈ Q and for each pair d1, d2 ∈ R+ such that diq ∈ R(i) (d1q)+ Λ (d2q)- = 0, and(ii) d1 Λ d2 = 0 implies (d1q)+ Λ d2 = 0.We remark that this characterization holds even when R does not have an identity element.