Left QF-3 Research Articles

A characterisation of reflexive modules

We characterise reflexive modules over the rings R such that each finitely generated submodule of E(RR) is torsionless (left QF-3″ rings) by means of a suitable linear compactness condition relative to the Lambek torsion theory.

Endomorphism Algebras of [formula omitted](Δ) over Quasi-hereditary Algebras

Let A be a finite-dimensional algebra over an algebraically closed field. If A is an F(Δ)-finite quasi-hereditary algebra, then the endomorphism algebra of the direct sum of all non-isomorphic indecomposable Δ-good modules over A is quasi-hereditary. Moreover, this endomorphism algebra is left QF-3 if and only if the injective direct summand of the characteristic module T cogenerates T.

Morita dualities associated with the R-dual functors

We study the dualities induced by a ring R such that every finitely generated submodule of E( R R) is torsionless (left QF-3″ rings) and use them to obtain new information about these rings. The left-right symmetry of this condition is characterized in terms of a linear compactness property, and we use this fact to study QF-3″ rings with certain chain conditions. In particular, we show that a QF-3″ maximal quotient ring with ACC or DCC on essential left ideals is left artinian and QF-3.

Reflexive Modules Over QF-3 Rings

AbstractWe give a characterization of reflexive modules over QF-3 rings generalizing the concept of linearly compact modules. Further, we study necessary and sufficient conditions for left QF-3 rings to be right QF-3.

QF-3 rings and torsion theories

AbstractIn this paper, the rings which have a torsion theory τ with associated torsion radical τ such that R/t(R) has a minimal τ-torsionfree cogenerator are studied. When τ is the trivial torsion theory these are precisely the left QF-3 rings. For τ = τL, the Lambek torsion theory, this class of rings is wider but, with an additional hypothesis on τL 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.

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].

Frobenius extensions of QF-3 rings

We investigate the inheritance of QF-3 property for ring extensions, mainly, for Frobenius extensions. Let A be a ring with identity. It is proved that a group ring A [ G ] A[G] of A with a finite group G is left QF-3 iff A is left QF-3 and that in case A is a. G-Galois extension of the fixed subring A G {A^G} relative to a finite group G of ring automorphism of A, A is left QF-3 iff A G {A^G} is left QF-3.

The Structure of QF-3 Rings

Introduction. In '[12] Thrall introdcuced three generalizations of the quasiFrobenius (=QF) algebras of Nakayama [9], [10]. In this paper we shall be concerned with ring theoretic generalizations of two of these Thrall algebras-namely QF-2 algebras and QF-3 algebras. If R is a ring then a one-sided ideal of R is primitive in case it is generated by a primitive idempotent, and an R-module is minimalfaithful in case it is faithful and has no proper faithful direct summand. Extending Thrall's original definitions to (two-sided) artinian rings we have: QF-2 rings: An artinian ring is QF-2 in case each of its primitive one-sided ideals has a simple socle. QF-3 rings: An artinian ring is QF-3 in case it has (to within isomorphism) a unique minimal faithful module. It is not difficult to show that QF rings are both QF-2 and QF-3 (see [2, ??58-59]). Moreover, Thrall [12] has shown that QF-2 algebras are QF-3 but not necessarily QF. Most of the information about QF-2 and QF-3 rings is limited to finite dimensional algebras (see [8], [12], [13]). Two notable exceptions generalize to QF-3 rings results known to hold for QF-3 algebras almost from their inception. Specifically, Jans [7] has characterized QF-3 rings as those artinian rings whose injective hulls are projective and Harada [5] has shown that the QF-3 property is actually two-sided (i.e., a QF-3 ring has a unique minimal faithful right module). In ?2 of this paper we obtain ideal theoretic characterizations of the injective projective modules (and hence of the unique minimal faithful module) over left QF-3 rings. Our main results appear in ?3. With the aid of Morita's duality theorems [8] we obtain characterizations of QF-3 rings that are analogous to Nakayama's original definition of QF rings in terms of socles of primitive one-sided ideals [10], his characterization of QF-rings in terms of the double annihilator property for one-sided ideals [10], and the fact (see [8, ?14]) that QF rings are precisely those artinian rings for which the functor HomR ( , R) provides a duality between the categories of finitely generated and finitely generated right R-modules.

Semi-Primary QF-3 Rings

A ring R (with identity) is semi-primary if it contains a nilpotent ideal N with R/N semi-simple with minimum condition. R is called a left QF-3 ring if it contains a faithful projective injective left ideal. If R is semi-primary and left QF-3, then there is a faithful projective injective left ideal of R which is a direct summand of every faithful left R-module [5], in agreement with the definition of QF-3 algebra given by R.M. Thrall [6]. Let Q(M) denote the injective envelope of a (left) R-module M. We call R left QF-3+ if Q(R) is projective. J.P. Jans showed that among rings with minimum condition on left ideals, the classes of QF-3 and QF-3+ rings coincide [5].

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.

A Characterization of QF-3 Rings

A left QF-3 ring R is one in which RR, the ring considered as a left module over itself, can be embedded in a projective infective left R module Q(RR). QF-3 rings were introduced by Thrall [14] and have been studied and characterized by a number of authors [5, 8, 9, 12, 13, 15] usually restricted to the case of algebras over a field. In such a case, the concept of left QF-3 and right QF-3 coincide.The study of QF-3 rings and algebras and many other such classes of rings had its origin in the now classic papers of Nakayama [10, 11]. He was an outstanding pioneer in algebra for many years, and we acknowledge our great debt to him and to his many excellent papers.