QF Rings Research Articles

Duality in QF-3 rings

Duality in QF-3 rings

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].

The structure of 𝑄𝐹-3 rings

The structure of 𝑄𝐹-3 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.