Relative QF-3' modules and QF-3 modules

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.

The idea of duality occurs in many parts of mathematics. Its earliest appearance (some 150 years ago) was in projective geometry where it permitted to halve the number of theorems to be proved. It also played a most important role in analysis and topology, for example in Banach spaces and algebraic topology. In algebra, beginning with the duality of finite abelian groups as well as that of finite dimensional vector spaces, its role has been no less important. As one of the most natural generalizations of the latter for modules, Morita duality has become an important part of textbooks or a topic of lecture notes on rings and modules (cf. Anderson and Fuller [2], Cohn [23], Faith [34], Kasch [47] and Xue [84]). Since the results concerning Morita duality are very numerous and there is the excellent survey paper of Muller [70], it cannot be the purpose of this paper to make a complete report on all aspects and results relating to Morita duality. In the present survey we intend to highlight the basic ideas and observations of Morita duality (in particular, we will stress the importance of linear compactness and Grothendieck’s condition AB5*) and to show how much of this theory can be deduced from more general results about Baer duality, as well as from results of a purely category-theoretical nature. Furthermore, we will touch upon aspects of duality in some important special classes of rings, however, without aiming at completeness.

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.

We characterise artinian left duo rings via Morita duality, and show that there is a large class of such rings that possess a duality.

We study classical right quotient rings and ordinary extensions of various kinds of 2-primal rings, constructing examples for situations that raise naturally in the process. We show: (1) Let R be a right Ore ring with P(R) left T-nilpotent. Then Q is a 2-primal local ring with P(Q)=J(Q) = {ab-1 ∈ Q | a ∈ P(R), b ∈ C(0)} if and only if C(0)=C(P(R))=R∖P(R), where Q is the classical right quotient ring of R. (2) Let R be a right Ore ring. Then R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R). As a consequence, if R is a right Noetherian ring, then R[[x]] is a domain whose classical right quotient ring is a division ring if and only if R[x] is a domain whose classical right quotient ring is a division ring if and only if R is a right p.p. ring with C(P(R))=R∖P(R).

On duality theory and AB5 ∗ modules

Finiteness conditions of reflexive objects of a Morita duality of Grothendieck category is studied. It is observed that the relation between coproduct and product is a fundamental fact. We show that for coalgebras with a duality, the class of reflexive objects coincides with the class of quasi-finite comodules.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

For a non-commutative ring there exists not always the classical quotient ring, while there exists always the maximal quotient ring. In the case of a semiprime ring, howevel; to have a semi-simple Artinian maximal quotient ring implies the existence of the classical quotient ring (cf. Johnson [6, 4.4 Theorem] and Sandomierski [17, Theorem 1.6]). Therefore avoiding the use of the Ore condition it may be possible to find some conditions, under which the existence of the classical quotient ring is assured by requiring the chain condition on a ring which is constructed by a suitable localization. The purpose of the present paper is to provide such conditions and localizations. For a ring with an identity every idempotent topologizing filter ~is defined by a finitely cogenerating injective right R-module W in the following way: ~ = {right ideal D of R I HomR (R/D, W) = 0}, and we have the localizing functor H such that H(X)=~ Horn R (D, X/X~), where X is a right R-module DE~ and X ~ = {xeXIAnnR x e ~ } . H(R) has a ring-structure and H(X) becomes a right H(R)-module in a natural way. Recently, Morita [14] has shown that H(R) and the double centralizer Q of W R are identical and H is exact. With these notations we can state our main theorem in w 3: The following statements I to III are equivalent.

Centers and prime ideals in group algebras of polycyclic-by-finite groups

While the classical right ring of quotients Q cl r (R) exists (if and) only if R is a right Ore ring, a maximal right ring of quotients Qmaxr(R) exists for any ring R. To define Q max r (R), one must first understand the structure of the endomorphism ring of a QI (quasi-injective) module.

Given a polynomial ring $S = \Bbbk[x_1, \dots, x_n]$ over a field $\Bbbk$, and a monomial ideal $M$ of $S$, we say the quotient ring $R = S/M$ is Macaulay-Lex if for every graded ideal of $R$, there exists a lexicographic ideal of $R$ with the same Hilbert function. In this paper, we introduce a class of quotient rings with combinatorial significance, which we call colored quotient rings. This class of rings include Clements-Lindstr\"om rings and colored squarefree rings as special cases that are known to be Macaulay-Lex. We construct two new classes of Macaulay-Lex rings, characterize all colored quotient rings that are Macaulay-Lex, and give a simultaneous generalization of both the Clements-Lindstr\"om theorem and the Frankl-F\"uredi-Kalai theorem. We also show that the $f$-vectors of $(a_1, \dots, a_n)$-colored simplicial complexes or multicomplexes are never characterized by "reverse-lexicographic" complexes or multicomplexes when $n>1$ and $(a_1, \dots, a_n) \neq (1, \dots, 1)$.

In this paper we characterize commutative rings with finite dimensional classical ring of quotients. To illustrate the diversity of behavior of these rings we examine the case of local rings and FPF rings. Our results extend earlier work on rings with zero-dimensional rings of quotients.

Torsion theories over commutative rings

In her paper [3], Osofsky exhibited an example of a ring R containing 16 elements which (i) is equal to its left complete ring of quotients, (ii) is not self-injective and (iii) whose injective hull HR = H(RR) allows a ring structure extending the R-module structure of HR. In the present note, we offer a general method of constructing such rings; in particular, given a non-trivial split Frobenius algebra A and a natural n ≧ 2, a certain ring of n x n matrices over A provides such an example. Here, taking for A the semi-direct extension of Z/2Z by itself and n = 2, one gets the example of Osofsky. Thus, our approach answers her question on finding a non-computational method for proving the existence of such rings.