Kent R. Fuller

Algebras from diagrams

Abstract A method of constructing an algebra directly from a diagram representing a preconceived structure is described, and various conditions sufficient to insure the feasibility of the construction are presented.

Equivalence and Duality for Module Categories

So far our emphasis has been on studying rings in terms of the module categories they admit—that is, in terms of the representations of the rings as endomorphism rings of abelian groups. As we shall see the Wedderburn Theorem for simple artinian rings can be interpreted as asserting that a ring R is simple artinian if and only if the category R M is “the same” as the category D M for some division ring D. On the other hand, if D is a division ring, then the theory of duality from elementary linear algebra asserts that the categories D FM and FM D of finitely generated left D-vector spaces and right D-vector spaces are “duals” of one another.

Tilting objects in abelian categories and quasitilted rings

D. Happel, I. Reiten and S. Smarlo initiated an investigation of quasitilted artin K-algebras that are the endomorphism rings of tilting objects in hereditary abelian categories whose Hom and Ext groups are all finitely generated over a commutative artinian ring K. Here, employing a notion of *-objects, tilting objects in arbitrary abelian categories are defined and are shown to yield a version of the classical tilting theorem between the category and the category of modules over their endomorphism rings. This leads to a module theoretic notion of quasitilted rings and their characterization as endomorphism rings of tilting objects in hereditary cocomplete abelian categories.

Modules over ring extensions

Conditions under which a *-module AU extends to a *-module RR⊗AU over a ring extension R of A are presented, along with con¬nections to quasi-progenerators, tilting modules, and torsion theory counter equivalences.

Algebras whose projective modules are reflexive

The algebras of the title are investigated. They are shown to include l-hereditary algebras, monomial algebras whose indecomposable projective representations are reflexive, and certain binomial algebras.

Natural and Doubly Natural Dualities

ABSTRACT Representable dualities between categories of modules satisfying natural closure properties are investigated and shown to be determined by homological properties shared by various classical and recently investigated types of dualities.

On quasi-duaiity modules

Regarding a generalization of Morita duality, as in [F] we call a module I/Ck with endomorphism ring S a quasi-dualzty module if it represents a duality between the finitely closed categories of modules generated by WR and sS. While Morita dualities are the duals of Morita equivalences, quasi-duality modules behave as duals of the quasi-progenerators studied in detail in [F2]. They have been investigated under various names in this context by several authors, e.g., [GGW], [Me], [MOI, [ZJI . In this paper, we study quasi-duality rnoduies and i i~ei i reflexive modtilea, giving particular attention to questions regarding closure of their categories of reflexive modules raised in [F], and providing several relevant new examples. Let R be a fixed ring and WR a right R-module with S = End(WR) and B = End(sW). Then a right B-module is automatically a right R-module via the natural ring homomorphism p : R -+ B. Throughout this paper, these notations will be fixed, unless otherwise specified. En Section 1, we give relations between the s WR-reflexive modules and the WB-reflexive modules. Using topological ~nethods, Menini [Me] gave a useful characterization of quasi-duality modules (Theorem 2.1). We present an algebraic proof of her

Direct sums of reflexive modules

Abstract The problem of determining when reflexivity is inherited by direct sums of reflexive modules over finite dimensional split algebras is addressed. We show that this holds if the algebra is left locally distributive, a class of algebras which includes serial algebras, as well as for those algebras R with radical J and a basic set of idempotents such that dim( eJf ) ≤ 3 for all idempotents e , f in the set. In the opposite direction, it is proved that a necessary condition for direct sums of reflexive modules to be reflexive is that the quiver of the algebra should contain no triple arrows.

Weak Morita Duality

Abstract The notion of a weak Morita duality between subcategories of Mod-R and S-Mod is introduced and investigated. This concept includes dualities induced by cotilting modules over finite dimensional algebras, the R-dual for both noetherian rings of injective dimension ≤ 1 and the Matlis duality domains, as well as the first authors generalized Morita duality. Moreover it yields a “cotilting theorem” dual to the classical tilting theorem, and a characterization of generalized Morita duality.

Classical Artinian Rings

In our concluding chapter we present basic results on several types of artinian rings that have come to be regarded as classical due to their natural origins and the influence they have had on the literature of ring and module theory. These include artinian rings with duality, quasi-Frobenius (or QF) rings, QF-3 rings, and serial rings.