Frank W. Anderson

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.

Square-free algebras and their automorphism groups

A finite-dimensional algebra A over a field K is square-free in case for every pair e,f of primitive idempotents in Adim K (e A f) ≤ 1. For example, every incidence algebra of a finite pre-ordered set over a field is square-free. The automorphism groups of the latter have been studied by Stanley, Scharlau, Baclawski, and more recently by Coelho. In this paper we characterize all finite-dimensional square-free K-algebras A as certain semigroup algebras A ≅ K ξ S over a square-free semigroup S twisted by some ξ ∈Z 2 (S K *), a two-dimensional cocycle of S with coefficients in the group of units A * of K. We prove that for each such A ≅ K ξ S, its outer automorphism group Out A is the middle term of a short exact sequence where H 1 (S K *) is the first cohomology group of S with coefficients in K *Aut0 S is the group of “normal” automorphisms of the semigroup S, and Stabξ(Auto S) is the stabilizer in Auto S of ξ under the action of Auto S on H 2 (S K * ). Finally, if ξ ≅ 1, so that A ss KS is untwisted, then...

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.

Automorphosms of infinite dimensional souare-free algebras

Recently, the authors characterized all finite-dimensional square-free algebras A over a field K, and gave a detailed description of their automorphism groups. Using the notion of rings with local units we now extend the definition of square-freealgebras to possibly infinite dimensional ones and observe that the earlier characterization carries over to the infinite dimensional case. Then relying on an appropriate version of the recent notion of inner automorphism introduced by Beattie and del Rio, we extend the description of the automorphism groups to the enlarged class of square-free algebras.

Direct Sums and Products

For each ring R we have derived several module categories—among these the category R M of left R-modules. This derivation is not entirely reversible for, in general, R M does not characterize R. However, as we shall see in Chapter 6 it does come close. Thus, we can expect to uncover substantial information about R by mining R M. So in this chapter we start to probe more deeply into the structure of the modules themselves. In so far as possible we propose to do this in the context of the category R M for in this way at any subsequent stage we shall be able to apply the general machinery of category theory.

Perfect Rings: A Local Approach

ABSTRACT We analyze how the left perfect condition on a ring A (not necessarily with identity) relates to the left perfect condition on unital subrings of the form eAe (where e is idempotent). We then use this analysis to examine a class of graded left perfect rings.

Functors Between Module Categories

it should now be clear that the structure of the category R M determines to a significant extent the structure of the ring R. Thus in this chapter we turn to the direct studies of these categories R M. Our starting point will be the study of certain natural “functors” or “homomorphisms” between pairs of these categories.

Rings, Modules and Homomorphisms

The subject of our study is ring theory. In this chapter we introduce the fundamental tools of this study. Section 1 reviews the basic facts about rings, subrings, ideals, and ring homomorphisms. It also introduces some of the notation and the examples that will be needed later.

Finiteness Conditions for Modules

The first round of generalities is over, and it is now time for us to apply this formal machinery to the study of specific classes of rings and modules. We begin in this chapter with an investigation of the structure of classes of modules having certain natural finiteness properties. In the next chapter we return to the rings themselves.

Injective Modules, Projective Modules, and Their Decompositions

In this chapter we return to the study of decompositions of modules—specifically of injective and projective modules. First we examine characterizations of noetherian rings in terms of the structure of injective modules. Then, after considering the decomposition theory of direct sums of countably generated modules, we proceed to the study of semiperfect and perfect rings (those over which all finitely generated modules and, respectively, all modules have projective covers). In the final section we show that the structure of the endomorphism ring of a finitely generated module determines whether direct sums of copies of that module have decompositions that complement direct summands.