Introduction

This chapter is devoted to the study of certain classes of rings and that of modules over these rings. There are several underlying themes behind this study, such as classification or decomposition questions or the description of subobjects and of sets of morphisms.

We essentially only deal with these questions under reasonable finiteness assumptions, which is why the chapter opens with the notions of module and of Noetherian and Artinian rings.

As far as rings are concerned, we present several results that allow one to understand Artinian rings:

a) The Jacobson radical of an Artinian ring is its nilradical (§10), and the corresponding quotient is a semisimple ring (§8).

b) A semisimple ring is isomorphic to the product of a finite family of simple rings (§7).

c) By Wedderburn’s theorem (VIII, p. 120, Theorem 1), a simple ring is isomorphic to a matrix algebra over a field.

Up to Morita equivalence (§6), an algebra of finite degree that is a simple ring is determined by its class in an abelian group called the Brauer group. We give several descriptions of this group (§15 and §16), as well as examples (§18 and §19).

For modules, we present two natural decompositions: The first, in terms of composition series (I, §4, No. 7, p. 41, Definition 9), is provided by the Jordan–Hölder theorem (I, §4, No. 7, p. 43, Theorem 6). The second, which corresponds to direct sums, is given in the case of modules of finite length by the Krull–Remak–Schmidt theorem (VIII, p. 37, Theorem 2), which, here, we deduce from a result of Azumaya on semiprimordial modules (VIII, p. 32,

xvii

Theorem 1). We also consider the invariants associated with modules that behave additively for the decompositions mentioned above; the Grothendieck groups described in §11 are solutions of universal problems for these invariants. When we study the structure of modules over a ring, the notion of ring isomorphism is, to our advantage replaced by Morita equivalence.

For semisimple modules, that is, direct sums of simple modules, the notion of description of a module (VIII, p. 69, Definition 5) allows us to describe the homomorphisms arising from the module as well as its submodules.

By way of illustration, in §21, we consider the case of the algebra of a finite group, whose modules correspond to the linear representations of the group.

A historical note included at the end of the volume, copied from the previous edition, retraces how many of the notions developed here arose.