In Chapter 11 we considered modules over arbitrary right artinian rings, and found that there are a number of simplifying features in that case. Modules over artin algebras are even better behaved, and their theory has been much developed. The main examples of artin algebras are algebras finite-dimensional over a field, and their theory has been developed farthest: this is reflected in this chapter in that, although we begin by considering general artin algebras, we soon focus on the case of algebras over a field.

Recall that a ring R is an artin algebra if it is finitely generated as a module over its centre and its centre is artinian. Such a ring is, in particular, right and left artinian. Examples are algebras finite-dimensional over a field and finite rings. Perhaps the most significant property of these rings is that there is a good duality between mod-R and R-mod.

In section 1, from the existence of almost split sequences, we deduce that the isolated points of the space of indecomposables are precisely the indecomposable finitely generated modules.

The second section contains background material on representations of quivers and representation type, as well as descriptions of the finitely generated indecomposable modules over certain algebras.

In the third section, I outline the classification of the infinitely generated indecomposable pure-injectives over the path algebras of the extended Dynkin quivers, and indicate what is known about IR for certain tame non-domestic algebras.

The space of indecomposables

Let R be an artin algebra. Since every finitely generated module is totally transcendental (11.15), every finitely generated indecomposable is a point of the space, I(R), of indecomposable pure-injectives.