Modules of finite length over their endomorphism rings

Summary

Given a ring R (associative, with 1) one can define the endolength of an R–module M to be its length when it is regarded in the natural way as an End_R (M)–module, and thus one can consider the class of modules of finite endolength. The aim of this paper is to show that this is a useful concept. Briefly, the contents are as follows. In §§1–3 we cover some background machinery, in §§4–6 we discuss the modules of finite endolength for a general ring, and in §§7–9 we show how these modules control the behaviour of the finite length modules for noetherian and artin algebras. Although much of this paper has a survey nature, there are some new results proved here, the main ones being the characterization of the pure-injective modules which occur as the source of a left almost split map in §2, the character theory for modules of finite endolength in §5, and the characterization of the artin algebras with an indecomposable module of infinite length and finite endolength (a generic module) proved in §§8–9.

The author is supported by an SERC Advanced Research Fellowship. I would like to thank M. Prest and A. H. Schofield for some useful discussions.

THE FUNCTOR CATEGORY

If R is a ring, we denote by R–Mod the category of left R–modules, and by mod–R be the category of finitely presented (f.p.) right R–modules. We denote by D(R) the category of additive functors from mod–R to ℤ–Mod (the category of abelian groups). This category has a very rich structure, and is an invaluable tool for the study of R–modules.