Date of this Version
Published in Transactions of the American Mathematical Society, Vol. 135. (Jan., 1969), pp. 127-141.
Since D. G. Higman proved that bounded representation type and finite representation type are equivalent for group algebras at prime characteristic, there has been a renewed interest in the Brauer-Thrall conjecture that bounded representation type implies finite representation type for arbitrary algebras. The main purpose of this paper is to present a new approach to this conjecture by showing the relevance (when the base field is algebraically closed) of questions concerning the structure of indecomposable modules of certain special types, namely, the stable (every maximal submodule is indecomposable), the costable (having the dual property), and the stable-costable (having both properties) indecomposable modules. The main tools are the Sandwich Lemma (1.2) which is proved using an old observation of É. Goursat, an observation of A. Heller, C. W. Curtis, and D. Zelinsky concerning quasifrobenius (QF) rings (Proposition 2.1), and a general interlacing technique similar to methods used by Jans, Tachikawa, and Colby for building up large indecomposable modules of finite length which has validity in any abelian category (Theorem 3.1).