On algebras with a finite number of indecomposable modules

Abstract

Let A be an associative algebra over a field K such that the dimension, [A: K], of A over K is finite. Following the terminology of [8], we shall say that A is of bounded module type if there exists an integer n such that for every indecomposable left A-module M, the inequality [M: K] ? n holds. The algebra A is of finite module type if A has only a finite number of nonisomorphic indecomposable left modules. Clearly, algebras of finite module type are of bounded module type. The converse to this statement has been conjectured for some time [8], and it appears to be quite difficult to prove. There are some classes of algebras (more general than semi-simple algebras) which are known to be of finite module type. Higman has shown [7] that the group algebra KG of a finite group G over a field K of characteristic p is of finite module type if and only if the p-Sylow groups of G are cyclic. Nakayama [13] introduced the class of generalized uniserial algebras, and showed that they are all of finite module type. It has been known for some time, however, that these two classes do not contain all algebras of finite module type [16]. Before stating the main theorem of this paper, we introduce some notations and definitions. We shall be concerned only with finite dimensional algebras. We shall denote the radical of an algebra A by N. By the socle S(M) of a left A-module M, we mean the sum of the simple submodules of M; S(M) can also be characterized as the set of all elements m in M such that Nm = 0. The socle S(M) of M is the maximal semi-simple submodule of M, and is a direct sum of simple modules. In general, a semisimple module will be called square-free if it is a direct sum of simple modules, no two of which are isomorphic. Note that in a square-free semisimple module M, two simple submodules are isomorphic if and only if they coincide. We can now state the main theorem of this paper.