On artinian rings of finite representation type

Abstract

In [5], Roiter has solved the Brauer-Thrall conjecture for finite-dimensiona algebras over fields, which states that, if the lengths of the finitely generated indecomposable modules are bounded, then there is only a finite number of finitely generated indecomposablc modules. Rcccntly Auslander [l] has proved it for ilrtinian rings. In this paper, we show how we construct all indecomposable modules from simple modules over an Artinian ring of finite representation type. That is, if A is an Artinian ring of finite representation type, then every indecomposable A-module appears as a direct summand of (a) the radical of a projective indecomposable A-module or (b) the middle term of an almost split sequence [2], which is successively obtained from a simple ,4-module. Simultaneously, we give a module-theoretical, self-contained, and simple proof for the conjecture though Auslander’s proof is categorical. Throughout this paper ;I will be a right Artinian ring with identity and all modules will be finitely generated right A-modules. Let M be an indecomposable module and N a module. Following Auslander [ 11, a homomorphism f : N ---+ M is said to be aZmst splittable if (a) it is not a splittable epimorphism and (b) for any homomorphism g: S + M which is not a splittable epimorphism, there is a homomorphism h: S -+ N such that g == fh. In the following, an almost splittable homomorphism f : N --+ JZ will be called almost split extemion ozw AI provided that (a) if JZ is projective, then N is the unique maximal submodule of A!! andfis the inclusion, or (b) if M is not project&c, thenfis an epimorphism and Ker f is indecomposable, in which case 0 -+ KerfN f, AI --z 0 is called ahost split sequence in the sense of [2]. It is known [I, 21 that an almost split extension is uniquely determined up to isomorphism and that if the ring A is an Artin algebra or is of finite representation type, then there is an almost split extension over any indecomposable A-module. But it is an open question whethet almost split extensions always exist for arbitrary right Artinian rings. In the following, [M] denotes the isomorphism class of a given module M. For an indecomposable module M, we define a set E,(M) (n 2 0) of finitely many isomorphism classes of indecomposable modules as follows: