On local right pure semisimple rings of length two or three

Abstract

We investigate in this paper non-commutative local rings of the smallest length that are potential counter-examples to the pure semisimplicity conjecture. Throughout the paper is an associative ring with an identity element. We call local, if the Jacobson radical ( ) of is a two-sided maximal ideal. We denote by mod( ) the category of finitely generated right -modules. Given a right -module of finite length we denote by ( ) the length of . We recall that a ring is said to be of finite representation type, if is artinian and the number of the isomorphism classes of finitely generated indecomposable right (and left) -modules is finite. Following [24] we call a ring right pure semisimple, if every right -module is a direct sum of finitely generated modules. It is well known that a ring is of finite representation type if and only if is right pure semisimple and is left pure semisimple (see [2], [11], [18], [22]–[24]). It is still an open question, called the pure semisimplicity conjecture, if a right pure semisimple ring is of finite representation type (see [2] and [24], [25], [28]). In [13] the question is answered in affirmative for rings satisfying a polynomial identity and for self-injective rings (see also [7], [19] and [31]). The reader is referred to [42] and to the author’s expository papers [30] and [32] for a basic background and historical comments on the pure semisimplicity conjecture. It was shown by the author in [28] and [33] that there is a chance to find a counter-example to the pure semisimplicity conjecture and might be hereditary with two simple non-isomorphic modules. The existence of a counter-example depends on a generalized Artin problem on division ring extensions. In the present paper we are mainly interested in the existence of counter-examples to pure semisimplicity conjecture that are local of the smallest length, that is, of length ( ) two or three. This continues our study started in [28], [35] and [33]. It is shown in Lemma 3.1 that every such a local ring has ( )2 = 0. Therefore we study representation-infinite right pure semisimple local rings with ( )2 = 0 such that the Auslander-Reiten quiver (mod ) is of the form · · · → •→ •→ •→