Some properties of purely simple Kronecker modules, I

Abstract

Let K be an algebraically closed field. A K 2-system is a pair of K-vector spaces ( V, W) together with a K-bilinear map from K 2× V to W. The category of systems is equivalent to the category of right modules over some K-algebra, R. Most of the concepts in the theory of modules over the polynomial ring K[ξ] have analogues in Mod- R. Unlike the purely simple K[ξ]-modules, which are easily described, purely simple R-modules are quite complex. If M is a purely simple R-module of finite rank n then any submodule of M of rank less than n is finite-dimensional. The following corollaries are derived from this fact: 1. 1.|Every non-zero endomorphism of M is monic. 2. 2.|Every torsion-free quotient of M is purely simple. 3. 3.|An ascending union of purely simple R-modules of increasing rank is not purely simple. It is also shown that a large class of torsion-free rank one modules can occur as the quotient of a purely simple system of rank n, n any positive integer. Moreover, starting from a purely simple system another purely simple module M' of the same rank is constructed and M' is shown to be both a submodule of M and a submodule of a rank 1 torsion-free system. Since the category of right R-modules is a full subcategory of right S-modules, where S is any finite-dimensional hereditary algebra of tame type, the paper provides a way of constructing infinite-dimensional indecomposable S-modules.