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Some properties of purely simple Kronecker modules, I

1983; Elsevier BV; Volume: 27; Issue: 1 Linguagem: Inglês

10.1016/0022-4049(83)90028-2

1873-1376

Frank Okoh,

Rings, Modules, and Algebras

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.