Direct sums and direct products of canonical pencils of matrices

Abstract

A theorem of Kulikov characterizes the K[ x]-modules which are direct sums of finite-dimensional (as a K-vector space) indecomposable modules, where K[ x] is the polynomial ring over the field K. In this paper an analogous characterization is given for modules over the ring R, arising from pairs of linear transformations between a pair of complex vector spaces, ( V, W). R is a certain subring of the ring of 3 × 3 complex matrices. The equivalence between the category of right R-modules and the category of systems enables one to work entirely in the category of systems. (A pair of complex vector spaces is a system if and only if there is a C -bilinear map from C 2 × V to W). R-modules that are direct sums of finite-dimensional indecomposable subsystems are called pure-projective. The above characterization of pure-projective R-modules is used to prove that an R-module M is projective if and only if Ext( M, R) = 0. Direct products of finite-dimensional indecomposable R-modules are also studied, and a theorem pinpoints those that are pure-projective. An example of an R-module M that is not pure-projective, but with the property that every finite subset of M is contained in a pure-projective direct summand of M, is given. A by-product of this example is a class of matrices that generalizes the Vandermonde matrices.