Indecomposable modules constructed from Liouville numbers (Q1820851)

The submodules of the polynomial Kronecker module are investigated. Attention is focussed on indecomposable submodules. The main result is: For each positive integer \(n>1\), there is a family of vector spaces \(\{V_ s:\) \(s\in S\}\), Card S\(=2^{\aleph_ 0}\), of indecomposable submodules of the Kronecker module P of rank n with the following properties: (a) \(Hom(V_{s_ 1},V_{s_ 2})=0\) if \(s_ 1\neq s_ 2;\) (b) \(End(V_ s)=K\) for every s in S. (c) dim Ext(V\({}_{s_ 1},V_{s_ 2})\geq 2^{\aleph_ 0}\) for any \(s_ 1,s_ 2\) in S. This result is proved by constructing extensions of finite-dimensional modules by P using Liouville numbers. Each extension, V, is shown to share with P a common submodule which reflects properties of V. A consequence of this is that, for each positive integer \(n>1\), P contains a nonterminating descending chain of nonisomorphic indecomposable submodules of rank n.