Uniserial modules: sums and isomorphisms of subquotients

Abstract

Let R be an associative ring with 1. A left R module M is uniserial i f the lattice L(M) of its submodules is totally ordered under inclusion. We give an example of a uniserial module M with the property of having two submodules 0 < H < K < M such that M is isomorphic to K/H (we call a module M with this property shrinkable). Then we give an example of a uniserial module M isomorphic to all its nonzero quotients M/N, N<M, and with L(M) isomorphic to ω2+1; this solves a problem of Hirano and Mogami [7]. Finally we show that for uniserial modules the property of being shrinkable is connected to the problem of deciding whether a module, which is both a homomorphic image of a finite direct sum of uniserial modules and a submodule of a finite direct sum of uniserial modules, is a finite direct sum of uniserial modules