The isomorphism problem for uniserial modules over an arbitrary ring

Abstract

Firstly, we give a partial solution to the isomorphism problem for uniserial modules of finite length with the help of the morphisms between these modules over an arbitrary ring. Later, under suitable assumptions on the lattice of the submodules, we give a method to partially solve the isomorphism problem for uniserial modules over an arbitrary ring. Particular attention is given to the natural class of uniserial modules defined over algebras given by quivers.

Hopfian and co-Hopfian modules over Artinian rings

“…It is even an open problem at the level of Artin algebras, algebras over a commutative Artinian ring S that are finitely generated as S-modules (see Problem 2, p. 411, in the list of open problems listed in [3]). D'este, Kaynarca, and Tütüncü [9] give a summary of progress in the Artin algebra case in their introduction, and present an example of an Artin algebra having two nonisomorphic uniserial modules of length two having the same composition factors. B. Huisgen-Zimmermann [18] provides the tools to answer the question in the case in which the ring S is an algebraically closed field.…”