If $R$ is a ring and $M$ an $R$-module, $M$ is *uniserial* if its lattice of submodules is a chain. Over an Artinian $R$, the chain will be finite. From what I understand, deciding when two uniserial modules are isomorphic is an open problem. D'Este, Kaynarca, and Tutuncu point out in the introduction to their paper *Isomorphism problem for uniserial modules over an arbitrary ring,* arXiv:1910.06173v1[math.RT], that B. Huisgen-Zimmerman (*The geometry of uniserial representations of finite dimensional algebras,* J. Pure Appl. Alg. 127 (1998), 39-72) solved the problem for finite dimensional algebras over algebraically closed fields. D'E-K-T also provide an example of an Artin algebra having two non-isomorphic uniserial modules of length two with the same socle and top.


However, what I am more interested in is knowing whether an Artinian ring has *finite uniserial type,* that is, only finitely many isomorphism classes of uniserial modules, especially in the case that the Artinian ring is also a principal ideal ring (two-sided Artinian and two-sided principal). Whether the answer is positive, or negative, it will be of use to me as I put the final touches on an article. If someone has a reference for the answer, that would be great.



Rings of finite uniserial type
It is an open problem which Artin algebras have only finitely many uniserial indecomposable modules.
This is stated for example as problem 2 in the open problems section in the book "Representation theory of Artin algebras" by Auslander-Reiten-Smalo.
No good characterisation exists as far as I know or an algorithm to check that property for a given algebra.


