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.