Rings of finite uniserial type

Question

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.

Answer 1

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.

Answer 2

Only not to leave unanswered the well known and easy part of the question, i.e.

especially in the case that the Artinian ring is also a principal ideal ring (two-sided Artinian and two-sided principal

The case of Artinian PIR (and more generally of Artinian serial rings) was solved (Köthe, Asano and others) in the years just before WW2 and was included in the first of Jacobson's books (The theory of rings, 1943). These are of finite representation type, all irreducible module are uniserial (and if I remember correctly even modules of infinite lenght are direct sums of the uniserials of finite length, but I think this was shown after WW2. Faith, Algebra II and "rings and things" should have modern references).

The case of Artinian PIR (which are the same as proper homomorphic images of Dedekind domains or PID, even non commutative ones, so the finitely generated "bounded" modules over such domains are exactly the f.g. modules over Artinian PIR): Artinian PIR are exactly the finite direct sum of matrix rings over CPU rings (CPU: a ring where each ideal is two sided, and they form a finite chain. They are the proper homomorphic images of discrete valuation domains, and conversely these domains are the inverse limit of a sequence of CPU).

The category of (f.g.) modules over such a ring is then, by Morita equivalence, the category of (f.g.) modules over a finite direct sum of CPU. If a CPU has length $n$ then there are exactly $n$ indecomposable nonzero modules, and they are exactly the nonzero submodules of the CPU and also exactly the proper homomorphic images. Two distinct CPU in the direct decomposition of the PIR (even two copies of the "same" CPU up to isomorphism) give indecomposable modules with zero Hom among them.

[My interest in such matters comes from the Baer / Inaba / J'onsson - Monk coordinatization theorem. The fact that "primary lattices" are simple or finite chains gives, together with the basic theory of such lattices, a nice "incidence geometry" description of the above facts about indecomposables]

[For the more general Artinian serial rings, the description of the ring and of the indecomposable modules is more complicated, but quite explicit; there are several cases to discuss].