Modules that have only finitely many submodules Drawing the lattice of submodules of a given module helps me to gain some intuition about the structure of module. Sometimes, however, it is not possible to draw in neat manner; For example vector spaces may have infinitely many subspaces. What is a good class of modules to draw the lattice of submodules in neat manner? (Uniserial modules are candidate which I know of.) An answer to this question might be somewhere, but I don't know. Lastly, let me make my questions clear.


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*What is a (large enough) class of modules that have only finitely many submodules (NOT up to isomorphic)?

*What is a (large enough) class of rings or algebras that every (finitely generated) right modules have only finitely many submodules? Solved owing to Jeremy Rickard's answer.

 A: For the second question, the finite rings are precisely those for which every finitely generated module has finitely many submodules.
For a ring $R$ and $r\in R$, let $M_r$ be the submodule of $R\oplus R$ generated by $(1,r)$. Then $(1,r)$ is the only element of $M_r$ whose first coordinate is $1$, and so $M_r\neq M_s$ for $r\neq s$, and so if $R$ is infinite then the modules $\{M_r\mid r\in R\}$ form an infinite set of submodules of $R\oplus R$.
A: It seems like your best bet for both questions will be to consider finite rings and their finitely generated modules. These at least will be closed under products.
A: A satisfactory answer to my question 1 was given in [Jan57, Lemma 1.2, Corollary 1.3] (or [Ful79, Proposition 1.3]). In particular, for a finite dimensional algebra over an infinite field, a finitely generated module has finitely many submodules if and only if it is distributive (i.e., the lattice of submodules is distributive). Such modules are characterized by the socle of factors [Cam75, Theorem 1]: A module is distributive if and only if every factor has square-free socle.


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*[Cam75] V. CAMILLO, 'Distributive modules', J. Algebra 36 (1975) 16-25, doi:10.1016/0021-8693(75)90151-9. MathSciNet zbMATH

*[Ful79] K. FULLER, 'Biserial rings', Ring Theory Waterloo 1978, Lecture Notes in Mathematics 734 (eds D. Handelman and J. Lawrence; Springer-Verlag, 1979) 64-90, doi:10.1007/BFb0103154. MathSciNet zbMATH

*[Jan57] J. JANS, 'On the indecomposable representations of algebras', Ann. of Math. 66 (1957) 418-429, doi:10.2307/1969899. MathSciNet zbMATH
A: There is a paper on modules with finitely many submodus, by
Akbari, Khalashi Ghezelahmad and Yaraneri.
