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Suppose $R$ is an integral domain, and let $M$ be a finitely generated torsion $R$-module. Must it have a composition series?

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    What is a composition series for you? – Mohan Jan 16 '18 at 02:10
  • @Mohan That's a good question. I suppose for now I want to consider possibly infinite composition series, though I'm not sure how to best define them. I suppose I want to consider stuff like a set of submodules $M_i$ indexed by a totally ordered set $I$ which admits a successor function such that every successive quotient is simple. Though, I'd also appreciate a counterexample in the case where I'm asking about finite composition series. –  Jan 16 '18 at 05:42

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It is a theorem (I think it appears in Eisenbud's book on commutative algebra) that an $R$-module has a finite composition series if and only if it is both Artinian and Noetherian. This will be true for example if $R$ is an Artinian ring. However, this doesn't need to hold. For example, it is also known that a module is Noetherian if and only if every submodule is finitely generated. But there are examples of finitely generated modules with infinitely generated submodules, which are laying around on this site. Here's one such example.

A. Thomas Yerger
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