Suppose $R$ is an integral domain, and let $M$ be a finitely generated torsion $R$-module. Must it have a composition series?
2026-03-25 04:18:44.1774412324
Do finitely generated torsion modules have a composition series?
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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.