アルティン環
It is well known that a module has a finite composition series if and only if it is both an Artinian module and a Noetherian module. If "R" is an Artinian ring, then every finitely generated "R"-module is Artinian and Noetherian, and thus has a finite composition series. In particular, for any field "K", any finite-dimensional module for a finite-dimensional algebra over "K" has a composition series, unique up to equivalence.