アルティン環
As a consequence, any finitely-generated module over an Artinian ring is Artinian. Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring "R", any finitely-generated "R"-module is both Noetherian and Artinian, and is said to be of finite length; however, if "R" is not Artinian, or if "M" is not finitely generated, there are counterexamples.