Artinian and Noetherian implies finite length

Artinian and Noetherian implies finite length: Let $M$ be an $R$-module. If $M$ is both Artinian and Noetherian, then $M$ admits a finite composition series; equivalently, $M$ has finite length.

This combines Artinian and Noetherian chain conditions to produce a finite composition series , whose size is measured by module length .