The length of all composition series of a module M. According to the Jordan-Hölder theorem for modules, if M has any composition series, then all such series are equivalent. The length of a module without composition series is conventionally set equal to ∞. A module has finite length iff it is both Artinian and Noetherian; this includes the case where M is finite. An abstract vector space has finite length iff it is finite-dimensional, and in this case the length coincides with the dimension.

composition series | dimension | Jordan-Hölder theorem