Finitelength

Finite length, in algebra, refers to a property of a module that indicates it has a finite composition structure. A module M over a ring R has finite length if there exists a finite chain of submodules 0 = M0 < M1 < ... < Mn = M such that each successive quotient Mi/Mi-1 is a simple module. The integer n is called the length of M, denoted length(M) or l(M). The finite length condition is equivalent to M being both Noetherian (every ascending chain of submodules stabilizes) and Artinian (every descending chain stabilizes). By the Jordan–Hölder theorem, any two composition chains have the same length and the same multiset of simple factors, up to isomorphism.

Examples and special cases help clarify the concept. If R is a field, then an R-module is just a vector space, and having finite length is the same as being finite-dimensional; the length equals the dimension. More generally, if A is a finite-dimensional algebra over a field, a finite-length A-module is one with a finite filtration by submodules whose successive quotients are simple. Finite length is a central notion in representation theory, where many representations of interest are finite length.

Relation to semisimplicity is nuanced. If M is semisimple, it decomposes as a finite direct sum of simple modules, and its length equals the total number of simple summands (counted with multiplicity). Finite length does not necessarily imply semisimplicity, since the filtration may not split.

See also: composition series, simple modules, Noetherian and Artinian modules, Jordan–Hölder theorem, finite-length representations.