Length of a module

In algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. ^{page 153} It is defined to be the length of the longest chain of submodules.

The modules of finite length are finitely generated modules, but as opposite to vector spaces, many finitely generated modules have an infinite length. Finitely generated modules of finite length are also called Artinian modules and are at the basis of the theory of Artinian rings.

For vector spaces, the length equals the dimension. This is not the case in commutative algebra and algebraic geometry, where a finite length may occur only when the dimension is zero.

The degree of an algebraic variety is the length of the ring associated to the algebraic set of dimension zero resulting from the intersection of the variety with generic hyperplanes. In algebraic geometry, the intersection multiplicity is commonly defined as the length of a specific module.