Length of a module

In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size.[1] page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It is defined to be the length of the longest chain of submodules. Modules with finite length share many important properties with finite-dimensional vector spaces.

Other concepts used to 'count' in ring and module theory are depth and height; these are both somewhat more subtle to define. Moreover, their use is more aligned with dimension theory whereas length is used to analyze finite modules. There are also various ideas of dimension that are useful. Finite length commutative rings play an essential role in functorial treatments of formal algebraic geometry and Deformation theory where Artin rings are used extensively.

A module M has finite length if and only if it has a (finite) composition series, and the length of every such composition series is equal to the length of M.

For the need of Intersection theory, Jean-Pierre Serre introduced a general notion of the multiplicity of a point, as the length of an Artinian local ring related to this point.

The first application was a complete definition of the intersection multiplicity, and, in particular, a statement of Bézout's theorem that asserts that the sum of the multiplicities of the intersection points of n algebraic hypersurfaces in a n-dimensional projective space is either infinite or is exactly the product of the degrees of the hypersurfaces.

This definition of multiplicity is quite general, and contains as special cases most of previous notions of algebraic multiplicity.

which is similar to defining the order of zeros and poles in Complex analysis.

The order of vanishing is a generalization of the order of zeros and poles for meromorphic functions in Complex analysis. For example, the function

of submodules.[6] More generally, using the Weierstrass factorization theorem a meromorphic function factors as

which is a (possibly infinite) product of linear polynomials in both the numerator and denominator.