Length of a module

In algebra, the length of a module over a ring ${\displaystyle R}$ is a generalization of the dimension of a vector space which measures its size.^{[1] page 153} It is defined to be the length of the longest chain of submodules. For vector spaces (modules over a field), the length equals the dimension. If ${\displaystyle R}$ is an algebra over a field ${\displaystyle k}$, the length of a module is at most its dimension as a ${\displaystyle k}$-vector space.

In commutative algebra and algebraic geometry, a module over a Noetherian commutative ring ${\displaystyle R}$ can have finite length only when the module has Krull dimension zero. Modules of finite length are finitely generated modules, but most finitely generated modules have infinite length. Modules of finite length are called Artinian modules and are fundamental to the theory of Artinian rings.

The degree of an algebraic variety inside an affine or projective space is the length of the coordinate ring of the zero-dimensional intersection of the variety with a generic linear subspace of complementary dimension. More generally, the intersection multiplicity of several varieties is defined as the length of the coordinate ring of the zero-dimensional intersection.