Editing Length of a module (section)

{{Short description|In algebra, integer associated to a module}}
In [[algebra]], the '''length''' of a [[module (mathematics)|module]] over a ring <math>R</math> is a generalization of the [[dimension (vector space)|dimension]] of a [[vector space]] which measures its size.<ref name=":0">{{Cite web|title=A Term of Commutative Algebra|url=http://www.centerofmathematics.com/wwcomstore/index.php/commalg.html|website=www.centerofmathematics.com|pages=153–158|url-status=live|archive-url=https://web.archive.org/web/20130302125321/http://www.centerofmathematics.com/wwcomstore/index.php/commalg.html|archive-date=2013-03-02|access-date=2020-05-22}} [https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/exercises/Altman_-Kleiman---A-term-of-commutative-algebra-_2017_.pdf Alt URL]</ref> <sup>page 153</sup> It is defined to be the length of the longest chain of [[submodule]]s. For vector spaces (modules over a field), the length equals the dimension. If <math>R</math> is an algebra over a field <math>k</math>, the length of a module is at most its dimension as a <math>k</math>-vector space. 

In [[commutative algebra]] and [[algebraic geometry]], a module over a [[Noetherian ring|Noetherian]] commutative ring <math>R</math> can have finite length only when the module has [[Krull dimension]] zero. Modules of finite length are [[finitely generated module]]s, but most finitely generated modules have infinite length. Modules of finite length are [[Artinian module]]s and are fundamental to the theory of [[Artinian ring]]s.

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 [[General position|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.