Editing Length of a module (section)

== Examples ==

=== Finite dimensional vector spaces ===
Any finite dimensional vector space <math>V</math> over a field <math>k</math> has a finite length. Given a basis <math>v_1,\ldots,v_n</math> there is the chain<math display=block>0 \subset \text{Span}_k(v_1) \subset \text{Span}_k(v_1,v_2) \subset \cdots \subset \text{Span}_k(v_1,\ldots, v_n) = V</math>which is of length <math>n</math>. It is maximal because given any chain,<math display=block>V_0 \subset \cdots \subset V_m</math>the dimension of each inclusion will increase by at least <math>1</math>. Therefore, its length and dimension coincide.

=== Artinian modules ===
Over a base ring <math>R</math>, [[Artinian module]]s form a class of examples of finite modules. In fact, these examples serve as the basic tools for defining the order of vanishing in [[intersection theory]].<ref name=":1">{{Cite book|last=Fulton, William, 1939-|url=https://www.worldcat.org/oclc/38048404|title=Intersection theory|date=1998|publisher=Springer|isbn=3-540-62046-X|edition= 2nd|location=Berlin|pages=8–10|oclc=38048404}}</ref>

==== Zero module ====
The zero module is the only one with length 0.

==== Simple modules ====
Modules with length 1 are precisely the [[simple module]]s.

==== Artinian modules over Z ====
The length of the [[cyclic group]] <math>\mathbb{Z}/n\mathbb{Z}</math> (viewed as a module over the [[integer]]s '''Z''') is equal to the number of [[prime number|prime]] factors of <math>n</math>, with multiple prime factors counted multiple times. This follows from the fact that the submodules of <math>\mathbb{Z}/n\mathbb{Z}</math> are in one to one correspondence with the positive divisors of <math>n</math>, this correspondence resulting itself from the fact that <math>\Z</math> is a [[principal ideal ring]].