Editing Length of a module (section)

== Properties ==

=== Finite length and finite modules ===
If an <math>R</math>-module <math>M</math> has finite length, then it is [[finitely generated module|finitely generated]].<ref>{{Cite web|title=Lemma 10.51.2 (02LZ)—The Stacks project|url=https://stacks.math.columbia.edu/tag/02LZ|website=stacks.math.columbia.edu|access-date=2020-05-22}}</ref> If ''R'' is a field, then the converse is also true.

=== Relation to Artinian and Noetherian modules ===
An <math>R</math>-module <math>M</math> has finite length if and only if it is both a [[Noetherian module]] and an [[Artinian module]]<ref name=":0" /> (cf. [[Hopkins' theorem]]). Since all Artinian rings are Noetherian, this implies that a ring has finite length if and only if it is Artinian.

=== Behavior with respect to short exact sequences ===
Suppose<math display=block>0\rarr L \rarr M \rarr N \rarr 0</math>is a [[short exact sequence]] of <math>R</math>-modules. Then M has finite length if and only if ''L'' and ''N'' have finite length, and we have <math display=block>\text{length}_R(M) = \text{length}_R(L) + \text{length}_R(N)</math> In particular, it implies the following two properties

* The direct sum of two modules of finite length has finite length 
* The submodule of a module with finite length has finite length, and its length is less than or equal to its parent module.

=== Jordan–Hölder theorem ===
{{main|Jordan–Hölder theorem}}
A [[composition series]] of the module ''M'' is a chain of the form

:<math>0=N_0\subsetneq N_1 \subsetneq \cdots \subsetneq N_n=M</math>

such that

:<math>N_{i+1}/N_i \text{ is simple for }i=0,\dots,n-1</math>

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''.