Namespaces
Variants
Actions

Difference between revisions of "Noetherian module"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
Line 1: Line 1:
 +
{{TEX|done}}
 
A [[Module|module]] for which every submodule has a finite system of generators. Equivalent conditions are: Every strictly ascending chain of submodules breaks off after finitely many terms; every non-empty set of submodules ordered by inclusion contains a maximal element. Submodules and quotient modules of a Noetherian module are Noetherian. If, in an [[Exact sequence|exact sequence]]
 
A [[Module|module]] for which every submodule has a finite system of generators. Equivalent conditions are: Every strictly ascending chain of submodules breaks off after finitely many terms; every non-empty set of submodules ordered by inclusion contains a maximal element. Submodules and quotient modules of a Noetherian module are Noetherian. If, in an [[Exact sequence|exact sequence]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066830/n0668301.png" /></td> </tr></table>
+
$$0\to M'\to M\to M''\to0,$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066830/n0668302.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066830/n0668303.png" /> are Noetherian, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066830/n0668304.png" />. A module over a [[Noetherian ring|Noetherian ring]] is Noetherian if and only if it is finitely generated.
+
$M'$ and $M''$ are Noetherian, then so is $M$. A module over a [[Noetherian ring|Noetherian ring]] is Noetherian if and only if it is finitely generated.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,  "Algebra" , Addison-Wesley  (1974)</TD></TR></table>

Revision as of 08:35, 1 August 2014

A module for which every submodule has a finite system of generators. Equivalent conditions are: Every strictly ascending chain of submodules breaks off after finitely many terms; every non-empty set of submodules ordered by inclusion contains a maximal element. Submodules and quotient modules of a Noetherian module are Noetherian. If, in an exact sequence

$$0\to M'\to M\to M''\to0,$$

$M'$ and $M''$ are Noetherian, then so is $M$. A module over a Noetherian ring is Noetherian if and only if it is finitely generated.

References

[1] S. Lang, "Algebra" , Addison-Wesley (1974)
How to Cite This Entry:
Noetherian module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_module&oldid=32634
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article