Difference between revisions of "Noetherian module"
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| − | A [[ | + | A [[module]] for which every submodule has a finite system of generators. Equivalent conditions are: the ascending [[chain condition]] for [[submodule]]s (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]] |
$$0\to M'\to M\to M''\to0,$$ | $$0\to M'\to M\to M''\to0,$$ | ||
| − | $M'$ and $M''$ are Noetherian, then so is $M$. A module over a [[ | + | $M'$ and $M''$ are Noetherian, then so is $M$. A module over a [[Noetherian ring]] is Noetherian if and only if it is finitely generated. A module has a [[Composition sequence|composition series]] if and only if it is both [[Artinian module|Artinian]] and Noetherian. |
====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> | ||
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| + | [[Category:Associative rings and algebras]] | ||
Latest revision as of 19:05, 25 October 2014
A module for which every submodule has a finite system of generators. Equivalent conditions are: the ascending chain condition for submodules (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. A module has a composition series if and only if it is both Artinian and Noetherian.
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
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