View source for Noetherian module

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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]]

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

$M'$ 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,   "Algebra" , Addison-Wesley  (1974)</TD></TR></table>