Noetherian moduleNoetherianModule2001-10-15 18:23:082007-11-30 11:12:21DefinitionNoetherianNoetherian left moduleNoetherian right moduleleft Noetherian moduleright Noetherian modulecommutative algebra algebraic geometry\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}\PMlinkescapeword{equivalent}
\PMlinkescapephrase{generated by}
\PMlinkescapeword{left}
\PMlinkescapephrase{left noetherian}
\PMlinkescapeword{property}
\PMlinkescapeword{right}
\PMlinkescapephrase{right noetherian}
\PMlinkescapeword{similar}
\PMlinkescapeword{simple}
A (left or right) module $M$ over a ring $R$ is said to be \emph{Noetherian}
if the following equivalent conditions hold:
\begin{enumerate}
\item Every submodule of $M$ is finitely generated over $R$.
\item The ascending chain condition holds on submodules.
\item Every nonempty family of submodules has a maximal element.
\end{enumerate}
For example, the $\mathbb{Z}$-module $\mathbb{Q}$ is not Noetherian,
as it is not finitely generated,
but the $\mathbb{Z}$-module $\mathbb{Z}$ is Noetherian,
as every submodule is generated by a single element.
Observe that changing the ring can change whether a module is Noetherian or not:
for example, the $\mathbb{Q}$-module $\mathbb{Q}$ is Noetherian,
since it is \PMlinkname{simple}{SimpleModule}
(has no nontrivial submodules).
There is also a notion of \PMlinkname{Noetherian for rings}{Noetherian}:
a ring is left Noetherian if it is Noetherian as a left module over itself,
and right Noetherian if it is Noetherian as a right module over itself.
For non-commutative rings, these two notions can differ.
The corresponding property for groups is usually called the maximal condition.
Finally, there is the somewhat related notion of a
\PMlinkname{Noetherian topological space}{NoetherianTopologicalSpace}.