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'noetherian module'
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| Title of object: |
noetherian module |
| Canonical Name: |
NoetherianModule |
| Type: |
Definition |
| Created on: |
2001-10-15 18:23:08 |
| Modified on: |
2006-03-31 10:33:51 |
| Classification: |
msc:13E05 |
| Keywords: |
commutative algebra algebraic geometry |
| Defines: |
left noetherian module, right noetherian module |
Revision comment (for changes between this and next version):
Preamble:
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Content:
A left module $M$ over $R$ is said to be \emph{left 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}
If $M$ is a right $R$-module, the conditions for being a right noetherian module are analogous (in fact, we can interpret $M$ as a left module over the opposite ring $R^{\operatorname{op}}$; then being a noetherian right $R$-module is the same as being a noetherian left $R^{\operatorname{op}}$-module).
If $R$ is a commutative ring, then left and right modules are the same, and we call a module that is left noetherian simply \emph{noetherian}.
For example, as $\mathbb{Z}$-modules, $\mathbb{Q}$ is not noetherian, since the submodule generated by $\{1/p, p\text{ prime}\}$ is not finitely generated, but $\mathbb{Z}$ is noetherian: in fact, every submodule is generated by a single element.
Observe that changing the ring can change whether a module is noetherian or not: considered as a $\mathbb{Q}$-module, $\mathbb{Q}$ is noetherian, since it is simple (has no nontrivial submodules).
There is also a notion of \PMlinkname{noetherian for rings}{Noetherian}: a ring is left noetherian if it is a left noetherian module over itself; similarly for right noetherian. These two notions can differ.
The corresponding property for groups is usually called the maximal condition.
Finally, there is a notion of \PMlinkname{noetherian topological space}{NoetherianTopologicalSpace} which is vaguely similar in intent. |
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