| Version 18 |
Version 17 |
| \PMlinkescapeword{equivalent} |
\PMlinkescapeword{equivalent} |
| \PMlinkescapephrase{generated by} |
\PMlinkescapephrase{generated by} |
| \PMlinkescapeword{left} |
\PMlinkescapeword{left} |
| \PMlinkescapephrase{left noetherian} |
\PMlinkescapephrase{left noetherian} |
| \PMlinkescapeword{property} |
\PMlinkescapeword{property} |
| \PMlinkescapeword{right} |
\PMlinkescapeword{right} |
| \PMlinkescapephrase{right noetherian} |
\PMlinkescapephrase{right noetherian} |
| \PMlinkescapeword{similar} |
\PMlinkescapeword{similar} |
| \PMlinkescapeword{simple} |
\PMlinkescapeword{simple} |
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A (left or right) module $M$ over a ring $R$ is said to be \emph{Noetherian}
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A (left or right) module $M$ over $R$ is said to be \emph{Noetherian}
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| if the following equivalent conditions hold: |
if the following equivalent conditions hold: |
| \begin{enumerate} |
\begin{enumerate} |
| \item Every submodule of $M$ is finitely generated over $R$. |
\item Every submodule of $M$ is finitely generated over $R$. |
| \item The ascending chain condition holds on submodules. |
\item The ascending chain condition holds on submodules. |
| \item Every nonempty family of submodules has a maximal element. |
\item Every nonempty family of submodules has a maximal element. |
| \end{enumerate} |
\end{enumerate} |
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For example, the $\mathbb{Z}$-module $\mathbb{Q}$ is not Noetherian,
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For example the $\mathbb{Z}$-module $\mathbb{Q}$ is not Noetherian,
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| as it is not finitely generated, |
as it is not finitely generated, |
| but the $\mathbb{Z}$-module $\mathbb{Z}$ is Noetherian, |
but the $\mathbb{Z}$-module $\mathbb{Z}$ is Noetherian, |
| as every submodule is generated by a single element. |
as every submodule is generated by a single element. |
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| Observe that changing the ring can change whether a module is Noetherian or not: |
Observe that changing the ring can change whether a module is Noetherian or not: |
| for example, the $\mathbb{Q}$-module $\mathbb{Q}$ is Noetherian, |
for example, the $\mathbb{Q}$-module $\mathbb{Q}$ is Noetherian, |
| since it is \PMlinkname{simple}{SimpleModule} |
since it is \PMlinkname{simple}{SimpleModule} |
| (has no nontrivial submodules). |
(has no nontrivial submodules). |
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| There is also a notion of \PMlinkname{Noetherian for rings}{Noetherian}: |
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, |
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. |
and right Noetherian if it is Noetherian as a right module over itself. |
| For non-commutative rings, these two notions can differ. |
For non-commutative rings, these two notions can differ. |
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| The corresponding property for groups is usually called the maximal condition. |
The corresponding property for groups is usually called the maximal condition. |
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| Finally, there is the somewhat related notion of a |
Finally, there is the somewhat related notion of a |
| \PMlinkname{Noetherian topological space}{NoetherianTopologicalSpace}. |
\PMlinkname{Noetherian topological space}{NoetherianTopologicalSpace}. |
