Difference between revisions of "Noetherian"

 
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We say that a ring <math>R</math> is left (right) noetherian if it is noetherian as a left (right) <math>R</math>-module. If <math>R</math> is both left and right noetherian, we call it simply noetherian.
 
We say that a ring <math>R</math> is left (right) noetherian if it is noetherian as a left (right) <math>R</math>-module. If <math>R</math> is both left and right noetherian, we call it simply noetherian.
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[[Category:Ring theory]]
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Revision as of 18:03, 9 September 2008

Let $R$ be a ring and $M$ a left $R$-module. Then we say that $M$ is a noetherian module if it satisfies the following property, known as the ascending chain condition (ACC):

  • For any ascending chain $M_1\subseteq M_2\subseteq M_3\subseteq\cdots$ of submodules of $M$, there exists an integer $n$ so that $M_n=M_{n+1}=N_{n+2}=\cdots$ (i.e. the chain eventually terminates).

Theorem. The following conditions are equivalent for a left $R$-module:

(The second condition is also frequently used as the definition for noetherian.)

We also have right noetherian modules with the appropriate adjustments.

We say that a ring $R$ is left (right) noetherian if it is noetherian as a left (right) $R$-module. If $R$ is both left and right noetherian, we call it simply noetherian.

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