# Noetherian

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:

• $M$ is noetherian.
• Every submodule $N$ of $M$ is finitely generated (i.e. can be written as $Rm_1+\cdots+Rm_k$ for some $m_1,\ldots,m_k\in N$).
• For every collection of submodules of $M$, there is a maximal element.

(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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