# 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

$$M_0\subseteq M_1\subseteq M_2\subseteq\cdots$$

of submodules of $M$, there exists an integer $n$ so that $M_n=M_{n+1}=M_{n+2}=\cdots$ (i.e. the chain eventually stabilizes, or terminates).

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.

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

1. $M$ is Noetherian.
2. Every submodule $N$ of $M$ is finitely generated (i.e. can be written as $Rm_1+\cdots+Rm_k$ for some $m_1,\ldots,m_k\in N$).
3. Every collection of submodules of $M$ has 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.

Proof. In general, condition 3 is equivalent to ACC. It thus suffices to prove that condition 2 is equivalent to ACC.

Suppose that condition 2 holds. Let $M_0 \subseteq M_1 \subseteq \dotsb$ be an ascending chain of submodules of $M$. Then $$\bigcup_{n \ge 0} M_n$$ is a submodule of $M$, so it must be finitely generated, say by elements $a_1, \dotsc, a_n$. Each of the $a_k$ is contained in one of $M_0, M_1, \dotsc$, say in $M_{t(k)}$. If we set $N = \max t(k)$, then for all $n \ge N$, $$\{ a_1, \dotsc, a_n \} \subset M_n ,$$ so $$M_n = M_N = \bigcup_{n\ge 0} M_n .$$ Thus $M$ satisfies ACC.

On the other hand, suppose that condition 2 does not hold, that there exists some submodule $M'$ of $M$ that is not finitely generated. Thus we can recursively define a sequence of elements $(a_n)_{n=0}^{\infty}$ such that $a_n$ is not in the submodule generated by $a_0, \dotsc, a_{n-1}$. Then the sequence $$(a_0) \subset (a_0, a_1) \subset (a_0, a_1, a_2) \subset \dotsb$$ is an ascending chain that does not stabilize. $\blacksquare$

Note: The notation $(a,b,c \dotsc)$ denotes the module generated by $a,b,c, \dotsc$.

Hilbert's Basis Theorem guarantees that if $R$ is a Noetherian ring, then $R[x_1, \dotsc, x_n]$ is also a Noetherian ring, for finite $n$. It is not a Noetherian $R$-module.