[proofplan]
We treat the noetherian case; the artinian case is identical with "noetherian" replaced by "artinian" and ACC replaced by DCC throughout. The argument has three ingredients: (1) a finitely generated $R$-module $M$ is a quotient of a free module $R^{\oplus \ell}$ for some $\ell \geq 1$; (2) since $R$ is noetherian as a module over itself, the finite direct sum $R^{\oplus \ell}$ is noetherian by the [Direct Sums of Noetherian Modules](/theorems/2812); (3) a quotient of a noetherian module is noetherian, which follows from the [Noetherian Property in Short Exact Sequences](/theorems/2901).
[/proofplan]
[step:Express $M$ as a quotient of a free module $R^{\oplus \ell}$]
Let $M$ be a finitely generated $R$-module with generators $m_1, \ldots, m_\ell$. Define the $R$-module homomorphism
\begin{align*}
\pi: R^{\oplus \ell} &\to M, \quad (r_1, \ldots, r_\ell) \mapsto r_1 m_1 + \cdots + r_\ell m_\ell.
\end{align*}
Since $m_1, \ldots, m_\ell$ generate $M$, every element of $M$ is an $R$-linear combination of the $m_i$, so $\pi$ is surjective. Setting $K = \ker \pi$, the first isomorphism theorem gives an isomorphism $R^{\oplus \ell} / K \cong M$, or equivalently a short exact sequence
\begin{align*}
0 \to K \xrightarrow{\iota} R^{\oplus \ell} \xrightarrow{\pi} M \to 0
\end{align*}
where $\iota: K \hookrightarrow R^{\oplus \ell}$ is the inclusion.
[/step]
[step:Show that $R^{\oplus \ell}$ is noetherian using the direct sum theorem]
Since $R$ is a noetherian ring, it is noetherian as a module over itself (ACC on ideals is exactly ACC on $R$-submodules of $R$). The modules $M_1 = \cdots = M_\ell = R$ are each noetherian $R$-modules. By the [Direct Sums of Noetherian Modules](/theorems/2812), $R^{\oplus \ell} = R \oplus \cdots \oplus R$ ($\ell$ copies) is a noetherian $R$-module.
[/step]
[step:Conclude that $M$ is noetherian as a quotient of a noetherian module]
From the short exact sequence $0 \to K \to R^{\oplus \ell} \to M \to 0$, we have that $R^{\oplus \ell}$ is noetherian (established in the previous step). By the [Noetherian Property in Short Exact Sequences](/theorems/2901), $M$ is noetherian if and only if both $K$ and $M$ are noetherian -- but more precisely, the forward direction of that theorem states: if the middle term $R^{\oplus \ell}$ is noetherian, then both $K$ and $M$ are noetherian. In particular, $M$ is noetherian.
[guided]
We have the short exact sequence $0 \to K \xrightarrow{\iota} R^{\oplus \ell} \xrightarrow{\pi} M \to 0$ and we know $R^{\oplus \ell}$ is noetherian. The [Noetherian Property in Short Exact Sequences](/theorems/2901) states: the middle term of a short exact sequence of $R$-modules is noetherian if and only if both end terms are. Applying the forward direction ($\Rightarrow$), since $R^{\oplus \ell}$ is noetherian, both $K$ (the kernel, playing the role of $N$) and $M$ (the cokernel, playing the role of $L$) are noetherian.
Alternatively, one can see directly why a quotient of a noetherian module is noetherian. Let $M_0 \subseteq M_1 \subseteq \cdots$ be an ascending chain of submodules of $M$. The preimages $\pi^{-1}(M_0) \subseteq \pi^{-1}(M_1) \subseteq \cdots$ form an ascending chain in $R^{\oplus \ell}$, which stabilises because $R^{\oplus \ell}$ is noetherian. Since $\pi$ is surjective, $\pi(\pi^{-1}(M_i)) = M_i$ for each $i$, so the chain in $M$ stabilises as well.
The artinian case is identical: replace "noetherian" with "artinian" and "ACC" with "DCC." The same short exact sequence applies, and the [Noetherian Property in Short Exact Sequences](/theorems/2901) covers both chain conditions.
[/guided]
[/step]
