## If R is a Noetherian ring, then Rⁿ is a Noetherian R-module

Let $R$ be a Noetherian ring. Prove that $R^n$, as an $R$-module in the usual way, is also Noetherian.

Recall that a ring is Noetherian if every ideal is finitely generated, and a module is Noetherian if every submodule is finitely generated. (That is, a ring is Noetherian (as a ring) if it is Noetherian (as a module) over itself.)

We will proceed by induction on $n$.

For the base case, $n = 1$, $R^1$ is certainly Noetherian as a module over $R$.

For the inductive step, suppose $R^n$ is Noetherian. Now let $M \subseteq R^{n+1}$ be a submodule; our goal is to show that $M$ must be finitely generated. To that end, let $A = \{r \in R \ |\ (m_i)_0 = r\ \mathrm{for\ some}\ (m_i) \in M \}$. That is, $A$ is the collection (in $R$) of all zeroth coordinates of elements in $M$.

We claim that $A \subseteq R$ is an ideal. If $a,b \in A$, then there exist elements $(m_i)$ and $(n_i)$ in $M$ such that $m_0 = a$ and $n_0 = b$. Now since $M \subseteq R^{n+1}$ is a submodule, we have $(m_i) + r(n_i) \in M$ for all $r \in R$, so that $a+rb \in A$. We clearly have $0 \in A$, so that by the submodule criterion, $A$ is an ideal of $R$.

In particular, since $R$ is Noetherian, the ideal $A$ is finitely generated – say by $\{a_0, a_1, \ldots, a_k\}$. We will let $\alpha_i$ be an element of $M$ whose zeroth coordinate is $a_i$. Now let $m = (m_0,\ldots,m_{n+1}) \in M$. Now $m_0 = \sum c_i a_i$ for some $c_i \in R$, and so $m - \sum c_i m_i = (0,t_1,\ldots,t_{n+1})$ is an element of $M$ whose first coordinate is 0. In particular, we have $M = (m_0,\ldots,m_k) + B$, where $B = M \cap (0 \times R^n)$. (We showed the $(\subseteq)$ direction, and the $(\supseteq)$ inclusion is clear.) Now $M \cap (0 \times R^n)$ is an ideal of $0 \times R^n \cong_R R^n$, which is Noetherian by the induction hypothesis. So $B$ is finitely generated as an $R$-module, and thus $M$ is finitely generated over $R$.

Since $M$ was an arbitrary submodule of $R^{n+1}$, $R^{n+1}$ is Noetherian as a module.