The Stacks project

Proof. Observe that $(R, \mathfrak m)$ is not Artinian if and only if $V(\mathfrak m) \subset \mathop{\mathrm{Spec}}(R)$ is nowhere dense. See Proposition 10.60.7. We assume this from now on.

Let $J \subset R$ be the largest ideal killed by a power of $\mathfrak m$. If $J \not= 0$ then $R \to R/J$ shows that $(R, \mathfrak m)$ is as in (4).

Otherwise $J = 0$. In particular $\mathfrak m$ is not an associated prime of $R$ and we see that there is a nonzerodivisor $x \in \mathfrak m$ by Lemma 10.63.18. If $\mathfrak m$ is not an associated prime of $R/xR$ then $\text{depth}(R) \geq 2$ by the same lemma. Thus we are left with the case when there is a $y \in R$, $y \not\in xR$ such that $y \mathfrak m \subset xR$.

If $y \mathfrak m \subset x \mathfrak m$ then we can consider the map $\varphi : \mathfrak m \to \mathfrak m$, $f \mapsto yf/x$ (well defined as $x$ is a nonzerodivisor). By the determinantal trick of Lemma 10.16.2 there exists a monic polynomial $P$ with coefficients in $R$ such that $P(\varphi ) = 0$. We conclude that $P(y/x) = 0$ in $R_ x$. Let $R' \subset R_ x$ be the ring generated by $R$ and $y/x$. Then $R \subset R'$ and $R'/R$ is a finite $R$-module annihilated by a power of $\mathfrak m$. Thus $R$ is as in (4).

Otherwise there is a $t \in \mathfrak m$ such that $y t = u x$ for some unit $u$ of $R$. After replacing $t$ by $u^{-1}t$ we get $yt = x$. In particular $y$ is a nonzerodivisor. For any $t' \in \mathfrak m$ we have $y t' = x s$ for some $s \in R$. Thus $y (t' - s t ) = x s - x s = 0$. Since $y$ is not a zero-divisor this implies that $t' = ts$ and so $\mathfrak m = (t)$. Thus $(R, \mathfrak m)$ is regular of dimension 1. $\square$