Let $R$ be a commutative local Noetherian ring and $\mathfrak{m}$ its maximal ideal.
Prove that, if $\mathfrak{m}$ is principal, then $\mathrm{dim}(R)\leq 1$ (the Krull dimension of the ring).
Thank you.
Let $R$ be a commutative local Noetherian ring and $\mathfrak{m}$ its maximal ideal.
Prove that, if $\mathfrak{m}$ is principal, then $\mathrm{dim}(R)\leq 1$ (the Krull dimension of the ring).
Thank you.
First, let me explicitly assume $R$ is a noetherian local domain with a principal maximal ideal $\mathfrak{m}$.
Proposition. The Krull dimension of $R$ is at most $1$.
Proof. Let $\mathfrak{m} = (t)$, and let $\mathfrak{p}$ be prime. We know $\mathfrak{p} \subseteq \mathfrak{m}$, so it is enough to show that either $\mathfrak{p} = \mathfrak{m}$ or $\mathfrak{p} = (0)$. Suppose $\mathfrak{p} \ne \mathfrak{m}$: then $t \notin \mathfrak{p}$. Let $a_0 \in \mathfrak{p}$. For each $a_n$, because $\mathfrak{p}$ is prime, there exists $a_{n+1}$ in $\mathfrak{p}$ such that $a_n = a_{n+1} t$. By the axiom of dependent choice, this yields an infinite ascending sequence of principal ideals: $$(a_0) \subseteq (a_1) \subseteq (a_2) \subseteq \cdots$$ Since $R$ is noetherian, for $n \gg 0$, we must have $(a_n) = (a_{n+1}) = (a_{n+2}) = \cdots$. Suppose, for a contradiction, that $a_0 \ne 0$. Then, $a_n \ne 0$ and $a_{n+1} \ne 0$, and there is $u \ne 0$ such that $a_{n+1} = a_n u$. But then $a_n = a_{n+1} t = a_n u t$, so cancelling $a_n$ (which we can do because $R$ is an integral domain), we get $1 = u t$, i.e. $t$ is a unit. But then $\mathfrak{m} = R$ – a contradiction. So $a_n = 0$. $\qquad \blacksquare$
Here's an elementary proof which shows why we can reduce to the case where $R$ is an integral domain.
Proposition. Any non-trivial ring $A$ has a minimal prime.
Proof. By Krull's theorem, $A$ has a maximal ideal, which is prime. Let $\Sigma$ be the set of all prime ideals of $A$, partially ordered by inclusion. The intersection of a decereasing chain of prime ideals is a prime ideal, so by Zorn's lemma, $\Sigma$ has a minimal element. $\qquad \blacksquare$
Thus, we can always assume that a maximal chain of prime ideals starts at a minimal prime and ends at a maximal ideal. But if $R$ is a noetherian local ring with principal maximal ideal $\mathfrak{m}$ and $\mathfrak{p}$ is a minimal prime of $R$, then $R / \mathfrak{p}$ is a noetherian local domain with a principal maximal ideal $\mathfrak{m}$, and $\dim R = \sup_\mathfrak{p} \dim R / \mathfrak{p}$, as $\mathfrak{p}$ varies over the minimal primes.
Update. Georges Elencwajg pointed out in a comment that the first proof actually works without the assumption that $R$ is a domain, because $(1 - u t)$ is always invertible.
So, following Zhen Lin:
Let $\mathfrak{p}$ be a prime ideal of $R$. Then it is contained in $\mathfrak{m}$, the maximal ideal.
But $\mathfrak{m}$ is principal, say $\mathfrak{m}=(m)$ and $\mathfrak{p}$ must be f.g., since $R$ is Noetherian, say $\mathfrak{p}=(p_1,\dots,p_n)$.
Now $\mathfrak{p} \subseteq \mathfrak{m}$ implies $m \mid p_i, \ \forall i=1,\dots,n$, so $\mathfrak{p}$ is either zero or $(m)=\mathfrak{m}$.
So the chain that gives the Krull dimension contains either nothing (if $\mathfrak{m}=0$) or $\mathfrak{m}$ only.
Hope I got it right...