# Rings whose spectrum is Hausdorff

Let $A$ be a commutative ring with $1$ and consider the Zariski topology on $\operatorname{Spec}(A)$. When will $\operatorname{Spec}(A)$ be a Hausdorff space?

If $A$ has positive or infinite Krull dimension, this can never happen because there are points which will be a proper subset of their closure. In dimension $0$, any Noetherian ring is also Artinian and thus has a discrete spectrum, which is therefore Hausdorff.

What about the non-Noetherian, zero-dimensional case? I suspect that there are such rings with a non-Hausdorff spec, but I failed to find an example.

• $\text{Spec}(A)$ is Hausdorff if and only if $A$ is zero-dimensional. – Qiaochu Yuan Feb 5 '12 at 18:48

For an affine scheme $S=\operatorname{Spec}(A)$ the following are equivalent:

1) $A$ is zero-dimensional
2) $S$ has all its points closed (i.e. $S$ is $T_1$)
3) $S$ is Hausdorff
4) $S$ is compact Hausdorff

If moreover the ring $A$ is noetherian the above are also equivalent to:
5) $A$ is Artinian
6) $S$ has the discrete topology
7) $S$ is finite and has the discrete topology

If moreover the ring $A$ is finitely generated over a field $k$ (and thus noetherian) the above are also equivalent to:

8) $A$ is algebraic over $k$
9) $\text {dim}_k(A)\lt \infty$
10) $\text {Card}(S)\lt \infty$

Examples of non-noetherian rings satisfying 1) to 4):
Any infinite product of any fields $A=\Pi_{i\in I} K_i$ ($I$ infinite)

Examples of noetherian rings which are not algebras over a field but satisfy 1) to 7):
$\mathbb Z/(n)$ with $n\gt 1$ and not prime.

• On the other hand, rings with their maximal spectrum (the set of maximal ideals with the relative topology) Hausdorff are far more common. For example, if $X$ is a completely regular Hausdorff space, then the ring of continuous and bounded real functions on $X$ is a ring such that the maximal spectrum is Hausdorff; this maximal spectrum is indeed homeomorphic to the Stone-Čech compactification of $X$. – egreg Mar 7 '15 at 11:27
• @alex alexeq: Zero dimensional rings, page 8 for the equivalence of 1),3),4). And $1)\leftrightarrow 2)$ is trivial by definition of Krull dimension, since closed points correspond to maximal ideals – Georges Elencwajg May 10 '19 at 9:06

Actually, any Artinian Rings have Hausdorff spectrum.

• See this for writing better answers. – ASB Mar 7 '15 at 10:44