On pg. 13 of Atiyah-Macdonald's "Introduction to Commutative Algebra":
18.ii) Prove that $\overline{\{ x \}}=V(p_x)$
What is $\overline{\{ x \}}$? Is it the closure of prime ideal $x$? I assumed it was so, and landed up in a lot of weird conclusions.
The set of prime ideals of a commutative ring $A$ forms a topological space, it is denoted by Spec$(A)$. The closed subsets for this topology are given by $$V(I):=\{\mathfrak{p}\in Spec(A) | I \subset \mathfrak{p} \} $$
where $I$ runs through all the ideals $I\subset A$.
Now if $x$ is a point in Spec$(A)$, it is a prime ideal of $A$, let say $x=\mathfrak{p}_x$. Then $\{x\}\subset Spec(A)$, so it has a meaning to ask what is its closure $\overline{\{ x \}}$ with respect to the topology on $Spec(A)$ described above.
I understand that, especially at the beginning, the use of $x$ and $\mathfrak{p}_x$ to denote the same thing may be confusing. In any case you can rephrase everything using just a given prime ideal $\mathfrak{q}$ instead of $x$. And so the question will be: prove that $\overline{\{ \mathfrak{q} \}}=V(\mathfrak{q})$.
