When affine schemes are separated?

Question

Let $X = \mathrm{Spec} A$ be an affine scheme where $A$ is a $k$-algebra. Is there any nice if and only if condition for $A$ such that $X$ is separated scheme over $k$? I know that every affine scheme is quasi-separated.

Answer 1

Any morphism of affine schemes is separated.

Recall that being separated means that the diagonal is a closed immersion. For a ring morphism $A\to B$, the diagonal morphism $\operatorname{Spec}(B)\to \operatorname{Spec}(B\otimes_A B)$ of the induced morphism of schemes $\operatorname{Spec}(B)\to \operatorname{Spec}(A)$ corresponds to the ring morphism $B\otimes_A B\to B$ given by multiplication. Since the latter is surjective, the corresponding map on schemes is a closed immersion.