The pullback is a subset of the cartesian product in the category of commutative rings with unit.
What is the pullback in the category of commutative $k$-algebras? Is it the same set as in rings?
The pullback is a subset of the cartesian product in the category of commutative rings with unit.
What is the pullback in the category of commutative $k$-algebras? Is it the same set as in rings?
Yes.
More generally, fix a category $\mathcal{C}$ that contains finite limits. Given an object $A \in \mathcal{C}$, we can construct the category $\mathcal{C}_A$ of objects "under $A$." That is, an object of this category $\mathcal{C}_A$ is a morphism $A \to B$ and a morphism is a commutative triangle.
The claim is that $\mathcal{C}_A$ admits finite limits, and the limits are the same as in $\mathcal{C}$. This is basically formal. Given a functor $F:I \to \mathcal{C}_A$ from a finite category $I$, we get a functor $G: I \to \mathcal{C}$ which has a limit by assumption. Moreover, for each $i \in I$, we have a morphism $A \to Gi$. By the universal property, this becomes a morphism $A \to \lim G$.
I claim that the object $A \to \lim G$ in $\mathcal{C}_A$ is a limit of $F$. This can be checked directly from the definitions.