The following is Problem 2.21 in Gouvea's lecture notes on deformations of Galois representations:
Suppose we work in some subcategory $\mathcal Z$ of the category of commutative (unital) rings. Suppose we are given objects $A, B$ and $C$ and morphisms $\alpha\colon A \to C$ and $\beta \colon B \to C$. Let $A \times_C B$ be the ring-theoretical fiber product, i.e., the ring defined by $$A \times_C B=\{(a,b) \in A\times B \,\vert \, \alpha(a)=\beta(b) \}$$ with the natural operations. Question (Q): Is it true that if $A \times_C B$ is not an object of $\mathcal Z$ then there is no fiber product of $A$ and $B$ over $C$ in $\mathcal Z$?
Here is what I figured so far: Suppose the forgetful functor from $\mathcal Z$ to Set is representable. Then it preserves limits (including pullback). Hence the pullback should be the set-theoretic pullback with the natural commutative ring structure. Corollary: If $\mathcal Z=$Noetherian commutative rings (so the forgetful functor is represented by $\mathbb Z[X]$) the answer to (Q) is "true".