Fiber product in the (sub)category of commutative rings

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".

• There's definitely no way this will work in an arbitrary subcategory $\mathcal{Z}$: take such a category with only one ring and only the identity arrow for instance. Moreover, you need to change "if $A\times_C B$ is not an object of $\mathcal{Z}$" in order to admit isomorphic rings, which will also work as fiber products (otherwise you're again subject to trivialities) – Max May 28 '18 at 12:42
• I don't know whether there's a large "class of subcategories" for which this would work. Certainly the subcategories that are defined by equations (subvarieties) will. – Max May 28 '18 at 12:43
• Of course we want to understand everything up to isomorphism, i.e. "if $A×_C B$ is not isomorphic to any object of $\mathcal Z$". Then your example above doesn't work. – Layer Cake May 28 '18 at 12:50
• I trust that this fails for an arbitrary subcategory, I'd like a nice example though. – Layer Cake May 28 '18 at 12:59
• For a concrete example, I believe $A= \mathbb{Z}, B=\mathbb{Q}\times \mathbb{Z}, C= \mathbb{Q}$, $f$ the injection, $g$ the projection onto the first factor should do it. Indeed, $A\times_C B$ should be isomorphic to $\mathbb{Z}\times \mathbb{Z}$ if I'm not mistaken, which is not $A,B,C$ or $P$ up to isomorphism – Max May 28 '18 at 15:36