Working through homework problems so I would appreciate just a small nudge in the right direction. Note: all rings are commutative with unit.
I showed that pullbacks for Ab and $\mathbb{Z}$-Alg are just $[(a,b) \in A \times_{C} B: f(a) = g(b)]$, where $f: A \to C$ and $g: B \to C$. So, by dropping the ring structure via the forgetful functor, we have that pullbacks are preserved by the forgetful functor. Is this correct?
For pushouts, I am slightly stuck. In the category Ab, pushouts are simply $A \oplus B$ modded by the group generated by $\langle f(c), -g(c)\rangle $, where $f: C \to A$ and $g: C \to B$. However, in $\mathbb{Z}$-Alg, the pushout is just $A \otimes_{C} B$. My initial guess is that they are not the same since tensor products are generally much larger than direct sums. Am I heading in the right direction?