This problem is taken from Vakil's Foundations of Algebraic Geometry (Exercise 1.3.K. (b)):
Problem: If $\varphi: A \to B$ and $\psi: A \to C$ are two morphisms of rings, show that $B \otimes_A C$ has a natural structure of a ring.
In his notes, Vakil even gives us the multiplication: $$(b_1 \otimes c_1) (b_2 \otimes c_2) := (b_1 b_2) \otimes (c_1 c_2).$$
Now I do not understand why we need the maps in the first place. I know that $B$ and $C$ can obtain an $A$-module structure via $\varphi$ resp. $\psi$, so the notion $B \otimes_A C$ makes sense. However, why can we not just take arbitrary rings $B$ and $C$. As I tried to go through all ring axioms with the multiplication above, it seems like every axiom is still correct.
Could you let me know what I am missing here? Thanks!
You are right, you don't need the maps $A \to B$ and $A \to C$. But the ring you get will (in general) not be the same: it will be $B \otimes_{\mathbb Z} C$, associated to the (canonical) maps $\mathbb Z \to B$, $\mathbb Z \to C$.