# Tensor product terminology in category theory?

Say that I have any homomorphisms of commutative rings, $$A \rightarrow B, A \rightarrow C.$$ I recently read that $$B \otimes_A C$$ is the pushout of the morphisms in the category of commutative rings. However, I understood tensor products as defined for modules over a commutative ring. Can we somehow realize $$B, C$$ as modules over $$A$$ via the homomorphisms or is $$B \otimes_A C$$ as defined by the pushout a generalization of the 'normal' definition of tensors? It seems unlikely that it is a generalization as it seems to depend on these morphisms whereas the tensors of modules doesn't depend on any sort of morphisms.

Take all rings here to be commutative. A ring homomorphism $$f:A\to B$$ makes $$B$$ into an $$A$$-module. In detail, the module action is $$a\cdot b=f(a)b$$. With another ring homomorphism $$g:A\to C$$ then we have two $$A$$-modules, and can form the tensor product $$B\otimes_A C$$.
At first $$B\otimes_A C$$ is just a module. But it has a multiplication, defined as the composition $$(B\otimes_A C)\times(B\otimes_A C)\to (B\otimes_A C)\otimes_A (B\otimes_A C) \to (B\otimes_A B)\otimes_A (C\otimes_A C)\to B\otimes_A C.$$ The middle map is just permuting the factors, and the last map is induced by $$(b\otimes b')\otimes(c\otimes c')\mapsto bb'\otimes cc'$$. In terms of elements: $$(b\otimes c)(b'\otimes c')=bb'\otimes cc'.$$
Then $$B\otimes_A C$$ is a ring. There are ring homomorphisms from $$B$$ and $$C$$ to it, the first given by $$a\mapsto a\otimes 1_C$$. Now one sits down with a large sheet of paper, and proves that the map $$A\to B\otimes_A C$$ is the pushout of $$A\to B$$ and $$A\to C$$.