Tensor product of R-algebras

In mathematics, there is a construction in abstract algebra of the tensor product of commutative rings; which puts a ring structure on the tensor product as abelian groups of two commutative rings R and S. This structure on R $\otimes$ _ZS then makes it a coproduct in the category of commutative rings.

More generally, if R is a commutative ring and A and B are commutative R-algebras, we can make A $\otimes$ _RB into a commutative R-algebra by the same formula, getting a coproduct in the same way: the previous construction being the case R = Z. We observe the multilinear nature of the product a $\otimes$ b.c $\otimes$ d = ac $\otimes$ bd, to have a well-defined product on A $\otimes$ _RB; the ring axioms and R-linearity can be checked too.

This construction is of constant use in algebraic geometry: working in the opposite category to that of commutative R-algebras, it provides pullbacks of affine schemes, otherwise known as fiber products.