More generally, if R is a commutative ring and A and B are commutative R-algebras, we can make A_{R}B 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 'b*

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.

See also tensor product of fields.