Tensor product of algebras

This ring is an R-algebra, associative and unital with the identity element given by 1A ⊗ 1B.

[citation needed] There are natural homomorphisms from A and B to A ⊗R B given by[4] These maps make the tensor product the coproduct in the category of commutative R-algebras.

There the coproduct is given by a more general free product of algebras.

Nevertheless, the tensor product of non-commutative algebras can be described by a universal property similar to that of the coproduct: where [-, -] denotes the commutator.

on the left hand side with the pair of morphisms

For affine schemes X, Y, Z with morphisms from X and Z to Y, so X = Spec(A), Y = Spec(R), and Z = Spec(B) for some commutative rings A, R, B, the fiber product scheme is the affine scheme corresponding to the tensor product of algebras: More generally, the fiber product of schemes is defined by gluing together affine fiber products of this form.