is also an R-module. The tensor product can be given the structure of a ring by defining the product on elements of the form a ⊗ b by
and then extending by linearity to all of A ⊗RB. This ring is an R-algebra, associative and unital with identity element given by 1A ⊗ 1B. where 1A and 1B are the identity elements of A and B. If A and B are commutative, then the tensor product is commutative as well.
The tensor product of commutative algebras is of constant use in algebraic geometry. For affine schemesX, Y, Z with morphisms from X and Z to Y, so X = Spec(A), Y = Spec(B), and Z = Spec(C) for some commutative rings A,B,C, 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.
The tensor product can be used as a means of taking intersections of two subschemes in a scheme: consider the -algebras , , then their tensor product is , which describes the intersection of the algebraic curvesf = 0 and g = 0 in the affine plane over C.
Tensor products can be used as a means of changing coefficients. For example, and .
Tensor products also can be used for taking products of affine schemes over a field. For example, is isomorphic to the algebra which corresponds to an affine surface in if f and g are not zero.