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.
There are natural homomorphisms from A and B to A ⊗RB given by
These maps make the tensor product the coproduct in the category of commutative R-algebras. The tensor product is not the coproduct in the category of all 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.
The natural isomorphism is given by identifying a morphism on the left hand side with the pair of morphisms on the right hand side where and similarly .
The tensor product of commutative algebras is of frequent use in algebraic geometry. For affine schemesX, 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.
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.
More generally, if is a commutative ring and are ideals, then , with a unique isomorphism sending to .
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.