Tensor product of algebras

In mathematics, the tensor product of two over a commutative ring "R" is also an "R"-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the product of algebra representations.
Definition.
Let "R" be a commutative ring and let "A" and "B" be "R"-algebras. Since "A" and "B" may both be regarded as "R"-modules, their tensor product
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 by
and then extending by linearity to all of . This ring is an "R"-algebra, associative and unital with identity element given by . where 1"A" and 1"B" 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 turns the category of "R"-algebras into a symmetric monoidal category.
Further properties.
There are natural homomorphisms from "A" and "B" to 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 formula_6 on the left hand side with the pair of morphisms formula_7 on the right hand side where formula_8 and similarly formula_9.
Applications.
The tensor product of commutative algebras is of frequent use in algebraic geometry. For affine schemes "X", "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.