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 and 1 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("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.