In mathematics, the tensor product of two algebras 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. In mathematics, the tensor product of two algebras 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. 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 a ⊗ b by and then extending by linearity to all of A ⊗R B. 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 turns the category of R-algebras into a symmetric monoidal category. There are natural homomorphisms of A and B to A ⊗R B 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: The natural isomorphism is given by identifying a morphism ϕ : A ⊗ B → X {displaystyle phi :Aotimes B o X} on the left hand side with the pair of morphisms ( f , g ) {displaystyle (f,g)} on the right hand side where f ( a ) := ϕ ( a ⊗ 1 ) {displaystyle f(a):=phi (aotimes 1)} and similarly g ( b ) := ϕ ( 1 ⊗ b ) {displaystyle g(b):=phi (1otimes b)} . The tensor product of commutative algebras is of constant 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: