In mathematics, the tensor product of two R-algebras is also an R-algebra. This gives us a tensor product of algebras. The special case R = Z gives us a tensor product of rings, since rings may be regarded as Z-algebras.
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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, we may form their tensor product
which is also an R-module. We can give the tensor product the structure of an algebra by defining the product on elements of the form a ⊗ b by
and then extending by linearity to all of A ⊗R B. This product is R-bilinear, associative, and unital with an identity element given by 1A ⊗ 1B, where 1A and 1B are the identities of A and B. If A and B are both commutative then the tensor product is commutative as well.
The tensor product turns the category of all R-algebras into a symmetric monoidal category.
Further properties
There are natural homomorphisms of A and B to A ⊗R B given by
These maps make the tensor product a 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 an universal property similar to that of the coproduct:
The natural isomorphism is given by identifying a morphism
Applications
The tensor product of algebras is of constant use in algebraic geometry: working in the opposite category to that of commutative R-algebras, it provides pullbacks of affine schemes, otherwise known as fiber products.