Tensor product of algebras

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In mathematics, the tensor product of two R-algebras is also an R-algebra in a natural way. 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.

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

A \otimes_R B

which is also an R-module. We can give the tensor product the structure of an algebra by defining

(a_1\otimes b_1)(a_2\otimes b_2) = a_1a_2\otimes b_1b_2

and then extending by linearity to all of A⊗RB. This product is easily seen to be 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 as well.

The tensor product turns the category of all R-algebras into a symmetric monoidal category.

There are natural homomorphisms of A and B to A⊗RB given by

a\mapsto a\otimes 1_B
b\mapsto 1_A\otimes b

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).

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.

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