# Tensor product of algebras

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Tensor product of algebras

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:$\left(a_1otimes b_1\right)\left(a_2otimes b_2\right) = a_1a_2otimes b_1b_2$and then extending by linearity to all of "A"&otimes;"R""B". This product is easily seen to be "R"-bilinear, associative, and unital with an identity element given by 1"A"&otimes;1"B", where 1"A" and 1"B" 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"&otimes;"R""B" given by:$amapsto aotimes 1_B$:$bmapsto 1_Aotimes b$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.

ee also

*tensor product of modules
*tensor product of fields

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