- Tensor product of algebras
In

mathematics , thetensor 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:$(a\_1otimes\; b\_1)(a\_2otimes\; b\_2)\; =\; a\_1a\_2otimes\; b\_1b\_2$and then extending by linearity to all of "A"⊗_{"R"}"B". This product is easily seen to be "R"-bilinear, associative, and unital with an identity element given by 1_{"A"}⊗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"⊗

_{"R"}"B" given by:$amapsto\; aotimes\; 1\_B$:$bmapsto\; 1\_Aotimes\; b$These maps make the tensor product acoproduct 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 generalfree product of algebras).The tensor product of algebras is of constant use in

algebraic geometry : working in theopposite category to that of commutative "R"-algebras, it provides pullbacks ofaffine scheme s, otherwise known asfiber product s.**ee also***

tensor product of modules

*tensor product of fields

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