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. Specifically, 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

$A\; \backslash otimes\_R\; B$

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

$(a\_1\backslash otimes\; b\_1)(a\_2\backslash otimes\; b\_2)\; =\; a\_1a\_2\backslash otimes\; b\_1b\_2$

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 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 special case R = Z gives us a tensor product of rings, since rings may be regarded as Z-algebras.

There are natural inclusions of A and B into A⊗_RB given by

$a\backslash mapsto\; a\backslash otimes\; 1\_B$

$b\backslash mapsto\; 1\_A\backslash otimes\; b$

These inclusions 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.

See also

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• tensor product of modules
• tensor product of fields

Ring theory | Commutative algebra | Multilinear algebra