Tensor product of algebras

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.

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 ⊗_RB. This product is 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 commutative 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

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:

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