# Tensor product of algebras

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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^{[1]}

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},^{[2]} 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^{[3]}

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:

The natural isomorphism is given by identifying a morphism on the left hand side with the pair of morphism on the right hand side where and similarly .

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

- Extension of scalars
- Tensor product of modules
- Tensor product of fields
- Linearly disjoint
- Multilinear subspace learning

## Notes

## References

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