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In mathematics, the tensor product of two algebras over a commutative ring *R* is also an *R*-algebra. This gives the **tensor product of algebras**. When the ring is a field, the most common application of such products is to describe the product of algebra representations.

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, their tensor product

is also an *R*-module. The tensor product can be given the structure of a ring by defining the product on elements of the form *a* ⊗ *b* by^{[1]}^{[2]}

and then extending by linearity to all of *A* ⊗_{R} *B*. This ring is an *R*-algebra, associative and unital with identity element given by 1_{A} ⊗ 1_{B}.^{[3]} where 1_{A} and 1_{B} are the identity elements of *A* and *B*. If *A* and *B* are commutative, then the tensor product is commutative as well.

The tensor product turns the category of *R*-algebras into a symmetric monoidal category.^{[citation needed]}

There are natural homomorphisms from *A* and *B* to *A* ⊗_{R} *B* given by^{[4]}

These maps make the tensor product the 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 a universal property similar to that of the coproduct:

where [-, -] denotes the commutator. The natural isomorphism is given by identifying a morphism on the left hand side with the pair of morphisms on the right hand side where and similarly .

The tensor product of commutative algebras is of frequent use in algebraic geometry. For affine schemes *X*, *Y*, *Z* with morphisms from *X* and *Z* to *Y*, so *X* = Spec(*A*), *Y* = Spec(*R*), and *Z* = Spec(*B*) for some commutative rings *A*, *R*, *B*, the fiber product scheme is the affine scheme corresponding to the tensor product of algebras:

More generally, the fiber product of schemes is defined by gluing together affine fiber products of this form.

- The tensor product can be used as a means of taking intersections of two subschemes in a scheme: consider the -algebras , , then their tensor product is , which describes the intersection of the algebraic curves
*f*= 0 and*g*= 0 in the affine plane over**C**. - More generally, if is a commutative ring and are ideals, then , with a unique isomorphism sending to .
- Tensor products can be used as a means of changing coefficients. For example, and .
- Tensor products also can be used for taking products of affine schemes over a field. For example, is isomorphic to the algebra which corresponds to an affine surface in if
*f*and*g*are not zero.