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.
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 1A ⊗ 1B, where 1A and 1B 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 can be used as a means of taking intersections of two subschemes in a scheme: consider the -algebras , , then their tensor product is .
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 point. For example, is isomorphic to the algebra which corresponds to an affine surface in (assuming this is a general situation)