Tensor product of algebras

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 ab by[1][2]

and then extending by linearity to all of AR B. This ring is an R-algebra, associative and unital with identity element given by 1A ⊗ 1B.[3] where 1A and 1B 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.

Further properties

There are natural homomorphisms of A and B to ARB 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 constant use in algebraic geometry. For affine schemes X, Y, Z with morphisms from X and Z to Y, so X = Spec(A), Y = Spec(B), and Z = Spec(C) for some commutative rings A,B,C, 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.
  • 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.

See also


  1. Kassel (1995), p. 32.
  2. Lang 2002, pp. 629-630.
  3. Kassel (1995), p. 32.
  4. Kassel (1995), p. 32.


  • Kassel, Christian (1995), Quantum groups, Graduate texts in mathematics, 155, Springer, ISBN 978-0-387-94370-1.
  • Lang, Serge (2002) [first published in 1993]. Algebra. Graduate Texts in Mathematics. 21. Springer. ISBN 0-387-95385-X.
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