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.[1]


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 ab by[2][3]

and then extending by linearity to all of AR B. This product is R-bilinear, associative, and unital with an identity element given by 1A ⊗ 1B,[4] 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 turns the category of all R-algebras into a symmetric monoidal category.[citation needed]

Further properties[edit]

There are natural homomorphisms of A and B to A ⊗RB given by[5]

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.


  • 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)

See also[edit]


  1. ^ Lang (2002), pp. 629-631.
  2. ^ Kassel (1995), p. 32.
  3. ^ Lang 2002, pp. 629-630.
  4. ^ Kassel (1995), p. 32.
  5. ^ 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.