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

A \otimes_R B ,

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]

(a_1\otimes b_1)(a_2\otimes b_2) = a_1 a_2\otimes b_1b_2

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]

a\mapsto a\otimes 1_B
b\mapsto 1_A\otimes b

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:

Hom(A\otimes B,X) \cong \lbrace (f,g)\in Hom(A,X)\times Hom(B,X) \mid \forall a\in A, b\in B: [f(a),g(b)] = 0\rbrace

The natural isomorphism is given by identifying a morphism \phi:A\otimes B\to X on the left hand side with the pair of morphism (f,g) on the right hand side where f(a):=\phi(a\otimes 1) and similarly g(b):=\phi(1\otimes b).


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[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.