# 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.

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[1]

$(a_1\otimes b_1)(a_2\otimes b_2) = a_1a_2\otimes b_1b_2$

and then extending by linearity to all of A ⊗RB. This product is easily seen to be R-bilinear, associative, and unital with an identity element given by 1A ⊗ 1B,[2] where 1A and 1B are the identities of A and B. If A and B are both commutative then the tensor product is as well.

The tensor product turns the category of all R-algebras into a symmetric monoidal category.

There are natural homomorphisms of A and B to A ⊗RB given by[3]

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

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.

## Notes

1. ^ Kassel (1995), p. 32.
2. ^ Kassel (1995), p. 32.
3. ^ Kassel (1995), p. 32.

## References

• Kassel, Christian (1995), Quantum groups, Graduate texts in mathematics 155, Springer, ISBN 978-0-387-94370-1.