# Tensor product of algebras

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{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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

${\displaystyle A\otimes _{R}B}$

which is also an R-module. We can give the tensor product the structure of an algebra by defining[1]

${\displaystyle (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=a_{1}a_{2}\otimes b_{1}b_{2}}$

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]

${\displaystyle a\mapsto a\otimes 1_{B}}$
${\displaystyle b\mapsto 1_{A}\otimes b}$

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:

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

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

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.