Tensor product of algebras

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.

Definition

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

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

and then extending by linearity to all of A ⊗_R B. This product is easily seen to be R-bilinear, associative, and unital with an identity element given by 1_A ⊗ 1_B, where 1_A and 1_B 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.

Further properties

There are natural homomorphisms of A and B to A ⊗_R B given by

$a\mapsto a\otimes 1_{B}$

$b\mapsto 1_{A}\otimes b$

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 $g(b):=\phi (1\otimes b)$.

Applications

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

• Extension of scalars
• Tensor product of modules
• Tensor product of fields
• Linearly disjoint
• Multilinear subspace learning

References

• Kassel, Christian (1995), Quantum groups, Graduate texts in mathematics, vol. 155, Springer, ISBN 978-0-387-94370-1 {{citation}}: Invalid |ref=harv (help).
• Lang, Serge (2002) [first published in 1993]. Algebra. Graduate Texts in Mathematics. Vol. 21. Springer. ISBN 0-387-95385-X. {{cite book}}: Invalid |ref=harv (help)