In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the product of algebra representations.

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, their tensor product

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

is also an R-module. The tensor product can be given the structure of a ring by defining the product on elements of the form a ⊗ b by^[1][2]

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

and then extending by linearity to all of A ⊗_R B. This ring is an R-algebra, associative and unital with identity element given by 1_A ⊗ 1_B.^[3] where 1_A and 1_B are the identity elements of A and B. If A and B are commutative, then the tensor product is commutative as well.

The tensor product turns the category of R-algebras into a symmetric monoidal category.^{[citation needed]}

Further properties

There are natural homomorphisms from A and B to A ⊗_R B given by^[4]

${\displaystyle a\mapsto a\otimes 1_{B))$

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

These maps make the tensor product the 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 a universal property similar to that of the coproduct:

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

where [-, -] denotes the commutator. 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 morphisms ${\displaystyle (f,g)}$ on the right hand side where ${\displaystyle f(a):=\phi (a\otimes 1)}$ and similarly ${\displaystyle g(b):=\phi (1\otimes b)}$.

Applications

The tensor product of commutative algebras is of frequent use in algebraic geometry. For affine schemes X, Y, Z with morphisms from X and Z to Y, so X = Spec(A), Y = Spec(R), and Z = Spec(B) for some commutative rings A, R, B, the fiber product scheme is the affine scheme corresponding to the tensor product of algebras:

${\displaystyle X\times _{Y}Z=\operatorname {Spec} (A\otimes _{R}B).}$

More generally, the fiber product of schemes is defined by gluing together affine fiber products of this form.

Examples

• The tensor product can be used as a means of taking intersections of two subschemes in a scheme: consider the ${\displaystyle \mathbb {C} [x,y]}$-algebras ${\displaystyle \mathbb {C} [x,y]/f}$, ${\displaystyle \mathbb {C} [x,y]/g}$, then their tensor product is ${\displaystyle \mathbb {C} [x,y]/(f)\otimes _{\mathbb {C} [x,y]}\mathbb {C} [x,y]/(g)\cong \mathbb {C} [x,y]/(f,g)}$, which describes the intersection of the algebraic curves f = 0 and g = 0 in the affine plane over C.
• More generally, if ${\displaystyle A}$ is a commutative ring and ${\displaystyle I,J\subseteq A}$ are ideals, then ${\displaystyle {\frac {A}{I))\otimes _{A}{\frac {A}{J))\cong {\frac {A}{I+J))}$, with a unique isomorphism sending ${\displaystyle (a+I)\otimes (b+J)}$ to ${\displaystyle (ab+I+J)}$.
• Tensor products can be used as a means of changing coefficients. For example, ${\displaystyle \mathbb {Z} [x,y]/(x^{3}+5x^{2}+x-1)\otimes _{\mathbb {Z} }\mathbb {Z} /5\cong \mathbb {Z} /5[x,y]/(x^{3}+x-1)}$ and ${\displaystyle \mathbb {Z} [x,y]/(f)\otimes _{\mathbb {Z} }\mathbb {C} \cong \mathbb {C} [x,y]/(f)}$.
• Tensor products also can be used for taking products of affine schemes over a field. For example, ${\displaystyle \mathbb {C} [x_{1},x_{2}]/(f(x))\otimes _{\mathbb {C} }\mathbb {C} [y_{1},y_{2}]/(g(y))}$ is isomorphic to the algebra ${\displaystyle \mathbb {C} [x_{1},x_{2},y_{1},y_{2}]/(f(x),g(y))}$ which corresponds to an affine surface in ${\displaystyle \mathbb {A} _{\mathbb {C} }^{4))$ if f and g are not zero.
• Given ${\displaystyle R}$-algebras ${\displaystyle A}$ and ${\displaystyle B}$ whose underlying rings are graded-commutative rings, the tensor product ${\displaystyle A\otimes _{R}B}$ becomes a graded commutative ring by defining ${\displaystyle (a\otimes b)(a'\otimes b')=(-1)^{|b||a'|}aa'\otimes bb'}$ for homogeneous ${\displaystyle a}$, ${\displaystyle a'}$, ${\displaystyle b}$, and ${\displaystyle b'}$.