Tensor product of algebras
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In mathematics, the tensor product of two algebras over a commutative ring R is also an Ralgebra. 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[edit]
Let R be a commutative ring and let A and B be Ralgebras. Since A and B may both be regarded as Rmodules, their tensor product
is also an Rmodule. 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]}
and then extending by linearity to all of A ⊗_{R} B. This ring is an Ralgebra, 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 Ralgebras into a symmetric monoidal category.^{[citation needed]}
Further properties[edit]
There are natural homomorphisms from A and B to A ⊗_{R} B given by^{[4]}
These maps make the tensor product the coproduct in the category of commutative Ralgebras. The tensor product is not the coproduct in the category of all Ralgebras. There the coproduct is given by a more general free product of algebras. Nevertheless, the tensor product of noncommutative algebras can be described by a universal property similar to that of the coproduct:
where [, ] denotes the commutator. The natural isomorphism is given by identifying a morphism on the left hand side with the pair of morphisms on the right hand side where and similarly .
Applications[edit]
The tensor product of commutative algebras is of constant use in algebraic geometry. For affine schemes X, Y, Z with morphisms from X and Z to Y, so X = Spec(A), Y = Spec(B), and Z = Spec(C) for some commutative rings A, B, C, the fiber product scheme is the affine scheme corresponding to the tensor product of algebras:
More generally, the fiber product of schemes is defined by gluing together affine fiber products of this form.
Examples[edit]
 The tensor product can be used as a means of taking intersections of two subschemes in a scheme: consider the algebras , , then their tensor product is , which describes the intersection of the algebraic curves f = 0 and g = 0 in the affine plane over C.
 Tensor products can be used as a means of changing coefficients. For example, and .
 Tensor products also can be used for taking products of affine schemes over a field. For example, is isomorphic to the algebra which corresponds to an affine surface in if f and g are not zero.
See also[edit]
 Extension of scalars
 Tensor product of modules
 Tensor product of fields
 Linearly disjoint
 Multilinear subspace learning
Notes[edit]
References[edit]
 Kassel, Christian (1995), Quantum groups, Graduate texts in mathematics, 155, Springer, ISBN 9780387943701CS1 maint: ref=harv (link).
 Lang, Serge (2002) [first published in 1993]. Algebra. Graduate Texts in Mathematics. 21. Springer. ISBN 038795385X.CS1 maint: ref=harv (link)