# Tensor product

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The tensor product of two unitary modules and over an associative commutative ring with a unit is the -module together with an -bilinear mapping which is universal in the following sense: For any -bilinear mapping , where is an arbitrary -module, there is a unique -linear mapping such that The tensor product is uniquely defined up to a natural isomorphism. It always exists and can be constructed as the quotient module of the free -module generated by the set modulo the -submodule generated by the elements of the form     then . If one gives up the requirement of commutativity of , a construction close to the one described above allows one to form from a right -module and a left -module an Abelian group , also called the tensor product of these modules . In what follows will be assumed to be commutative.

The tensor product has the following properties:    for any -modules , and .

If and are bases of the free -modules and , then is a basis of the module . In particular, if the are free finitely-generated modules (for instance, finite-dimensional vector spaces over a field ). The tensor product of cyclic -modules is computed by the formula where and are ideals in .

One also defines the tensor product of arbitrary (not necessarily finite) families of -modules. The tensor product is called the -th tensor power of the -module ; its elements are the contravariant tensors (cf. Tensor on a vector space) of degree on .

To any pair of homomorphisms of -modules , , corresponds their tensor product , which is a homomorphism of -modules and is defined by the formula This operation can also be extended to arbitrary families of homomorphisms and has functorial properties (see Module). It defines a homomorphism of -modules  which is an isomorphism if all the and are free and finitely generated.

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How to Cite This Entry:
Tensor product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tensor_product&oldid=12437
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article