Tensor product of modules over a noncommutative ring

Unlike in the special case of the -tensor product of modules, the general tensor product of modules may itself lack module structure. Let be a (noncommutative) ring, be a right -module and be a left -module. The tensor product is an abelian group such that the -balanced maps from are in correspondence with the group homomorphisms from , as defined by the Universal property.

Universal property

Let be a right -module and be a left -module. The tensor product is a pair consisting of an abelian group together with an -balanced map such that any -balanced map factorizes uniquely through #m/def/module

such that is a group homomorphism.

Construction

Let be a free -module free abelian group on with the natural inclusion function . Let denote the -Submodule (subgroup) of generated by any elements of the form

for any , , . We construct the tensor product as the quotient -module

with its natural projection , so that the map

Proof of the universal property

By construction is -balanced. Let be -balanced. By the universal property of the free module we have a unique -linear map such that the following commutes:

and by the -balance of it follows , so by the universal property of the quotient module factors uniquely through , yielding the commutative diagram

as required.

Tensor product of bimodules

Note that if is a -bimodule and is a -bimodule then is naturally equipped with the structure of a -bimodule. If is commutative, then we recover the -tensor product of modules by considering -bimodules and this way.

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