Module theory MOC

-tensor product of modules

Let be modules over a commutative ring . The tensor product is an -module such that the -bilinear maps from are in correspondence with the -linear maps from , as defined by the Universal property.¹ A generalization for the Tensor product of modules over a noncommutative ring exists, but is not necessarily a module. We recover this notion of tensor product as the Tensor product of bimodules.

Universal property

Let be -modules. The tensor product is a pair consisting of an -module together with an -bilinear map such any -bilinear map factorizes uniquely through comm

such that is -linear.

Construction

Let be the free module on with the natural inclusion function . Let denote the -Submodule of generated by elements of the form

for all , , . 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 -bilinear. Let be -bilinear. By the universal property of the free module we have a unique -linear map such that the following commutes:

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

as required.

Properties

• A particular case is the Tensor product of vector spaces over a field
• Tensor-Hom adjunction

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tidy | en | sembr

Footnotes

1. 2009. Algebra: Chapter 0, §VIII.2.1, p. 501 ↩