Module theory MOC

K-tensor product of modules

Let 𝑀,𝑁 be modules over a commutative ring K. The tensor product 𝑀 ⊗K𝑁 is an K-module such that the K-bilinear maps from 𝑀 ×𝑁 are in correspondence with the K-linear maps from 𝑀 ⊗K𝑁, 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 𝑀,𝑁 be K-modules. The tensor product is a pair consisting of an K-module 𝑀 ⊗K𝑁 together with an K-bilinear map ( ⊗) :𝑀 ×𝑁 →𝑀 ⊗K𝑁 such any K-bilinear map 𝜑 :𝑀 ×𝑁 →𝑃 factorizes uniquely through ( ⊗) comm

such that ――𝜑 is K-linear.

Construction

Let 𝑅(𝑀×𝑁) be the free module on 𝑀 ×𝑁 with the natural inclusion function 𝜄 :𝑀 ×𝑁 ↪𝑅(𝑀×𝑁). Let 𝐾 denote the K-Submodule of 𝑅(𝑀×𝑁) generated by elements of the form

𝜄(𝑚,𝛼𝑛1+𝛽𝑛2)−𝛼𝜄(𝑚,𝑛1)−𝛽𝜄(𝑚,𝑛2);𝜄(𝛼𝑚1+𝛽𝑚2,𝑛)−𝛼𝜄(𝑚1,𝑛)−𝛽𝜄(𝑚2,𝑛);

for all 𝑚,𝑚1,𝑚2 ∈𝑀, 𝑛,𝑛1,𝑛2 ∈𝑁, 𝛼,𝛽 ∈𝑅. We construct the tensor product as the quotient module

𝑀⊗K𝑁=K(𝑀×𝑁)/𝐾

with its natural projection 𝜋 :K(𝑀×𝑁) ↠𝑀 ⊗K𝑁, so that the map

(⊗)=𝜋∘𝜄:𝑀×𝑁→𝑀⊗K𝑁

Proof of the universal property

By construction ( ⊗) is K-bilinear. Let 𝜑 :𝑀 ×𝑁 →𝑃 be K-bilinear. By the universal property of the free module we have a unique K-linear map ˜𝜑 such that the following commutes:

and by the K-bilinearity of 𝜑 it follows 𝐾 ≤𝑅𝖬𝗈𝖽ker⁡˜𝜑, so by the universal property of the quotient module ˜𝜑 factors uniquely through 𝜋, 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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Footnotes

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