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 𝑅-tensor product of modules 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 ( ⊗) #m/def/module/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

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

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

Properties

• A particular case is the 𝕂-tensor product of vector spaces
• Tensor-Hom adjunction

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