Definition of Tensor Product: M Tensor N

Let R be a commutative ring with 1 and let M and N be R-modules. This blog post will be about what is the R-module M\otimes_R N. The source of the material will primarily come from Abstract Algebra, 3rd Edition (by Dummit and Foote) which is a highly recommended Algebra book for undergraduates.

Motivation

The tensor product is a construction that, roughly speaking, allows us to take “products” mn of elements m\in M and n\in N.

Definition

(Following Dummit; there is another equivalent definition using “universal property”, see Wikipedia)

M\otimes_R N is the quotient of the free \mathbb{Z}-module over M\times N (also called module of the formal linear combinations of elements of M\times N) by the subgroup generated by elements of the form:

(m_1+m_2,n)-(m_1,n)-(m_2,n)

(m,n_1+n_2)-(m,n_1)-(m,n_2)

(mr,n)-(m,rn)

The outcome of the above definition is that the following nice properties hold:

(m_1+m_2)\otimes n=m_1\otimes n+m_2\otimes n

m\otimes (n_1+n_2)=m\otimes n_1+m\otimes n_2

mr\otimes n=m\otimes rn

 

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