## 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:

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

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

$mr\otimes n=m\otimes rn$