Let be a commutative ring with 1 and let and be -modules. This blog post will be about what is the -module . 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” of elements and .

## Definition

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

is the quotient of the free -module over (also called module of the formal linear combinations of elements of ) by the subgroup generated by elements of the form:

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