I am trying to study the tensor products of modules using Dummit & Foote (chapter 10.4). A general outline of what they are doing goes like this (some assumptions skipped for brevity):
Given a subring $R$ of the ring $S$, and a left $R$-module $N$, we wish to "extend the scalars" by defining an $S$-module structure on $N$ that extends the action of $R$ on $N$ to an action of $S$ on $N$.
They do this by taking the free Abelian group on $S \times N$ and taking the quotient of this group by a subgroup $H$ generated by a set of rules that include "$(sr,n) – (s,rn)$" (and others, omitted for brevity) to get the tensor product of $S$ and $N$ over $R$, denoted $S \otimes_R N$.
I am having a tough time developing an intuitive regarding what exactly the elements in $H$ and $S \otimes_R N$ are. Is there a small example that I can work out explicitly? Or simple examples that will illuminate this construction?
Best Answer
Here's a simple example of scalar extension. Let $R=\mathbb Z$ and let $N$ be the $R$-module $\mathbb Z[x]$. We have a natural map $\mathbb Z = R \hookrightarrow S=\mathbb Q$. Then $S \otimes_R N = \mathbb Q \otimes_{\mathbb Z} \mathbb Z[x]=\mathbb Q [x]$.
Or another, perhaps more natural example: let $V$ be a $\mathbb R$-vector space with a basis $\{ b_1, \cdots, b_n\}$. Then $V \otimes_{\mathbb R} \mathbb C$ is the $\mathbb C$-vector space spanned by the same basis (but taking $\mathbb C$-linear combinations instead).
Notice that the set $H$ is defined exactly so that taking the quotient $(S \times N)/H$ still gives us abelian groups with a module structure.