TITLE: $\mathbb Z[1/2]$ is not finitely generated?
QUESTION [5 upvotes]: $\mathbb Z[1/2]$ is not finitely generated ?

Maybe I misunderstood, what finitely generated means. Here it says, we need finitely many elements and I think $1$ and $1/2$ suffices as generators.
But here, at the end of Proposition $5.1.4$, from the fact that $\mathbb Z[1/2]$ is not finitely generated it is concluded that $1/2$ is not integral.

REPLY [3 votes]: As a $\Bbb{Z}$-algebra, $\Bbb{Z}[1/2]$ is finitely generated, but it is not a finitely generated $\Bbb{Z}$-module!
In fact, if $A=\left\{ \frac{a_i}{2^i} \right\}_{i=1}^N$ is finite subset of $\Bbb{Z}[1/2]$, the $\Bbb{Z}$-submodule generated by $A$ does not contain powers of $\frac{1}{2}$ bigger than $\frac{1}{2^N}$.