Ring notation, $R=\mathbb{Z}[1/N]$

Question

Regarding the notation $R=\mathbb{Z}[1/N]$, where $N$ is a positive integer, does $R$ refer to:

$R=\{a+b/N|a,b\in\mathbb{Z}\}$

or

$R=\{a_0 +a_1/N+a_2/N^2+\ldots +a_n/N^n|a_i\in\mathbb{Z}\}$

or others?

Thanks a lot.

Answer 1

As the comment suggests it's the second option, where you maybe should specify that $n\in\mathbb N$ may vary. You can think about it differently. Either it's "polynomials" in $\frac 1N$ with coefficients in $\mathbb Z$ or it is the smallest ring which contains $\mathbb Z$ and $\frac 1N$.