Meaning of the notations: ##\mathbb{Z}[\frac{1}{a}]##

[elias001]

The following is taken from Abstract Algebra: A First Course by Stephen Lovett

Background

Exercise Let $D=\{2^a 3^b\mid a,b,\mathbb{N}\}$ as a subset of $\mathbb{Z}$. Prove that $D^{-1}\mathbb{Z}$ is isomorphic to $\mathbb{Z}[\frac{1}{6}]$ even though $D\neq\{1,6,6^2,\ldots\}$


Question:

I would like to know how is the notation: $\mathbb{Z}[\frac{1}{6}]$ or $\mathbb{Z}[\frac{1}{a}]$ defined? Also I have seen $\mathbb{Z}[\frac{1}{2},\frac{1}{3}]$, and I would like to know how that is also defined.


Thank you in advance.




Thank you in advance

 

[fresh_42]

$\mathbb{Z}\left[\frac{1}{6}\right]$ are all polynomials in $1/6$ with integer coefficients. It is what you get if you substitute $x=1/6$ in polynomials from $\mathbb{Z}[x].$

The same for $\mathbb{Z}\left[\frac{1}{a}\, , \,\frac{1}{3}\right].$ Take $\mathbb{Z}[x,y]$ and replace $x=1/a$ and $y=1/3.$