Undergrad 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.