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

elias001
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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
 
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\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.
 
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