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
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