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Thread: [SOLVED] Notation: What is this group?

  1. #1
    MHF Contributor Swlabr's Avatar
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    [SOLVED] Notation: What is this group?

    Can someone please tell me what the group $\displaystyle \mathbb{Z}[1/n]$ is?

    Never mind - I was having a dull moment. If you are interested, it is just the integers in another guise...
    Last edited by Swlabr; Mar 5th 2010 at 03:17 AM.
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    Quote Originally Posted by Swlabr View Post
    Can someone please tell me what the group $\displaystyle \mathbb{Z}[1/n]$ is?

    Never mind - I was having a dull moment. If you are interested, it is just the integers in another guise...

    Well, not precisely...if $\displaystyle n=2$ , for example, and since the notation $\displaystyle \mathbb{Z}[1\slash n]$ usually denotes the set of all the values $\displaystyle p(1\slash n)\,,\,\,with\,\,\,p(x)\in\mathbb{Z}[x]$ , then

    with $\displaystyle p(x)=x$ we obtain $\displaystyle p(1\slash 2)=1\slash 2\notin\mathbb{Z}$ , so in fact $\displaystyle \mathbb{Z}[1\slash n]\neq \mathbb{Z}$

    Tonio
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by tonio View Post
    Well, not precisely...if $\displaystyle n=2$ , for example, and since the notation $\displaystyle \mathbb{Z}[1\slash n]$ usually denotes the set of all the values $\displaystyle p(1\slash n)\,,\,\,with\,\,\,p(x)\in\mathbb{Z}[x]$ , then

    with $\displaystyle p(x)=x$ we obtain $\displaystyle p(1\slash 2)=1\slash 2\notin\mathbb{Z}$ , so in fact $\displaystyle \mathbb{Z}[1\slash n]\neq \mathbb{Z}$

    Tonio
    I took it to be the set $\displaystyle \{\frac{a}{n} \mid a \in \mathbb{Z}\}$ (which is what you are taking it to be, I believe). So when I said "another guise" I meant "isomorphic" not "equal".

    $\displaystyle \frac{a}{n} \mapsto a$.
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