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I Rings Generated by Elements - Lovett, Example 5.2.1 ... ...

  1. Feb 16, 2017 #1
    I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Chapter 5 ...

    I need some help with Example 5.2.1 in Section 5.2: Rings Generated by Elements ...



    ?temp_hash=2e165cf6b1bf5270d1f5c27106307918.png



    In the Introduction to Section 5.2.1 (see text above) Lovett writes:

    " ... ... R[S] denotes the smallest (by inclusion) subring of ##A## that contains both ##R## and ##S## ... ... "


    Then, a bit later, in Example 5.2.1 concerning the ring ##\mathbb{Z} [ \frac{1}{2} ]## Lovett writes:


    " ... ... It is not hard to show that the set


    ##\{ \frac{k}{ 2^n} \ | \ k.n \in \mathbb{Z} \}##


    is a subring of ##\mathbb{Q}##. Hence, this set is precisely the ring ##\mathbb{Z} [ \frac{1}{2} ]## ... ... ... "


    BUT ...

    How has Lovett actually shown that the set ##\{ \frac{k}{ 2^n} \ | \ k.n \in \mathbb{Z} \}## as a subring of \mathbb{Q} is actually (precisely in Lovett's words) the ring ##\mathbb{Z} [ \frac{1}{2} ]## ... ...

    ... ... according to his introduction which I quoted Lovett says that the ring ##\mathbb{Z} [ \frac{1}{2} ]## is the smallest (by inclusion) subring of ##\mathbb{Q}## that contains ##\mathbb{Z}## and ##\frac{1}{2}## ... ...


    Can someone please explain to me exactly how Lovett has demonstrated this ... ...

    ... and ... if Lovett has not clearly proved this can someone please demonstrate a proof ...


    Just one further clarification ... is Lovett here dealing with ring extensions ... ... ???

    Hope someone can help ...

    Peter
     

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    Last edited by a moderator: Feb 16, 2017
  2. jcsd
  3. Feb 17, 2017 #2

    FactChecker

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    1) {k/2n | k,n∈ℤ} is a subring that contains ℤ and 1/2. Therefore, ℤ[1/2] ⊆ {k/2n | k,n∈ℤ}.
    2) Pick any element k/2n, k,n∈ℤ. For ℤ[1/2] to be a closed ring under an obvious series of operations, it must contain that element. Therefore {k/2n | k,n∈ℤ} ⊆ ℤ[1/2].
     
  4. Feb 18, 2017 #3
    Thanks FactChecker ... I think I now follow ...

    Appreciate your help ...

    Peter
     
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