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I Rings of Fractions ... Lovett, Section 6.2 ...

  1. Mar 11, 2017 #1
    I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Section 6.2: Rings of Fractions ...

    I need some help with some remarks following Definition 6.2.4 ... ... ...

    The remarks following Definition 6.2.4 reads as follows:



    ?temp_hash=fb73158f1e161ee8cd97c4fd29851acd.png




    In the above text from Lovett we read the following:

    " ... ... it is not hard to show that if we had taken ##D = { \mathbb{Z} }^{ \gt 0 }## we would get a ring of fractions that is that is isomorphic to ## \mathbb{Q}##. ... ... "


    Can someone please help me to understand this statement ... how is such an isomorphism possible ... in particular, how does one achieve a one-to-one and onto homomorphism from the positive integers to the negative elements of ##\mathbb{Q}## as well as the positive elements ...

    Hope someone can help ... ...

    Peter


    ==============================================================================

    To enable readers to understand Lovett's approach to the rings of fraction construction, I am providing Lovett Section 6.2 up to an including the remarks following Definition 6.2.4 ... as follows:


    ?temp_hash=fb73158f1e161ee8cd97c4fd29851acd.png
    ?temp_hash=fb73158f1e161ee8cd97c4fd29851acd.png
    ?temp_hash=fb73158f1e161ee8cd97c4fd29851acd.png
     
  2. jcsd
  3. Mar 11, 2017 #2
    There is no isomorphism between D and Q, but an isomorphism between a set of equivalence classes of pairs (r,d) (where r is in R and d is in D ) and Q.
    The equivalence class containing all pairs (-n,2n) will map to -1/2, for example.
     
  4. Mar 11, 2017 #3
    Hi willem2

    Thanks for the help ...

    Obviously I should have read the text more carefully ...

    Thanks again ...

    Peter
     
  5. Mar 28, 2017 #4

    WWGD

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    Should be Lovett and Leavitt or Lovett and Leavitt ;).
     
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