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Abstract Algebra - Properties of Q/Z

  1. Dec 3, 2011 #1
    1. The problem statement, all variables and given/known data
    Prove that the group Q/Z under addition cannot be isomorphic to the additive group of a commutative ring with a unit element, where Q is the field of rationals and Z is the ring of integers.


    2. Relevant equations
    The tools available are introductory-level group theory and ring theory, from a first course in Abstract Algebra.


    3. The attempt at a solution
    I was thinking that it might be helpful to show that Q/Z has no unit element (since 1 is in Z), and then show that if this were true, then Q/Z must have a unit element. However, I'm not quite sure how to get started, or if I'm even taking a correct approach.
     
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  3. Dec 3, 2011 #2

    micromass

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    Basically, what they want you to show that you cannot define a multiplication on Q/Z. So, assume that you do have a multiplication (with a unity), try to derive a contradiction.
     
  4. Dec 3, 2011 #3

    Hurkyl

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    Don't confuse 1 (the element of Z) with 1 (the unit element of a ring).
     
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