1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.
  2. jcsd
  3. Dec 3, 2011 #2
    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


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Don't confuse 1 (the element of Z) with 1 (the unit element of a ring).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook