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Homework Help: Ideals of Rings

  1. Apr 22, 2010 #1
    1. The problem statement, all variables and given/known data
    If R = Zn, show that every ideal of R has the form mR for some integer m.


    2. Relevant equations
    --


    3. The attempt at a solution
    Well, by a previous problem I showed mR is the principal ideal of the ring, but I don't know if that's relevant. I was given the hint to try to use GCDs somehow, but I really have no ideas.

    thanks so much!
     
  2. jcsd
  3. Apr 22, 2010 #2
    Use the division algorithm.
     
  4. Apr 22, 2010 #3
    You mean, say that n = qd + r, or for m?
    Sorry, I'm totally lost on this problem.
     
  5. Apr 22, 2010 #4

    TMM

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    Suppose you have an ideal of the form mR+nR. How can you express this as a principal ideal?
     
  6. Apr 22, 2010 #5
    Let a be the smallest positive element in your ideal I. If you have some element x in I, then x = aq + s for some s less then a.
     
  7. Apr 22, 2010 #6
    Would it be (m+n)R?
     
  8. Apr 22, 2010 #7
    So from here...can I say that s must equal 0 and x = aq for all x, since otherwise it's a contradiction because a is the smallest element?
     
  9. Apr 22, 2010 #8
    Yes. You can also reproduce this proof for other rings such as the Eisenstein integers and the Gaussian integers.
     
  10. Apr 22, 2010 #9
    Thank you so much for walking me through! :)
     
  11. Apr 22, 2010 #10
    No problem, cheers.
     
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