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Principal and Maximal Ideals

  1. Jan 11, 2010 #1
    1. The problem statement, all variables and given/known data
    1.Prove that every ideal of Zn is principal
    2. Relevant equations
    3. The attempt at a solution

    In 1-I've proved that if K is an ideal of Zn that contains an element k then it contains all the elements of the form mk (m in Zn)...But how can I prove that there are no elements that are not of the form mk?


    Thanks in advance...
     
  2. jcsd
  3. Jan 11, 2010 #2
    Let d be the least positive integer in your ideal K. You know all integers of the form md where m is an integer is in your ideal. To prove these are the only ones assume there is some integer d' in K that is not of the form md. Then we use the division algorithm to write it on the form:
    d' = qd + r
    for some integer q and 0<r<d. Now can you show that r is in K? If you can you will have a contradiction since r is positive and less than d.
    (this general approach works for Euclidean domains in general, and shows that Euclidean domains are principal ideal domains)
     
  4. Jan 11, 2010 #3
    Well... if d' is in our ideal K then d'-qd is also in our ideal (since an ideal is also a sub-ring)... But d'-qd=qd+r-qd=r... Hence r must also be in our ideal and then we get a contradiction :)

    THanks a lot!
     
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