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PID math problem

  1. Mar 30, 2007 #1
    1. The problem statement, all variables and given/known data
    If R is a PID then all its ideals can be generated by a single element. It dosen't imply that every element in R can generate an ideal does it?




    3. The attempt at a solution
    I have to agree that it dosen't but can't think of a proof.
     
  2. jcsd
  3. Mar 30, 2007 #2

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    Of course in any ring, any element can generate an ideal, the ideal generated by that element.
     
  4. Mar 30, 2007 #3
    You are right. Each element can always generate an ideal, namely a principle ideal. If R is a PID than it tells us that all the ideals in R are those that have been generated by a single element hence are all principle ideals.

    However one can turn any principle ideal into a nonprinciple ideal. i.e. in Z. <3> is a Principle ideal. Instead of denoting it by <3> we denote this ideal by {0, 3} or {3, 6} and either of those two will be equivalent to <3>.
     
  5. Mar 30, 2007 #4

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    It's still the same ideal, and is still a principal ideal, which is just an ideal that can be generated by a single element, regardless of whichever generating set you choose to focus on.
     
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