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Integral Domains

  1. Mar 13, 2017 #1
    1. The problem statement, all variables and given/known data
    Let ##R## be a principal ideal domain and suppose ##I_1,I_2,....## are ideals of ##R## with
    ## I_1 \subseteq I_2 \subseteq I_3 \subseteq ....##
    The Question has two parts: 1. to show that ##\cup _{i=0}^{\infty}I_i## is an ideal.
    2. to show that any ascending as above must stabilize, i.e. there is a positive integer ##n## with ##I_n=I_{n+1}=...##

    2. Relevant equations


    3. The attempt at a solution
    My problem is with the second question. I tried to assume for contradiction that for every positive integer ##n##, we have ##I_n \subsetneq I_{n+1}## which mean that there is a number ##x\in I_{n+1}## which is not in ##I_n##. Since we are in a PID, we can write ##I_n = (d), \quad I_{n+1}=(e)## ( where ##d,e## are the generators). I also got that ##d \nmid x##, and I tried to write ##\gcd(x,d)## as a linear combination of them... I have ran out of ideas...

    Any hint will be helpful!
    Thank you.
     
  2. jcsd
  3. Mar 13, 2017 #2

    fresh_42

    Staff: Mentor

    Doesn't part one help you out here?
     
  4. Mar 13, 2017 #3
    Unfortunately, not too much. The closest relationship between ##x## and ##d## that I have is ##x(1-s)=r\cdot d\cdot n##, where ##r,s## came from ##\gcd(x,d)=sx+rd##, and ##n## came from ##\gcd(x,d) \cdot n=x##

    Is it possible that it is related to the fact that we have unique factorization in PID?
     
    Last edited: Mar 13, 2017
  5. Mar 13, 2017 #4

    fresh_42

    Staff: Mentor

    If you can use this result, then it's the step in the right direction.
     
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