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Noetherian ring

  1. Aug 17, 2008 #1
    prove that a commutative noetherian ring in which all primes are maximal is artinian.
     
  2. jcsd
  3. Aug 17, 2008 #2

    morphism

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    So, what have you tried?
     
  4. Aug 17, 2008 #3
    well I dont have an idea how to start...
     
  5. Aug 17, 2008 #4
    Let there be given a decending chain of prime ideals I_1 \supset I_2 \supset I_3 \supset I_4 \supset.....


    Since all primes are maximal therefore for a natural number n we have I_n =I_{n+1}. Hence the ring is artinian.

    Is it correct? Please help thanks.
     
  6. Aug 18, 2008 #5

    morphism

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    Why is it sufficient to look at descending chains of prime ideals? Is it true that if a ring R satisfies the descending chain condition on its prime ideals then R is Artinian? (No: take R=[itex]\mathbb{Z}[/itex].) Also, how did you conclude that I_n = I_{n+1}? This doesn't follow from maximality.

    Try again!

    [Side note: Incidentally, one can prove that every commutative Noetherian ring satisfies the descending chain condition on prime ideals. So if this were sufficient to determine if a ring is Artinian, then we would be able to conclude that every commutative Noetherian ring is Artinian, which is definitely not the case.]
     
    Last edited: Aug 18, 2008
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