Proving Artinian of Commutative Noetherian Rings with Maximal Primes

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prove that a commutative noetherian ring in which all primes are maximal is artinian.
 
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So, what have you tried?
 
well I don't have an idea how to start...
 
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.
 
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=\mathbb{Z}.) 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.]
 
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