Proving Artinian of Commutative Noetherian Rings with Maximal Primes

[peteryellow]

prove that a commutative noetherian ring in which all primes are maximal is artinian.

 

[morphism]

So, what have you tried?

 

[peteryellow]

well I don't have an idea how to start...

 

[peteryellow]

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.

 

[morphism]

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.]