Proving Artinian of Commutative Noetherian Rings with Maximal Primes

In summary, the conversation discusses the proof that a commutative Noetherian ring with all maximal primes is Artinian. The proof involves showing that any descending chain of prime ideals in the ring will eventually stabilize, and thus the ring is Artinian. However, it is noted that this is not sufficient to determine if a ring is Artinian, as demonstrated by the example of the ring $\mathbb{Z}$. It is also mentioned that every commutative Noetherian ring satisfies the descending chain condition on prime ideals, but this does not necessarily imply that the ring is Artinian.
  • #1
peteryellow
47
0
prove that a commutative noetherian ring in which all primes are maximal is artinian.
 
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  • #2
So, what have you tried?
 
  • #3
well I don't have an idea how to start...
 
  • #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.
 
  • #5
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.]
 
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Related to Proving Artinian of Commutative Noetherian Rings with Maximal Primes

1. What is an Artinian ring?

An Artinian ring is a commutative ring that satisfies the descending chain condition on ideals. This means that any chain of ideals in the ring eventually stabilizes, or reaches a point where the next ideal in the chain is equal to the previous one.

2. What is a Noetherian ring?

A Noetherian ring is a commutative ring that satisfies the ascending chain condition on ideals. This means that any chain of ideals in the ring eventually stabilizes, or reaches a point where the next ideal in the chain contains the previous one.

3. What is the connection between Artinian and Noetherian rings?

A commutative ring is both Artinian and Noetherian if and only if it satisfies the descending chain condition on ideals and the ascending chain condition on ideals. This means that any chain of ideals in the ring eventually stabilizes in both directions.

4. How do we prove that a commutative ring is Artinian?

To prove that a commutative ring is Artinian, we can use the fact that a ring is Artinian if and only if all of its prime ideals are maximal. This means that we need to show that all prime ideals in the ring are maximal, which can be done by using the properties of prime and maximal ideals.

5. Can we prove that a commutative ring is Artinian using only its maximal primes?

Yes, we can prove that a commutative ring is Artinian by showing that all of its prime ideals are maximal. This can be done using the fact that any maximal prime ideal in a commutative ring is also a maximal ideal, and the fact that a ring is Artinian if and only if all of its maximal ideals are prime.

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