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