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August 11th, 2012, 10:17 AM   #1
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Prime Ideal

Prove that for the ring , is prime ideal if and only if is prime.
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August 11th, 2012, 07:21 PM   #2
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Re: Prime Ideal

We suppose that is prime. Then and thus . Now let and be two integers such that .
Then we have .
Since is prime, we have or .
That means : or .
Thus is a prime ideal.

Inversely, let be an integer such that the ideal is prime.
Let and be two integers such that .
That means : and .
But : and the ideal is prime.
So : or
Hence : or .
If , then (because we already have ) and thus .
If , then (because we already have ) and thus .
Hence is prime.
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August 12th, 2012, 12:17 AM   #3
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Re: Prime Ideal

Thanks a lot.
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