August 11th, 2012, 10:17 AM  #1 
Member Joined: Nov 2010 Posts: 48 Thanks: 0  Prime Ideal
Prove that for the ring , is prime ideal if and only if is prime.

August 11th, 2012, 07:21 PM  #2 
Member Joined: Aug 2011 From: Nouakchott, Mauritania Posts: 85 Thanks: 14 Math Focus: Algebra, Cryptography  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. 
August 12th, 2012, 12:17 AM  #3 
Member Joined: Nov 2010 Posts: 48 Thanks: 0  Re: Prime Ideal
Thanks a lot.


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