Results 1 to 3 of 3

Math Help - Show that the ideal is not prime.

  1. #1
    Junior Member
    Joined
    Sep 2010
    Posts
    42

    Show that the ideal is not prime.

    Show that the ideal (2 + \sqrt{2})  \subseteq \mathbb{Z}[ \sqrt{2} ] is not prime.

    Also, could you explain how to how show that any given ideal is not prime.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Rhymes with Orange Chris L T521's Avatar
    Joined
    May 2008
    From
    Santa Cruz, CA
    Posts
    2,844
    Thanks
    3
    Quote Originally Posted by joestevens View Post
    Show that the ideal (2 + \sqrt{2})  \subseteq \mathbb{Z}[ \sqrt{2} ] is not prime.

    Also, could you explain how to how show that any given ideal is not prime.
    Recall that an ideal P of a ring R is prime if

    * it doesn't equal the whole ring
    * If a,b\in R and ab\in P, then a\in P or b\in P.

    Now, consider the following example:

    Clearly, \sqrt{2},1+\sqrt{2}\in\mathbb{Z}[\sqrt{2}]

    So it follows that \sqrt{2}\cdot (1+\sqrt{2}) = 2+\sqrt{2}\in (2+\sqrt{2}), but \sqrt{2}\notin(2+\sqrt{2}) and 1+\sqrt{2}\notin(2+\sqrt{2}). If the ideal was prime, one of the factors \sqrt{2} or 1+\sqrt{2} must be contained in the ideal (2+\sqrt{2}), but that's not the case here.

    Therefore, (2+\sqrt{2}) is not a prime ideal in \mathbb{Z}[\sqrt{2}].

    Does this make sense?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2010
    Posts
    42
    Thanks!! That makes perfect sense
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. ideal,nil,nilpotent ideal in prime ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 24th 2011, 07:57 AM
  2. prove N is a maximal ideal iff N is a prime ideal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 20th 2011, 09:02 AM
  3. Ideal a is irreducible <--> a=p^n, p is prime ideal
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: July 3rd 2010, 10:54 PM
  4. Maximal Ideal, Prime Ideal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 28th 2008, 02:39 PM
  5. Prime ideal but not maximal ideal
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 14th 2007, 09:50 AM

Search Tags


/mathhelpforum @mathhelpforum