Results 1 to 3 of 3

Thread: 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 $\displaystyle (2 + \sqrt{2})  \subseteq \mathbb{Z}[ \sqrt{2} ]$ 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
    Chicago, IL
    Posts
    2,844
    Thanks
    5
    Quote Originally Posted by joestevens View Post
    Show that the ideal $\displaystyle (2 + \sqrt{2}) \subseteq \mathbb{Z}[ \sqrt{2} ]$ is not prime.

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

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

    Now, consider the following example:

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

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

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

    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: Mar 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: Jul 3rd 2010, 10:54 PM
  4. Maximal Ideal, Prime Ideal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Sep 28th 2008, 02:39 PM
  5. Prime ideal but not maximal ideal
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Nov 14th 2007, 09:50 AM

Search Tags


/mathhelpforum @mathhelpforum