Results 1 to 2 of 2

Math Help - Maximal Ideal

  1. #1
    Junior Member
    Joined
    May 2008
    Posts
    70

    Maximal Ideal

    Let N = {f in Z[x] | f has constant term 0}, clearly N is an ideal of Z[x], show that N is a maximal ideal of Z[x].
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by Coda202 View Post
    Let N = {f in Z[x] | f has constant term 0}, clearly N is an ideal of Z[x], show that N is a maximal ideal of Z[x].
    I don't beleive it is a maximal Ideal... Here is why

    Define a Homomorphism from \phi:\mathbb{Z}[x] \to \mathbb{Z} by \phi(f) \to f(0)

    Note that the kernel of \phi is N

    \phi is onto so we can use the first isomorphism theorem to get

    \mathbb{Z}/N \cong \mathbb{Z}

    Since \mathbb{Z} is an Integral Domain this implies that The Ideal is Prime and not maximal.

    This ideal in \mathbb{Z}[x] is contained in the Ideal gentered by (2, and the same set f) and this ideal is maximal
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. 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, 10:02 AM
  2. Maximal Ideal
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 27th 2010, 01:23 AM
  3. Prime ideal or maximal ideal
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: January 21st 2010, 06:42 AM
  4. Maximal Ideal, Prime Ideal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 28th 2008, 03:39 PM
  5. Prime ideal but not maximal ideal
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 14th 2007, 10:50 AM

Search Tags


/mathhelpforum @mathhelpforum