Results 1 to 6 of 6

Math Help - Prime ideal

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    10

    Prime ideal

    I need help with starting this problem. It says Show that the principal ideal in R[x] generated by x^3 - x^2 + x - 1 is not prime in R[x].


    I know that a prime ideal I in a ring R is an ideal such that a,b are elements of R and ab is an element of I implies that either a or b is an element of I. But how do I apply this to this problem.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by empressA88 View Post
    I need help with starting this problem. It says Show that the principal ideal in R[x] generated by x^3 - x^2 + x - 1 is not prime in R[x].


    I know that a prime ideal I in a ring R is an ideal such that a,b are elements of R and ab is an element of I implies that either a or b is an element of I. But how do I apply this to this problem.

    Note that x^3-x^2+x-1=(x-1)(x^2+1)\Longrightarrow (x-1)(x^2+1)\in <x^3-x^2+x-1> but x-1,\,x^2+1\notin <x^3-x^2+x-1>

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    10
    x-1,\,x^2+1\notin <x^3-x^2+x-1>

    Tonio[/QUOTE]

    I understand everything else besides this part.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by empressA88 View Post
    x-1,\,x^2+1\notin <x^3-x^2+x-1>

    Tonio
    I understand everything else besides this part.[/quote]


    The polynomials x-1\,\,\,and\,\,\,x^2+1 do not belong to the ideal generated by x^3-x^2+x-1 ...this is very easy to check if you first characterize the pol's that do belong to the ideal.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Apr 2010
    Posts
    10
    ok..let me see if am on the same page..the principal ideals are x-1, x^2 + 1?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Apr 2010
    Posts
    10
    ok..i get where you are coming from. Sorry..thank you
    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. Prime ideal or maximal ideal
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: January 21st 2010, 05:42 AM
  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