Results 1 to 4 of 4

Math Help - Ideals in polynomials ring

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    2

    Ideals in polynomials ring

    Let K be a (algebraically closed) field.
    Exists a characterization for not trivial ideals I \subset K[x_1,...,x_n] that satisfy the following property:
    (*) \forall f,g \in I, \ gcd(f,g) \ne 1.

    If an ideal satisfy (*), then is it contained in a principal ideal?

    The problem arises from an excercise of algebraic geometry that I tried to generalize.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Apr 2010
    Posts
    78
    I think the proof of K[x] being a PID could apply here, although you can only get one direction of containment, which is what you want
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Apr 2010
    Posts
    2
    Of course, if n=1 the claim is obviously true since K[x] is a PID.
    But I asked in the case n > 1, and in this case K[x_1,...,x_n] is not a PID.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Apr 2010
    Posts
    78
    I didn't mean the exact same proof. The proof that K[x] is a PID works because you can do an induction on the degree of the polynomial. In the multivariable case you can set up a lexicographic order on the \prod_{i=1}^n x_i^{c_i} by ordering the vector (c_1,...,c_n). This is Grobner basis theory, btw.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Ideals of a ring
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: October 17th 2011, 08:14 PM
  2. Ideals of ring and isomorphic ring :)
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 24th 2009, 03:23 AM
  3. Ring Homomorphism and Ideals
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 3rd 2009, 11:01 AM
  4. Ideals in a ring R
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 17th 2008, 09:25 PM
  5. Ideals in a ring
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 17th 2008, 04:17 PM

Search Tags


/mathhelpforum @mathhelpforum