Results 1 to 3 of 3

Math Help - Principal ideals

  1. #1
    Newbie dangkhoa's Avatar
    Joined
    Nov 2009
    Posts
    11

    Principal ideals

    Let a, b in R where R is a UFD.
    Is it true that (a) intersects (b) = (c) always? Where (a),(b) and(c) are principal ideals
    How do you prove it or give a counter-example?

    Thank you for your time
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Quote Originally Posted by dangkhoa View Post
    Let a, b in R where R is a UFD.
    Is it true that (a) intersects (b) = (c) always? Where (a),(b) and(c) are principal ideals
    How do you prove it or give a counter-example?

    Thank you for your time

    Well, a UFD is also a PID and since the intersection of two ideals is always an ideal then it MUST be that (a)\cap (b)=(c)....

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2009
    Posts
    19

    Pid <--> ufd ?

    A PID is a UFD, but the converse requires more info. See (for example)

    ftp://ftp.cis.upenn.edu/pub/papers/gallier/geomath3.pdf Proposition 5.40

    All is not lost, I think. Here is a suggestion: Suppose a and b have common factor k: a=k*a1, b=k*b1. Then (I believe, and invite verification) the intersection of (a) and (b) is (k*a1*b1)=(c). This may require the ideal to be two sided.

    Hope this helps....
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Product of non principal ideals is principal.
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: June 12th 2011, 01:17 PM
  2. principal and primary ideals
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 12th 2011, 08:19 PM
  3. Principal Ideals
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 11th 2010, 06:43 AM
  4. Principal Ideals of Rationals
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: December 7th 2009, 06:09 AM
  5. When are principal ideals prime ideals?
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 5th 2008, 12:18 PM

Search Tags


/mathhelpforum @mathhelpforum