1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Ideals with subsets and divides

  1. Oct 22, 2008 #1
    1. The problem statement, all variables and given/known data
    Let I = <f(x)>, J =<g(x)> be ideals in F[x]. prove that I[tex]\subset[/tex]J [tex]\leftrightarrow[/tex] g(x)|f(x)

    2. Relevant equations



    3. The attempt at a solution
    If I is a subset of J then does that mean that f is in J also and by definition of an ideal g*some b in J must equal something in J so g|f? because g|f means that f=bg for some b in J
     
  2. jcsd
  3. Oct 22, 2008 #2
    Correct.

    Sort of. I would just use the fact that J is generated by g(x).
     
  4. Oct 23, 2008 #3
    what does it mean that J is generated by g(x)? in layman's terms
     
  5. Oct 23, 2008 #4
    It means J = {a(x)g(x) : a(x) in F[x]}, in other words J is the set of all "multiples" of g(x).
     
  6. Oct 23, 2008 #5
    So if f(x) is in J and J = {a(x)g(x): a(x) in F[x]} then f(x) = a(x)g(x) therefore g(x)|f(x)?
     
  7. Oct 23, 2008 #6
    Correct.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook




Loading...
Similar Threads for Ideals subsets divides Date
Direct Products of Rings and Ideals ... Bland Problem 2(c) Saturday at 3:34 AM
Direct Products of Rings and Ideals ... Bland Problem 2(a) Apr 8, 2018
Powers of Prime Ideals Feb 22, 2018
Subsets in R^3 Jan 16, 2018
Primary Ideals in a PID Nov 19, 2017