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!

Abstract Algebra: a problem about ideal

  1. Mar 8, 2010 #1
    1. The problem statement, all variables and given/known data
    Let J be the set of all polynomials with zero constant term in Z[x]. (Z=integers)
    a.) Show that J is the principal ideal (x) in Z[x].
    b.) Show that Z[x]/J consists of an infinite number of distinct cosets, one for each n[tex]\in[/tex]Z.


    2. Relevant equations



    3. The attempt at a solution
    I have trouble understanding what a principal ideal is. Any help on how I should start would be great. Thanks!
     
  2. jcsd
  3. Mar 8, 2010 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Well, what's the set generated by {x} in Z(x)? Isn't it just x*Z(x)? What's that?
     
  4. Mar 8, 2010 #3
    An ideal is a subset of a ring that's closed under addition of its own elements and under multiplication by anything in the ring. A principal ideal is one that's generated by one element.

    So, take an element in Z[x] and multiply it by x. Whatever you get, by definition, is in the ideal (x). What do those things look like?

    Now, two things are in the same coset of Z[x]/J if their difference is in J. So, if I can subtract off any polynomial I want with zero constant term, what stays unchanged?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Abstract Algebra: a problem about ideal
Loading...