1. Not finding help here? Sign up for a free 30min 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!

Ideals and Residue class ring

  1. Sep 13, 2012 #1
    1. The problem statement, all variables and given/known data
    I am curious,
    if I,J, and M are ideals of the commutative ring R, and M/I[itex]\subseteq[/itex]J/I, then M[itex]\subseteq[/itex]J


    2. Relevant equations

    M/I = { m+I : m is in M}

    J/I = { j+I : j is in J}

    I[itex]\subseteq[/itex]R is an ideal if
    1.) if a and b are in I then a+b is in I
    2.) if r is in R and a is in I then a*r is in I



    3. The attempt at a solution

    Proof by contradiction:

    Assume M/I[itex]\subseteq[/itex]J/I,
    and M is not contained in J.

    Since M is not contained in J then there exists a m in M that is not in J.
    Then m+I is in M/I but not in J/I.

    This is a contradiction to the hypothesis that M/I[itex]\subseteq[/itex]J/I. Thus M is contained in J. (QED)

    Any flaws?

    Thank you for your time.
     
    Last edited: Sep 13, 2012
  2. jcsd
  3. Sep 15, 2012 #2

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    This step isn't necessarily obvious. The coset m+I can go by many different names. How do you know it isn't the same set as j+I for some j in J?

    I suggest assuming that m+I = j+I for some j in J, and deriving a contradiction.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Ideals and Residue class ring
  1. Ring Ideals (Replies: 1)

  2. Ideals of Rings (Replies: 9)

  3. Ideals in Rings (Replies: 1)

Loading...