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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


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    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.
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