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Show the image of an ideal is an ideal of the image

  1. May 4, 2017 #1
    1. The problem statement, all variables and given/known data
    Let ##\mu : R \to R'## be a ring homomorphism and let ##N## be an ideal of ##R##. Show that ##\mu [N]## is an ideal of ##\mu[R]##.

    2. Relevant equations


    3. The attempt at a solution
    For something to be an ideal of a ring it must be an additive subgroup ##N## such that ##aN \subseteq N## and ##Nb \subseteq N## for all ##a,b \in R##.

    Now, I know that ##\mu [N]## is a subgroup of ##R## under addition, but I don't necessarily know that it is a subgroup of ##\mu [R]##. How can I proceed if I can't establish this?
     
    Last edited: May 4, 2017
  2. jcsd
  3. May 5, 2017 #2

    andrewkirk

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    If a set A is a group and it is a subset of another group B, with the same group operation, it is a subgroup of B. Let ##A=\mu(N),\ B=\mu(R)##. First you need to show that ##\mu(R)## is a group. then you need to show that the criteria of the first sentence are met.
     
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