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

  • #1
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Homework Statement


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]##.

Homework Equations




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?
 
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Answers and Replies

  • #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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