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Math Help - Ideals and homomorphisms

  1. #1
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    Ideals and homomorphisms

    Let I and J be ideals of a ring R and let I be contained in J.

    Show that J/I is an ideal of R/I and also show that (R/I)/(J/I) is isomorphic to R/J.

    For the second part I am just trying to use a ring homomorphism but am having trouble defining a map.

    Also if there is a better way to do the first part than to show that J/I is a sub ring and then an ideal that would be awesome.

    Thanks.
    Last edited by mr fantastic; May 22nd 2009 at 04:14 PM. Reason: Restored original question deleted by OP
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  2. #2
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    Quote Originally Posted by curiousmuch View Post
    Let I and J be ideals of a ring R and let I be contained in J.

    Show that J/I is an ideal of R/I and also show that (R/I)/(J/I) is isomorphic to R/J.

    For the second part I am just trying to use a ring homomorphism but am having trouble defining a map.

    Also if there is a better way to do the first part than to show that J/I is a sub ring and then an ideal that would be awesome.

    Thanks.
    for the first part just do what we usually do to prove something is an ideal. for the second part define f: R/I \longrightarrow R/J by f(r+I)=r+J. this map is well-defined because I \subseteq J.

    it's obviously a surjective ring homomorphism and \ker f = J/I and the result follows.
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