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Math Help - Help Proving a Theorem (Ring Isomorphisms)

  1. #1
    Junior Member ginafara's Avatar
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    Help Proving a Theorem (Ring Isomorphisms)

    Can someone help me get started. I start by assuming 1-1 and know that i go the other direction, but can't seem to see how to use 1-1 to show anything...Thanks in advance.

    Theorem:
    Let \Phi :R \rightarrow S be a ring homomorphism. Show that \Phi is 1-1 iff ker(\Phi) is trivial
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  2. #2
    Lord of certain Rings
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    Quote Originally Posted by ginafara View Post
    Can someone help me get started. I start by assuming 1-1 and know that i go the other direction, but can't seem to see how to use 1-1 to show anything...Thanks in advance.

    Theorem:
    Let \phi :R \rightarrow S be a ring homomorphism. Show that \phi is 1-1 iff ker(\phi) is trivial
    1-1 \Rightarrow ker(\phi) is trivial

    Since for any ring, \phi(0) = 0, 1-1 nature of \phi forces 0 to be the only number of R to map to the 0 of S. Thus \phi(x) = 0 only for x=0. Hence ker(\phi) is trivial

    ker(\phi) = \{0\} \Rightarrow \phi is 1-1

    For any ring, \phi(0) = 0. If \phi(x) = \phi(y), then \phi(x-y) = 0. But since ker(\phi) = \{0\}, x=y
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