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Math Help - counter example

  1. #1
    ux0
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    counter example

    i just need a counter example, this has been bothering me all day...!

    If R and S are isomorphic commutative rings, then any ring homomorphism f: R \to S is an isomorphism.



    I wanted to say....

    if m \geq 2, then the map f:\mathbb{Z} \to \mathbb{Z}_m, given by f(n) = [n], is not injective, but it is surjective....... but my two rings don't seem to be isomophic..
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  2. #2
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    Quote Originally Posted by ux0 View Post
    i just need a counter example, this has been bothering me all day...!

    If R and S are isomorphic commutative rings, then any ring homomorphism f: R \to S is an isomorphism.
    for any ring R define the map f: R[x] \longrightarrow R[x] by f(a_0 + a_1 x + \cdots + a_n x^n)=a_0.
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  3. #3
    Newbie lepton's Avatar
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    The trivial map, sending all things to the (respective) identity, is an not an isomorphism between a ring and itself.
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