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Math Help - Ring homomorphism - isomorphism

  1. #1
    Junior Member
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    Exclamation Ring homomorphism - isomorphism

    Hello

    Could you please guys help me with the following exercise?

    Let  x: Z -->Zn a function defined by x(a)=[a]

    Show that  x es a homomorphism

    Is x a rings isomorphism?

    Thanks
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by osodud View Post
    Hello

    Could you please guys help me with the following exercise?

    Let  x: Z -->Zn a function defined by x(a)=[a]

    Show that  x is a homomorphism

    Is x a rings isomorphism?

    Thanks
    What do the brackets represent in this case? The floor function?

    To show that its a homomorphism, show that x(a+b)=\dots=x(a)+x(b) and x(ab)=\dots=x(a)x(b).

    Now try to see if \ker x=\{0\} and if x is onto. If both are satisfied, its a ring isomorphism.
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  3. #3
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by osodud View Post
    Hello

    Could you please guys help me with the following exercise?

    Let  x: Z -->Zn a function defined by x(a)=[a]

    Show that  x es a homomorphism

    Is x a rings isomorphism?

    Thanks
    The homomorphism part follows from the fact that addition of congruence classes modulo n is defined as [a]+[b]=[a+b] (and the operation is well defined).
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  4. #4
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by Chris L T521 View Post
    What do the brackets represent in this case? The floor function?
    The square brackets represent the equivalence class of the thing in the brackets - x, y \in [a] \iff x \equiv y \equiv a mod n (equivalently, x-y \equiv x-a \equiv y-a \equiv 0 mod n). Essentially,  [a] = a+\mathbb{Z}n (with \mathbb{Z}n = \{ni : i \in \mathbb{Z} \}).

    Hint for part 2: Notice that [a+n] = [a]...So, how many elements are in im(x), the image of the function x?
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