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Math Help - prime fields

  1. #1
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    prime fields

    Let R be a ring with unity 1 and n an integer. Define

    n 1 =

    {1 + 1 + . . . + 1 to n terms if n > 0

    0 if n = 0

    1 + (1) + . . . + (1) to |n| terms if n < 0


    (a) Show that ψ : Z R given by

    ψ (n) = n 1



    is a ring homomorphism.

    (b) Find know rings which are possible images of ψ.
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  2. #2
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    (a) Show that ψ : Z →R given by

    ψ (n) = n 1
    is a ring homomorphism.
    You need to show \phi (a+b) = \phi(a) + \phi(b). And this is true because (a+b)\cdot 1 = a\cdot 1 + b\cdot 1. You also need to show \phi(ab) = \phi(a)\phi(b). This is true because (a\cdot 1)(b\cdot 1) = (1+...+1)(b\cdot 1) = b\cdot 1 + ... + b\cdot 1 = (ab)\cdot 1. And of course \phi(1)=1. Thus, \phi is a (commutative) ring homomorphism.

    (b) Find known rings which are possible images of ψ.
    \mathbb{Z}_n.
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