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Math Help - Homorphism Problem

  1. #1
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    Homorphism Problem

    Let Y:R->S be a homomorphism from a ring R to a ring S. Prove if T is a subring of R, then the set Y(T) = {f in S : f = Y(T) , for an element t in T} is a subring of S.
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let Y:R->S be a homomorphism from a ring R to a ring S. Prove if T is a subring of R, then the set Y(T) = {f in S : f = Y(T) , for an element t in T} is a subring of S.
    Y[T]={f(x)|x in T}

    If x in T and y in T then xy in T (why?)
    Thus, phi(x+y)=phi(x)+phi(y) thus, phi(x)+phi(x) in Y[T].

    And phi(0) is identity element.

    And phi(x^-1) is inverse.

    And it is associative because R is associative.

    Finally, phi(x)+phi(y)=phi(x+y)=phi(y+x)=phi(y)+phi(x)

    So it is commutative.

    This show that <Y[T],+> forms abelian group.

    I leave it to you to show everything else.
    The same idea appiles.
    Last edited by ThePerfectHacker; March 28th 2007 at 11:20 AM.
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