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Math Help - Quotient Rings and Homomorphic images in F(R)

  1. #1
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    Quotient Rings and Homomorphic images in F(R)

    "let g be the function from F(R) to RxR defined by g(f)=( f(0), f(1) ). Prove that g is a homomorphism from F(R) onto RxR, and describe its kernel"

    I'm trying to find the kernel. so the kernel is kerf={x in F(R); g(x)= e_RxR} ? Wouldn't it just be (0,0) ? If this is right how do i prove this? If this is wrong then how do i find the kernel?
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  2. #2
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    Re: Quotient Rings and Homomorphic images in F(R)

    it would help if we know exactly what you meant by F(R) and R. if R is a ring, then the identity of (RxR,+) is indeed (0,0), so the kernel of g is:

    ker(g) = {f in F(R): f(0) = f(1) = 0}.

    my guess is that by F(R) you mean the set of all functions from R to R with:

    (f+h)(x) = f(x) + h(x) and

    (fh)(x) = f(x)h(x).

    note that the additive identity of F(R) is the function 0(x) = 0, for all x in R, and the multiplicative identity of F(R) is the (constant) function 1(x) = 1, for all x in R.

    (note that ker(g) is non-trivial, the function f(x) = x(x-1) is in it).
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