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Math Help - Ring Homomorphism and Ideals

  1. #1
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    Ring Homomorphism and Ideals

    Let f: R --> R' be a ring homomorphism and let N be an ideal of R.
    Show that f(N) is an ideal of f(R) and also show that f(N) doesn't need to be an ideal of R'
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  2. #2
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    Quote Originally Posted by Coda202 View Post
    Let f: R --> R' be a ring homomorphism and let N be an ideal of R.
    Show that f(N) is an ideal of f(R)
    To show f(N) is an ideal basically what you need to show is that rx,xr\in f(N) for all r \in f(R) and x\in f(N). Now since r\in f(R) it means r = f(r_0) for some r_0\in R and x = f(n_0) for some n_0\in N. Therefore, rx=f(r_0)f(n_0) = f(r_0n_0) \in f(N) since r_0n_0\in N for N is an ideal, similarly xr\in f(N). Thus, f(N) \triangleleft f(R).

    also show that f(N) doesn't need to be an ideal of R'
    Consider \phi: \mathbb{Q}[x] \to \mathbb{C} defined by \phi (f(x)) = f(i) (evaluating polynomials at i).
    Let N = \mathbb{Q}[x] then \phi(N) = \mathbb{Q}(i) = \{a+bi|a,b\in \mathbb{Q}\}.
    This is not an ideal of \mathbb{C} since the complex numbers are a field.
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