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Modern algebra

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I am stuck with this question.

If phi : F -> R is a nonzero homomorphism from a field F to a ring R, show that phi is one-to-one (Hint: recall that for a ring homomorphism, phi is one-to-one if and only if Ker(phi)={0}. )

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This solution is comprised of a detailed explanation to show that phi is one-to-one.

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We do not consider the trivial case: phi(a)=0 for any a in F. In this case, phi is not ...

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