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Math Help - Fields, Rings and Homomorphisms

  1. #1
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    Fields, Rings and Homomorphisms

    Let F be a field and R a ring and suppose phi: F --> R is a homomorphism.
    Prove that either phi is one-to-one or phi is the trivial homomorphism (that is, phi(a) = 0 for all a in F).
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    Quote Originally Posted by Coda202 View Post
    Let F be a field and R a ring and suppose phi: F --> R is a homomorphism.
    Prove that either phi is one-to-one or phi is the trivial homomorphism (that is, phi(a) = 0 for all a in F).
    Hint1: The kernel of a ring homorphism gives rise to an ideal.
    Hint2: What are the ideals of a field?
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  3. #3
    Senior Member JaneBennet's Avatar
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    Mind you, some authors e.g. D.J.H. Garling in A Course in Galois Theory (1986, Cambridge University Press) insist that a ring homomorphism should map the multiplicative identity of one ring to that of the other. If you adopt such a definition, the second possibility would not be possible: all homomorphisms from a field to a ring would then have to be injective.
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    Quote Originally Posted by JaneBennet View Post
    Mind you, some authors e.g. D.J.H. Garling in A Course in Galois Theory (1986, Cambridge University Press) insist that a ring homomorphism should map the multiplicative identity of one ring to that of the other. If you adopt such a definition, the second possibility would not be possible: all homomorphisms from a field to a ring would then have to be injective.
    I like to distinguish between "homomorphism between rings" and "homomorphism between commutative unitary rings". So when I see "ring homomorphism" all I think of is \phi(ab) = \phi(a)\phi(b),\phi(a+b)=\phi(a)+\phi(b). However, if I see "commutative ring homomorphism" (it is rather common to called commutative unitary rings simply by commutative rings) then I think of the additional condition \phi(1) =1.
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