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Homework Help: Ring homomorphism and subrings

  1. Feb 3, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that if f:R-R' is a ring homomorphism, then
    a) f(R) is a subring of R'
    b) ker f= f[tex]^{-1}[/tex](0) is a subring of R
    c) if R has 1 and f:R-R is a ring epimorphism, then f(1[tex]_{R}[/tex])=1[tex]_{R'}[/tex]

    2. Relevant equations
    For a ring homomorphism,
    f(a+b)= f(a) + f(b)
    f(ab)= f(a)f(b)

    3. The attempt at a solution
    In order to show something is a subring of something else, do I just show that the former satisfies the properties of a ring?
    Like in a), would I show that f(R) is a ring?
    I just need a little guidance please.
  2. jcsd
  3. Feb 3, 2010 #2
    A subring is a subset of a ring that is a ring under the same binary operations. You must show that your subset is closed under multiplication and subtraction if you want to show it's a subring
  4. Feb 3, 2010 #3
    So for part a), I would do
    let a,b[tex]\in R[/tex] and f(a),f(b)[tex]\in R'[/tex]
    So f(a)-f(b)= f(a-b) [tex]\in R'[/tex]
    and f(a)f(b)= f(ab)[tex]\in R'[/tex]?
  5. Feb 3, 2010 #4
    Yes, and similar for part (b)
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