Ring homomorphism and subrings

  • Thread starter gotmilk04
  • Start date
  • #1
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Homework Statement


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]


Homework Equations


For a ring homomorphism,
f(a+b)= f(a) + f(b)
f(ab)= f(a)f(b)


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.
 

Answers and Replies

  • #2
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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
 
  • #3
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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]?
 
  • #4
614
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Yes, and similar for part (b)
 

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