# Ring homomorphism and subrings

1. Feb 3, 2010

### gotmilk04

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$$^{-1}$$(0) is a subring of R
c) if R has 1 and f:R-R is a ring epimorphism, then f(1$$_{R}$$)=1$$_{R'}$$

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. Feb 3, 2010

### VeeEight

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. Feb 3, 2010

### gotmilk04

So for part a), I would do
let a,b$$\in R$$ and f(a),f(b)$$\in R'$$
So f(a)-f(b)= f(a-b) $$\in R'$$
and f(a)f(b)= f(ab)$$\in R'$$?

4. Feb 3, 2010

### VeeEight

Yes, and similar for part (b)