# Ring homomorphism and subrings

gotmilk04

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

## 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

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

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)$$\in R'$$?

VeeEight
Yes, and similar for part (b)