Ring Isomorphism Help

[cs0978]

I seem to be having a difficult time trying to figure out how to do this problem. It's from a non-graded homework assignment. I was able to get every other problem except for this one.

a.) Let R and S be commutative rings and let f: R->S be a ring isomorphism. Prove if x is a zero divisor in R, then f(x) is a zero divisor in S.
b.) Give an example to show that (a) is not necessarily true if R is not isomorphic to S.

Any help would be greatly appreciated! I don't usually do this, but I would really like to understand this topic better.

Thanks!

 

[pka]

a.) Let R and S be commutative rings and let f: R->S be a ring isomorphism. Prove if x is a zero divisor in R, then f(x) is a zero divisor in S.
b.) Give an example to show that (a) is not necessarily true if R is not isomorphic to S.

Do you know that any isomorphism is a bijection?
Do you know that \(\displaystyle f(0_R)=0_S\)?

If each of \(\displaystyle u~\&~v\) is a zero divisor in \(\displaystyle R\) then \(\displaystyle 0_S=f(0_R)=f(u\cdot_R v)=f(u)\cdot_S f(v)\)

What does that tell you?