Stuck on this problem.

Let f: R to S be a ring homomorphism.

-Let J-S be an ideal (the - is the triangular symbol). Prove that f^(-1) (J) (equal by definition to {r in R: f(r) in J}) is an ideal of R.

-Prove that if f is surjective and I-R is an ideal then f(I) is an ideal (where f(I)={f(i): i in I}).

-Show, by example, that if f is not surjective the assertion in the above need not hold.

I appreciate any help.