This is way over my head.

Let R and S be commutative runs and phi:R---->S be a ring homomorphism. a) Given an ideal J < S (that < means subset), define

={a (element of) R: phi(a) (element of) J] < R. (usually this is denoted by phi^-1(J)). PRove this is an ideal

b) Given an ideal I < R define

={phi(aO:a (element of) I] < S (usually denoted by phi(I)). Prove this is an ideal provided phi maps onto S.

c) In a case with b, then prove that if phi maps onto S then phi(<a>)=<phi(a)>.