Why don't you start with the easier part of the proof. Start with "If is commutative, then is an ideal in ."
Well, the center is every element in the ring that commutes with the whole ring. So if the whole ring is commutative, the center will be the whole ring. So Z(R)=R, so Z(R) is an ideal....the other way of the if and only if proof is what I have been struggling with...