three things that you need to recall first:

1) since is surjective, for some which is clearly in because

2) if and only if is a unit: this is a quick result of this fact that

3) for some if and only if for sime unit : this is an immediate result of 2).

now we want to show that is a principal ideal. first from 1) we have: thus hence suppose now that and then

hence by 3): this proves that and part (a) of your problem is solved.

for part (b), let obviously choose with then and so for some unit therefore