proper ideals in the principal ideal domain

Question

I'm to prove that every proper ideal is a product of maximal ideals which are uniquely determined up to order. I have no idea even how to start in the proof to solve this question :( May anybody help me ?

Answer 1

Hint: Every PID is a UFD. Does that help you get started?

Answer 2

Hints:

In such a ring, $(ab)=(a)(b)$, and the nonzero maximal ideals are the same thing as the nonzero prime ideals. Do you know what the prime ideals of PID's look like?