Show that any non zero ideal of a principal ideal domain is a unique product of prime ideals

Question

I know that in a PID all the ideals are principal ideals but how to connect this with the given question? I have no idea how to prove this.Any help will be highly appreciated.

Answer 1

Fact: Every PID is a UFD. See here for example or any book in the topic.

Also, some easy exercises:

• The ideal $(p)$ is prime if and only if $p$ is a prime element.
• If $I=(a)$ and $J=(b)$ then $IJ=(ab)$.
• The ideals $(a)$ and $(b)$ are equal iff there is a unit $u$ such that $a=ub$.
• In a UFD an element is prime if and only if it is irreducible.

Now let $I$ be a nonzero ideal on a PID $A$. Then there is an element $a\in A$ such that $I=(a)$. By the exercises above we have that $I=P_1\cdots P_n$ for some primes ideals $P_1,\dots,P_n$ if and only if $a=u\cdot p_1\cdots p_n$ for some prime elements $p_i$. But as each $p_i$ is irreducible this is precisely a factorization of $a$ in irreducible elements.

Hence, existence and uniqueness of the factorization into product of prime ideals are the direct translations of existence and uniqueness of the factorization of $a$ into irreducible elements. These are true because $A$ is a UFD.