In this question I will use the following definition of a Dedekind domain:
An integral domain $A$ is a Dedekind Domain if:
1) $A$ is a Noetherian Ring.
2) $A$ is integrally closed.
3) Every non-zero prime ideal of $A$ is maximal.
I also know that every non-zero ideal $I \subset A$ can be expressed uniquely as a product of powers of prime ideals (a proof that does not use results from localization or primary decomposition can be found in Pierre Samuel's Algebraic Theory of Numbers).
With this information, and without having been taught localization (this is for an undergrad class in ANT) is it possible to prove the following statement:
If $A$ is a Dedekind Domain and $I \subset A$ a non-zero ideal, then every ideal of $A/I$ is principal.
--I should point out that I can come up with a proof when I know that in a Dedekind domain $A$ if $p \subset A$ is a non-zero prime (hence maximal) ideal, then the localization $A_p$ is a P.I.D. But, as we have not been taught localization in class I cannot use it to solve the question. I also looked at Atiyah- Macdonald, and they basically prove this as Theorem 9.3, but they use results from localization and primary decomposition.