Other possible proof that the quotient of a Dedekind domain by a nonzero ideal is PID.

Question

In J.S. Chahal's Algebraic Number Theory book, p.39 he states in Theorem 3.72 that any ideal of a Dedekind domain can be generated by two elements. And in Corollary 3.73 he states that the quotient ring of a Dedekind domain by a nonzero ideal is a principal ideal domain. This statement is already proved in If $A$ is a Dedekind domain and $I \subset A$ a non-zero ideal, then every ideal of $A/I$ is principal. clearly. My question is, perhaps, can we prove the Corollary 3.73 from the Theorem 3.72 by other method than the linked post?

Answer 1

We do not say PID here since the quotient is not an integral domain all the time unless the ideal is prime.

The answer is "Yes", we do have a prove which involves what we call "Localization".

By Chinese Remainder Theory, the only thing we need to prove is that $A/\mathfrak p^n$ is a ring whose ideals are all principle. $A$ is a Dedekind domain and $\mathfrak p$ is a prime ideal of it.

One can show two facts.

1: $A_\mathfrak p$ is actually a discrete valuation ring, i.e. ,a local PID.

2:$A/\mathfrak p^n\cong A_\mathfrak p/\mathfrak p^nA_\mathfrak p$

Once you combine this two facts, the proof is done.

For (1),this is classic result of Dedekind domain. For (2), you can prove it by induction on $n$, take in mind that Dedekind implies prime ideals are all maximal.

And these result can be proved from any equivalent definition of Dedekind domain .Here is a good note for relating result.If you are not familiar with localization ,you can refer to this.