TITLE: Error in Neukirch's "Algebraic Number Theory"?
QUESTION [6 upvotes]: I found what I believe is an error in Neukirch's book, in Chapter 1 Section 3 (Ideals).  Exercise 5 states

The quotient ring $\mathcal{O}/\mathfrak{a}$ of a dedekind domain by an ideal $\mathfrak{a} \neq 0$ is a principal ideal domain.

I believe I can prove every ideal in the quotient $\mathcal{O}/\mathfrak{a}$ is principal, but this need not be a domain, correct? (i.e. $\mathbb{Z}/(4)$ )
This seems obvious, but I just need some validation otherwise I'll feel crazy.

REPLY [4 votes]: You're right. In this situation, the quotient ring is necessarily a principal Artinian ring, but not necessarily a domain. See Pete L. Clark's answer here.