TITLE: In a Dedekind domain every ideal is either principal or generated by two elements.
QUESTION [27 upvotes]: Prove that in a Dedekind domain every ideal is either principal or generated by two elements.

Help me some hints.
Thanks a lot!

REPLY [8 votes]: It is enough to show that R/(x) is a PIR for each nonzero nonunit x.  If you can use the structure theorem for rings in which every ideal is a product of prime ideals (called general ZPI-rings), this is immediate, since any zero-dimensional such ring is a PIR. (A general ZPI-ring is a finite direct product of Dedekind domains and PIR's with one prime.)  
Alternatively, it is not hard to directly prove that R/(x) is a PIR.  (This is a much easier special case of the theorem).  Being a zero-dimensional semi-quasi-local general ZPI-ring, it is a finite direct product of (necessarily quasi-local) connected such rings.  These factors are PIRs since each principal ideal is a power of the maximal ideal, so R/(x) is a PIR.