Let D be a principal ideal domain and N an ideal of D.?

Question

Using the ascending chain condition, show that if D doesn't equal N, then N is contained in a maximal ideal of D.

Answer 1

Isn't that by definition? If D is a principal ideal domain, D is necessarily Noetherian, so the proper ideals of D respects that ACC under set inclusion, and therefore by the maximal condition on ACC, there is a maximal element.