Let R be a principal ideal domain with a unique maximal ideal. Which of the following statements are necessarily true?
1
Every quotient ring of R is a principal ideal domain
2
There exists a quotient ring S of R and an ideal I ⊆ S which is not principal
3
R has countably many ideals
4
Every quotient ring S {≠ {0}) of R has a unique maximal ideal which is principal