TITLE: Ring Sandwiched between PIDs
QUESTION [6 upvotes]: If I have three commutative rings $R \subset S \subset T$, such that $R$ and $T$ are principal ideal domains, will this imply that $S$ itself is a principal ideal domain?

REPLY [2 votes]: $k[t^3]\subset k[t^2,t^3]\subset k[t]$ .