Question

Show that a Dedekind ring R has trivial Picard group iff R is a PID, iff R is a unique factorization domain.

Answer 1

If R is a PID, it is well-known that R is a UFD. If R is a UFD, then gives Pic(R) = {1}. Finally, if Pic(R) = {1},
then every invertible ideal in R is principal. Since R is a Dedekind ring, every nonzero
ideal is invertible, and hence principal. This shows that R is a PID