Question

Prove that a semilocal Dedekind ring is a PID

Answer 1

If R is (commutative) semilocal, we know that Pic(R) = {1}. If R is also Dedekind, then
next argument shows that R is a PID:
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.