Question #16364

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

Expert's answer

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

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

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