# Answer to Question #16364 in Algebra for Irvin

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

Need a fast expert's response?

Submit orderand get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

## Comments

## Leave a comment