I want to prove that the following definition of a Dedekind domain are equivalent:

$(i)$ $R$ is noetherian and integrally closed, and every nonzero prime ideal is maximal.

$(ii)$ $R$ is noetherian, and each localization $R_P$ is a discrete valuation ring.

I know that the valuation of $R_P$ is a $\pi$-adic valuation where $\pi$ is a uniformizer of $P$ element, but I want to prove the existance of that uniformizing element.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.