# Extension of prime ideals to completion are irreducible ideals

Let $$(R,\frak m)$$ be a Noetherian local ring. I am trying to prove that if the natural map $$\operatorname{Spec}(\widehat R)\rightarrow \operatorname{Spec}(R)$$ is bijective then $$\mathfrak{p}\widehat R$$ is irreducible for all $$\mathfrak p\in \operatorname{Spec}(R)$$.

Thank you for your proof or comment.

• What's an example of such an $R$? Dec 20 '19 at 20:54
• A DVR (or any local domain of dimension 1 whose completion is still a domain) would be an example. I can't think of anything in higher dimension though. Dec 21 '19 at 16:48
• Thank you for your comment Hailong Dao!
– TNAn
Jun 22 '20 at 3:31