How does one study Krull dimension of some Iadic completion of a ring or, more generally, a module? I know that in case of Noetherian local ring Krull dimension of its completion equals Krull dimension of the ring, but what can we say in general case?

2$\begingroup$ How general? Are you concerned primarily about the local hypothesis, or do you also want something that works in the nonNoetherian case? $\endgroup$ – Charles Staats Apr 17 '10 at 23:54

$\begingroup$ @Charles I'm mainly interested in Noetherian rings. If I'm not mistaken, Zariski rings also preserve Krull dimension under completion. Are there other wellbehaved types of rings? Is it true for modules over Zariski ring? Or, maybe, there is a general technique to find it out... $\endgroup$ – Andrew Penning Apr 18 '10 at 7:04
For a Noetherian ring R, the Krull dimension of its $I$adic completion, $\hat{R}$ is given by $\sup h(J)$, where $J$ ranges over all maximal ideals of $R$ containing $I$ and $h(J)$ is the height of $J$. Therefore $\dim \hat R\le \dim R$ with equality only when $I\subset \operatorname{rad} R$. A reference is "Topics in $\mathfrak m$adic topologies" by S.Greco, P.Salmon

$\begingroup$ Thanks, it was very helpful. This reasoning applies to modules as well, by the way. $\endgroup$ – Andrew Penning Apr 20 '10 at 14:29

$\begingroup$ What is an example of $I$ such that $I \not\subset {\mathrm{rad}}\,R$? $\endgroup$ – Pierre MATSUMI Jun 27 '16 at 1:52