We say that the ring $R$ is *topological*, when we are given neighbourhoods $\{O_{\lambda};\lambda \in \Lambda\}$ of $0_R$.

We say ${\cal I}$ is a *closed* ideal of $R$ if and only if for any sub-index set $\Delta \subset \Lambda$ such that

\begin{equation*} O_{\lambda} \subset {\cal I}, \phantom{A} \underset{\lambda \in \Delta}\cap O_{\lambda} = 0_R, \end{equation*} any Cauchy sequence converges in ${\cal I}$ as follows$\colon$ \begin{equation*} \underset{c_{\lambda} \in R,\,a_{\lambda} \in O_{\lambda} \\ \phantom{AAA}\lambda \in \Delta}{\Sigma} c_{\lambda}a_{\lambda} \in {\cal I}. \end{equation*}

Let ${\cal I}$ be a *non-closed* ideal of the compact ring $R$, and $\overline{{\cal I}}$ be the closure of ${\cal I}$. We have ${\cal I} \subsetneq \overline{\cal I}$.
We define the radical ${\sqrt{\cal I}} \colon= \{r \in R ; r^n \in {\cal I}~{\mathrm{for ~some}}~n > 0\}$.