Let $R$ be a complete DVR with maximal ideal $\mathfrak m$ and residue field $k = R/\mathfrak m$. Let $\mathfrak X$ be a formal scheme over $\mathrm{Spf}(R)$, formally of finite type if necessary. Such a formal scheme may be seen as a direct system $(X_n)_{n\in \mathbb N}$ where $X_n$ is a scheme over the ring $R/\mathfrak m^{n+1}$, and with a few compatibility properties.
The special fiber of $\mathfrak X$ is the scheme $\mathfrak X_0$ defined over $k$.
On the other hand, provided that $\mathfrak X$ is locally noetherian, it admits a largest ideal of definition which we call $\mathfrak J$. The reduction of $\mathfrak X$ is the scheme sharing the same underlying topological space as $\mathfrak X$ and with structure sheaf $\mathcal O_{\mathfrak X}/\mathfrak J$. It is denoted by $\mathfrak X_{\mathrm{red}}$, it is defined over $k$ and it is reduced.
Am I correct to think of $\mathfrak X_{\mathrm{red}}$ as the reduced $k$-scheme structure on the special fiber $\mathfrak X_0$ ? If so, what kind of properties on $\mathfrak X$ can insure that $\mathfrak X_0$ is already reduced ?