In this paper, we show that for any ${sigma}$-complete Boolean subalgebra $mathcal{M}$ of $mathcal{R}(X)$ containing $Z(X)^{sharp}$, the Stone-space $S(mathcal{M})$ of $mathcal{M}$ is a basically diconnected cover of ${eta}X$ and that the subspace {${alpha}{mid}{alpha}$ is a fixed $mathcal{M}$-ultrafilter} of the Stone-space $S(mathcal{M})$ is the the minimal basically disconnected cover of X if and only if it is a basically disconnected space and $mathcal{M}{subseteq}{Lambda_X(A){mid}A{in}Z({Lambda}X)^{sharp}}$.