Expanding Čech cohomology for quantales

We expand Čech cohomology of a topological space $X$ with values in a presheaf on $X$ to Čech cohomology of a commutative ring with unity $R$ with values in a presheaf on $R$. The strategy is to observe that both the set of open subsets of $X$ and the set of ideals of $R$ provide examples of a (semicartesian) quantale. We study a particular pair of (adjoint) functors $(\theta ,\tau )$ between the quantale of open subsets of $X$ and the quantale of ideals of $C(X)$, the ring of real-valued continuous functions on $X$. This leads to the main result of this paper: the $q$th Čech cohomology groups of $X$ with values on the constant sheaf $F$ on $X$ is isomorphic to the $q$th Čech cohomology groups of the ring $C(X)$ with values on a sheaf $F\circ \tau$ on $C(X)$.