  We expand \v{C}ech cohomology of a topological space $X$ with values in a
presheaf on $X$ to \v{C}ech 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 \v{C}ech cohomology groups of
$X$ with values on the constant sheaf $F$ on $X$ is isomorphic to the $q$th
\v{C}ech cohomology groups of the ring $C(X)$ with values on a sheaf $F \circ
\tau$ on $C(X)$.
