  Let $X$ be a completely regular topological space. We study closed ideals $H$
of $C_B(X)$, the normed algebra of bounded continuous scalar-valued mappings on
$X$ equipped with pointwise addition and multiplication and the supremum norm,
which are non-vanishing, in the sense that, there is no point of $X$ at which
every element of $H$ vanishes. This is done by studying the (unique) locally
compact Hausdorff space $Y$ associated to $H$ in such a way that $H$ and
$C_0(Y)$ are isometrically isomorphic. We are interested in various
connectedness properties of $Y$. In particular, we present necessary and
sufficient (algebraic) conditions for $H$ such that $Y$ satisfies (topological)
properties such as locally connectedness, total disconnectedness,
zero-dimensionality, strong zero-dimensionality, total separatedness or
extremal disconnectedness.
