  The main purpose of this paper is to introduce and study minimal and maximal
ideals defined on ideal topological spaces. Also, we define and investigate the
concepts of ideal quotient and annihilator of any subfamily of $2^X$, where
$2^X$ is the power set of $X.$ We obtain some of their fundamental properties.
In addition, several relationships among the above notions have been discussed.
Moreover, we get a new topology, called sharp topology via the sharp operator
defined in the scope of this study, finer than the old one. Furthermore, a
decomposition of the notion of open set has been obtained. Finally, we conclude
our work with some interesting applications.
