Topologies derived from the old one via ideals

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 $X\mathrm{.}$ 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.