Annihilator Conditions related to the quasi-Baer Condition

We call a ring R an EGE-ring if for each $I \leq R$, which is generated by a subset of right semicentral idempotents there exists an idempotent $e$ such that $r(I) = eR$. The class EGE includes quasi-Baer, semiperfect rings (hence all local rings) and rings with a complete set of orthogonal primitive idempotents (hence all Noetherian rings) and is closed under direct product, full and upper triangular matrix rings, polynomial extensions (including formal power series, Laurent polynomials, and Laurent series) and is Morita invariant. Also we call $R$ an AE-ring if for each $I \unlhd R$, there exists a subset $S \subseteq S_{r}(R)$ such that $r(I) = r(RSR)$. The class AE includes the principally quasi-Baer ring and is closed under direct products, full and upper triangular matrix rings and is Morita invariant. For a semiprime ring $R$, it is shown that $R$ is an EGE (resp., AE)-ring if and only if the closure of any union of clopen subsets of $Spec(R)$ is open (resp., $Spec(R)$ is an EZ-space).