We call a ring R an EGE-ring if for each$I \leq R$, which is generated by a subset of right semicentral idempotentsthere exists an idempotent $e$ such that $r(I) = eR$. The class EGE includesquasi-Baer, semiperfect rings (hence all local rings) and rings with a completeset of orthogonal primitive idempotents (hence all Noetherian rings) and isclosed under direct product, full and upper triangular matrix rings, polynomialextensions (including formal power series, Laurent polynomials, and Laurentseries) 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 underdirect products, full and upper triangular matrix rings and is Moritainvariant. 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).