Rings in which idempotents generate maximal or minimal ideals

We characterize rings in which every left ideal generated by an idempotent different from 0 and 1 is either a maximal left ideal or a minimal left ideal. In the commutative case, we give a characterization in terms of topological properties of the maximal spectrum with the Zariski topology. We also consider a strictly weaker variant of this property, defined almost similarly, and characterize those rings that have the property in question.