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.

中文翻译：

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幂等产生最大或最小理想的环

我们对环的特征进行了描述，其中由一个不同于0和1的幂等数生成的每个左理想都是最大左理想或最小左理想。在可交换情况下，我们使用Zariski拓扑结构根据最大光谱的拓扑特性进行了表征。我们还考虑了该属性的一个严格较弱的变体，其定义几乎类似，并描述了具有相关属性的那些环。