In this paper, new criteria for zero dimensional rings, Gelfand rings, clean rings and mp-rings are given. A new class of rings is introduced and studied, we call them purified rings. Specially, reduced purified rings are characterized. New characterizations for pure ideals of reduced Gelfand rings and mp-rings are provided. It is also proved that if the topology of a scheme is Hausdorff, then the affine opens of that scheme is stable under taking finite unions. In particular, every compact scheme is an affine scheme.

中文翻译：

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Gelfand 戒指的特征特别清洁戒指和它们的双戒指

在本文中，给出了零维环、Gelfand 环、清洁环和 mp 环的新标准。引入并研究了一类新的环，我们称它们为纯化环。特别地，特征在于还原的纯化环。提供了简化 Gelfand 环和 mp 环的纯理想的新特征。还证明了如果一个方案的拓扑是 Hausdorff，那么该方案的仿射开在取有限联合下是稳定的。特别是，每一个紧凑方案都是一个仿射方案。