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

Characterizations of Gelfand rings specially clean rings and their dual rings