  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.

Commutative Algebra