In this paper, new criteria for zero dimensional rings, Gelfand rings, clean rings and mp-rings are given. The equivalency of some of the classical criteria are also proved by new and simple methods. A new and interesting class of rings is introduced and studied, we call them purified rings. Specially, non-trivial characterizations for reduced purified rings are given. Purified rings are actually the dual of clean rings. The pure ideals of reduced Gelfand rings and mp-rings are characterized. 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 (and nonempty finite intersections). In particular, every compact scheme is an affine scheme.
In this paper, new algebraic and topological results on purely-prime ideals of a commutative ring are obtained. Some applications of this study are also given. In particular, the new notion of semi-noetherian ring is introduced and Cohen type theorem is proved.
Let R be a commutative Noetherian ring. In this paper, we study those finitely generated R-modules whose Cousin complexes provide Gorenstein injective resolutions.
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