Characterizations of Gelfand rings specially clean rings and their dual rings

Abstract

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.

Structural results on harmonic rings and lessened rings

“…Although the present paper is the continuation of our recent work [1], but it is self-contained. For all undefined notions we refer the reader to that paper.…”

“…In the following result, new criteria for zero dimensional rings are given. Also a new proof for the implication (i)⇒(ii) is provided which has geometric nature, for a purely algebraic proof of this implication see [1,Theorem 3.3].…”

“…It follows that g ∈ Ker π p . So by [1,Lemma 8.4], there exists an idempotent e ∈ Ker π p ⊆ p such that g(1 − e) = 0. This yields that 1 − e ∈ q.…”

“…In this paper, just like that [1], a combination of algebraic and topological methods are used to solve commutative algebra problems. Specially among them, it is proved that for a given mp-ring A, then the retraction map Spec(A) → Min(A) is Zariski continuous if and only if Min(A) is Zariski compact, see Theorem 3.26.…”

“…A rich literature there was dedicated to the Gelfand rings, clean rings and mprings (see e.g. the References and Introduction of [1]). In the literature, Gelfand rings and mp-rings are also called harmonic rings (note that in the classical usage, a ring A was called harmonic if Max(A) is a Hausdorff space with respect to the Zariski topology, also a Gelfand ring was called a strongly harmonic ring, see [1,Remark 4.5], [3, p. 99-315], [4] and [6].…”