Let A be a commutative ring with 1. Denote by Spec(A) the set of all prime ideals of A equipped with the hull-kernel topology, by Min(A) the subspace consisting of minimal prime ideals. We give a new characterization of the compactness of Min(A), which seems to give more light to the topological situation; this characterization, among other things, allows us to show that the class of (weakly) Baer rings coincides with the class of rings such that: 1) their minimal spectrum is compact, and 2) every prime ideal contains a unique minimal prime ideal. We always deal with rings without non-zero nilpotents; but of course all purely topological results are independent of this hypothesis.
On the compactness of minimal spectrum
ARTICO, GIULIANO;MARCONI, UMBERTO
1977
Abstract
Let A be a commutative ring with 1. Denote by Spec(A) the set of all prime ideals of A equipped with the hull-kernel topology, by Min(A) the subspace consisting of minimal prime ideals. We give a new characterization of the compactness of Min(A), which seems to give more light to the topological situation; this characterization, among other things, allows us to show that the class of (weakly) Baer rings coincides with the class of rings such that: 1) their minimal spectrum is compact, and 2) every prime ideal contains a unique minimal prime ideal. We always deal with rings without non-zero nilpotents; but of course all purely topological results are independent of this hypothesis.
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