The Space of Maximal Subrings of a Commutative Ring

Abstract

Let R be a commutative ring and X = RgMax R be the set of all maximal subrings of R. We give a topology on X by putting S = T ∈ X S ⊆ T , where S ranges over all subrings of R, as a subbase for closed subsets for X. We investigate the decomposition into irreducible components for this topology. It is shown that valuation domains behave similar to prime ideals in Zariski topology in our topology. Further we present an analogous form of the Prime Avoidance Lemma for valuation domains instead of prime ideals. The compactness of S for certain subrings S of R is determined. Moreover, we characterize fields E for which the space X = RgMax E is compact.

Prime Avoidance Property

“…Finally in this paper we want to give a valuation version of Davis Theorem. First we need some observation from [4]. Let V, V 1 , .…”

“…. , W m are also valuation for K and m i=1 W i ⊆ n i=1 V i , then there exist i and j such that W j ⊆ V i , see [4,Remark 3.11]. We refer the reader to [14, Theorem 6 and Corollary 8] for generalization of these facts.…”

“…If k = 0, then it suffices to take v = 0; and if k = n, then by Valuation Avoidance Lemma we infer that V V 1 ∪ • • • ∪ V n , for otherwise V ⊆ V i for some i and therefore V [x] ⊆ V i which is absurd. Thus there [4,Remark 3.11], either V ⊆ V i for some 1 ≤ i ≤ k, and therefore V [x] ⊆ V i which is impossible or V j ⊆ V i for some 1 ≤ i ≤ k and k + 1 ≤ j ≤ n, which again is impossible since V i 's are incomparable. Thus there exists v ∈ (…”

Commutative rings with infinitely many maximal subrings

“…Let R be a ring and X = RgM ax(R). Then we have a topology on X, by putting X(S) = {T ∈ X | S ⊆ T } as a subbase for closed subsets for X, where S ranges over all subrings of R, which is called K-space in [4]. In [4], it is shown that if E is a field then X = RgM ax(E) is compact if and only if E has only finitely many maximal subrings.…”

Finite unions of overrings of an integral domain

“…This investigation was initiated and inspired by a theorem of A. Azarang, proved as Corollary 3.10 in [1]. The theorem, by Azarang properly called the valuation avoidance lemma, states that if V, V 1 , V 2 , .…”