Amenable and Locally Amenable Algebraic Frames
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Amenable and Locally Amenable Algebraic Frames Themba Dube1 Received: 23 July 2019 / Accepted: 28 November 2019 / © Springer Nature B.V. 2019
Abstract We say a prime element of an algebraic frame is amenable if it comparable to every compact element. If every prime element of an algebraic frame L is amenable, we say L is an amenable frame. If the localization of L at every prime element is amenable, we say L is locally amenable. These concepts are motivated by notions of divided and locally divided commutative rings. We show that an algebraic frame is (i) amenable precisely when its prime elements form a chain, and (ii) locally amenable precisely when its prime elements form a tree. Given any prime element p of L, we construct a certain pullback in the category of algebraic frames with the finite intersection property on compact elements, and characterize (in terms of the localization Lp and the quotient ↑p of L) when this pullback is amenable and when it is locally amenable. Keywords Algebraic frame · Sublocale · Prime element · Chain · Tree · Pullback · Amenable frame · Locally amenable frame
1 Introduction It is perhaps best to introduce the topic that we study in this paper by first recounting an analogous one from ring theory. Throughout, by “ring” we mean a commutative ring with identity. In one of his several quests to find internal characterizations of integral domains that satisfy the going-down property, Dobbs [7] introduced the notion of divided prime ideals in integral domains. This concept was then extended to all rings by Badawi [2]. A prime ideal of a ring A is divided if it is comparable under inclusion to every ideal of A (or, equivalently, to every principal ideal of A). If every prime ideal of A is divided, then A is called a divided ring. If the localization of A at every prime ideal is a divided ring, then A is called a locally divided ring. The rings with the latter property are studied in detail in [3]. A strictly weaker requirement on a prime ideal than that it be comparable to all ideals is that it be comparable to all ideals that are radicals of finitely-generated ideals. Such prime ideals have not been named; so, for purposes of this introduction, let us call them nearly
Themba Dube
[email protected] 1
Department of Mathematics Sciences, University of South Africa, 0001 Preotria, South Africa
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divided. Rings in which all prime ideals are nearly divided were studied by Badawi [1], who showed that they are precisely the rings whose prime ideals are linearly ordered. Badawi did not name these rings, so let us call them nearly divided rings. The lattice of radical ideals of a ring A, denoted RId(A), is well known to be a compact algebraic frame in which the meet of any two compact elements is compact. Such lattices are called coherent frames. A fundamental theorem of Banaschewski [4] states that coherent frames are, up to isomorphism, precisely the frames RId(A), for rings A. The topological antecedent of this result was proved by Hochster [11] for spectral spaces. Coherent fr
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