Warren Wm. McGovern

Prüfer domains with Clifford class semigroup

Bazzoni’s Conjecture states that the Prüfer domain R has finite character if and only if R has the property that an ideal of R is finitely generated if and only if it is locally principal. In [4] the authors use the language and results from the theory of lattice-ordered groups to show that the conjecture is true. In this article we supply a purely ring theoretic proof. 1. Bazzoni’s Conjecture Throughout all integral domains are assumed to be commutative. For an integral domain R, F (R) denotes the semigroup of fractional ideals of R (under ideal multiplication) while P(R) denotes the subsemigroup consisting of principal ideals. The class semigroup of R is the factor semigroup F (R)/P(R) and is denoted S (R). A semigroup S is called a Clifford semigroup when every element is regular in the sense of von Neumann, that is, for every a ∈ S there is an s ∈ S for which as = a. The domain R is called a Clifford regular domain when S (R) is Clifford regular. In the article [1] S. Bazzoni proved that if a Prüfer domain has finite character (that is, every nonzero element belongs to a finite number of maximal ideals) then S (R) is a Clifford semigroup, and in turn, if S (R) is a Clifford semigroup, then R satisfies (∗) (defined below). In a later article, [2], she was able to show that if S (R) is a Clifford semigroup, then R has finite character. In [1] and then again in [2] she proposed the following Conjecture: A Prüfer domain satisfies property (*) if and only if R has finite character. Recently, the authors of [4] proved using techniques from the theory of latticeordered groups that the conjecture is indeed true. The main road used in their proof was to translate the concepts discussed above into the language of `-groups via the lattice-ordered group of invertible ideals of a Prüfer domain. Once the translations were made they used several old and well-known results to finish the proof. In this article we give a purely ring-theoretic proof of the validity of Bazzoni’s conjecture. Our proof is mainly a translation from `-groups to ring theory of the proof from [4] except in one crucial place. We elaborate on this matter. Given a Prüfer domain R and G its `-group of invertible ideals any information about an `-homomorphic image of G can be translated to information about an appropriate localization of R. On the other hand, and unfortunately, there is no known ring-theoretic construction that allows one to gather information about the 2000 Mathematics Subject Classification. Primary 13F05.

GENERALIZATIONS OF COMPLEMENTED RINGS WITH APPLICATIONS TO RINGS OF FUNCTIONS

It is well known that a commutative ring R is complemented (that is, given a ∈ R there exists b ∈ R such that ab = 0 and a + b is a regular element) if and only if the total ring of quotients of R is von Neumann regular. We consider generalizations of the notion of a complemented ring and their implications for the total ring of quotients. We then look at the specific case when the ring is a ring of continuous real-valued functions on a topological space.

Commutative nil clean group rings

In [A. J. Diesl, Classes of strongly clean rings, Ph.D. Dissertation, University of California, Berkely (2006); Nil clean rings, J. Algebra383 (2013) 197–211], a nil clean ring was defined as a ring for which every element is the sum of a nilpotent and an idempotent. In this short paper, we characterize nil clean commutative group rings.

Feebly Projectable Algebraic Frames and Multiplicative Filters of Ideals

In the article (Martinez and Zenk, Algebra Universalis, 50, 231–257, 2003.), the authors studied several conditions on an algebraic frame L. In particular, four properties called Reg(1), Reg(2), Reg(3), and Reg(4) were considered. There it was shown that Reg(3) is equivalent to the more familiar condition known as projectability. In this article we show that there is a nice property, which we call feebly projectable, that is between Reg(3) and Reg(4). In the main section of the article we apply our notions to the frame of multiplicative filters of ideals in a commutative ring with unit and give characterizations of several well-known classes of commutative rings.

Least Integer Closed Groups

An a-closure of a lattice-ordered group is an extension which is maximal with respect to preserving the lattice of convex l-subgroups under contraction. We describe the a-closures of some local singular archimedean lattice-ordered groups with designated weak unit. In particular, we provide explicit descriptions of all of the a-closures of groups that are singularly convex, such as the group C(X, ℤ) of continuous integer-valued functions on a zero-dimensional space.

Rings of quotients of f-rings by gabriel filters of ideals

The article examines the role of Gabriel filters of ideals in the ontext of semiprime f-rings. It is shown that for every 2-convex semiprime f-ring Aand every multiplicative filter B of dense ideals the ring of quotients of A by B, namely the direct limit of the Hom A (I, A) over all I∈ B, is an l-subring of QA, the maximum ring of quotients. Relative to the category of all commutative rings with identity, it is shown that for every 2-convex semiprime f-ring A qA, the classical ring of quotients, is the largest flat epimorphic extension of A. If Ais also a Prufer ring then it follows that every extension of Ain qA is of the form S -1A for a suitable multiplicative subset S. The paper also examines when a Utumi ring of quotients of a semiprime f-ring is obtained from a Gabriel filter. For a ring of continuous functions C(X), with Xcompact, this is so for each C(U) and C *(U), when Uis dense open, but not for an arbitrary direct limit of C(U),taken over a filter base of dense open sets. In conclusion, it is...

Commutative Singular f-Rings

This paper examines commutative f-rings A with identity 1, for which 1 is a singular element; (i.e., such that 0 ≤ s ≤ 1 implies that s ∧ (1 — s) = 0.) One of the main results is that for such f-rings the following are equivalent: (a) A is semihereditary; (b) A is a Prufer ring; (c) the weak dimension of A does not exceed 1; (d) every subring B of the maximum ring of quotients QA which contains A is flat over A; (e) the lattice of all ideals of A is distributive. In the final section singular f-rings which are I-rings are discussed. It is shown that C(X, ℤ) is an I-ring precisely when X is an extremally disconnected almost P-space.

The global dimension of an f-ring via its space of minimal prime ideals

This article examines hereditary f-rings, after first characterizingthe hereditary von Neumann regular commutative rings as those for which the space of maximal ideals is hereditarily paracompact. It is shown that if C ( X , Z ) , the ring of integer-valued continuous functions on a zero-dimensional space X, is hereditary, then X is finite. This is shown two ways; once as a consequence of the following: if A is any singular archimedean f-ring, then A/2A is a boolean ring, and gd(A) >= gd(A/2A)+ 1 , where gd(A) stands for the global dimension of A. As a consequence of this it is also shown that if A is a singular archimedean f-ring and gd(A) <= 2, then Min(A), the space of minimal prime ideals is hereditarily paracompact. The paper concludes with a calculation of the global dimension of a mihereditary singular archimedean f-ring A, in which the cellularityof Min(A) is “much less” than |A| if finite, it is k+2, where Nk = |A|.

Weakly least integer closed groups

An archimedean lattice-ordered groupA with distinguished weak unit has the canonical Yosida representation as an ℓ-group of extended real-valued functions on a certain compact Hausdorff spaceY A. Such an ℓ-groupA is calledleast integer closed, orLIC (resp.,weakly least integer closed, orwLIC) if, in the representation,a ∈A implies [a] ∈A (resp., there isa′ ∈A witha′=[a] on a dense set inY A), where [r] ≡ the least integer greater than or equal tor. Earlier, we have studiedLIC groups, with an emphasis on their a-extensions. Here, we turn towLIC groups: we give an intrinsic (though awk-ward) characterization in terms of existence of certain countable suprema. This results also in an intrinsic characterization ofLIC, previously lacking. Also,wLIC is a hull class (whichLIC is not), and the hullwlA is “somewhere near” the projectable hullpA. The best comparison comes from a (somewhat novel) factoringpA=loc(wpA), wherewpA is the “weakly projectable” hull (defined here), andlocB is the “local monoreflection”; then,wpA≤wlA≤loc(wpA), andpA≤loc(wlA), while with a strong unit, all these coincide. Numerous examples and special cases are examined.

When the maximum ring of quotients of C(X) is uniformly complete

A Tychonoff space X such that the maximum ring of quotients of C(X) is uniformly complete is called a uniform quotients space. It is shown that this condition is equivalent to the Dedekind– MacNeille completion of C(X) being a ring of quotients of C(X), in the sense of Utumi. A compact metric space is a uniform quotients space precisely when it has a dense set of isolated points. Extremally disconnected spaces and almost P -spaces are uniform quotients spaces. Also characterized are the compact spaces of dense constancies which are uniform quotients spaces.  2001 Elsevier Science B.V. All rights reserved. AMS classification:Primary 54H10; 06F25; 13B30, Secondary 54D35; 54G05