Groups of Automorphisms of Tournaments

By the Holland Representation Theorem, every lattice ordered group (l-group) is isomorphic to a subalgebra of the l-group of automorphisms of a chain. Since weakly associative lattice groups (wal-groups) and tournaments are non-transitive generalizations of l-groups and chains, respectively, the problem concerning the possibility of representation of wal-groups by automorphisms of tournaments arises. In the paper we describe the class of wal-groups isomorphic to wal-groups of automorphisms of tournament and show some of its properties.     

The theory of radicals in lattice-ordered rings

Special classes of associative lattice-ordered rings are introduced which are analogous to V. A. Andrunakievich's special classes of rings. The appropriate special radicals for them are defined. It is shown that the special classes ofl-rings are: 1) the class of alll-primaryl-rings; 2) the class of alll-primaryl-rings without locally nilpotentl-ideals (it is shown that the correspondingl-ideal is a union of nil-l-ideals of the ring); 3) the class ofl-rings not containing strictly positive divisors of zero; 4) the class of subdirectly indecomposablel-rings withl-idempotent core.Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 639–648, December, 1968.     

Groupable lattices

A lattice is called groupable provided it can be endowed with the structure of an l-group (lattice ordered group). The primary objective of this paper is to introduce an order theoretic property of groupable lattices which implies that all associated l-groups are subdirect products of totally ordered groups. This is an analog to Iwasawa's well-known result which asserts that a conditionally complete l-group is abelian. A secondary objective is to outline a general method for identifying classes of l-groups determined by order theoretic properties.     

k-metric spaces

In this paper, we give a new generalization of metric spaces called k-metric spaces. Our k-metrics are valued in lattice ordered groups, which allows us to talk about distance in non-abelian lattice ordered groups. We also discuss a class of (not necessarily abelian) lattice ordered groups in which every k-metric induces a topology. Then we show that every k-metric valued in the real numbers is metrizable. In the last section, we characterize intrinsic metrics on lattice ordered rings that are almost f-rings and prove that being an almost f-ring is necessary and sufficient for this characterization. Then we show that if a lattice ordered ring is representable, then every intrinsic metric is a k-metric.     

On QB ∞-Rings

In this article, we introduce a new class of rings, the QB _∞-rings. We establish various properties of this concept. These show that, in several respects, QB _∞-rings behave like QB-rings. We prove that the notion of QB _∞-rings is a Morita invariant property of rings and every finite subdirect product of QB _∞-rings is a QB _∞-ring. We also exhibit examples to point out that the class of QB _∞-rings is much larger than the class of QB-rings.     

Orderpotentf-rings

The class off-rings in which the product of anyn elements is comparable to zero (n-order-potentf-rings) generalizes the concept of both totally ordered and nilpotentf-rings. Necessary and sufficient conditions are found for anf-ring to ben-orderpotent. It is shown thatn-orderpotency is closely related to the ring having sufficiently many annihilating elements. Special consideration is given to generalized semigroup rings, a rich source of examples.Presented by J. Sichler.Dedicated to the memory of Alan DayParts of this paper based on this author's doctoral dissertation under the direction of Professor W. Charles Holland at Bowling Green State University.     

Integral Stone Rings

It is shown that certain partially ordered rings, defined by some of the properties of the totally ordered ring of integers, are exactly the bounded Z-rings, that is, the commutative f-rings with strong singular unit. The partially ordered rings in question amount to a discrete version of the rings introduced by M.H. Stone for his abstract characterization of the rings of real-valued continuous functions on compact Hausdorff spaces, and the function rings they correspond to are given by the integer-valued continuous functions on Boolean spaces.     

Formations of lattice ordered groups and of <Emphasis Type="Italic">GMV</Emphasis>-algebras

A class of lattice ordered groups is called a formation if it is closed with respect to homomorphic images and finite subdirect products. Analogously we define the formation of GMV-algebras. Let us denote by ℱ₁ and ℱ₂ the collection of all formations of lattice ordered groups or of GMV-algebras, respectively. Both ℱ₁ and ℱ₂ are partially ordered by the class-theoretical inclusion. We prove that ℱ₁ satisfies the infinite distributivity law and that ℱ₂ is isomorphic to a principal ideal of ℱ₁. This work was supported by VEGA grant 2/7141/27.     

REGULARITIES AND SUBDIRECT PRODUCTS OF RINGS

Abstract

Structure theorems are obtained for certain radical classes of rings (including the Brown-McCoy radical class, the class of λ-rings, the class of E ₅-rings, the class of E ₆-rings and the class of f-regular rings) by generalizing the concept of a prime ideal.     

Prime and semiprime lattice ordered lie algebras

The concepts of prime Lie algebras and semiprime Lie algebras are important in the study of Lie algebras. The purpose of this paper is to investigate generalizations of these concepts to lattice ordered Lie algebras over partially ordered fields. Some results concerning the properties of l-prime and l-semiprime lattice ordered Lie algebras are obtained. A necessary and sufficient condition for a lattice ordered Lie algebra to be an l-prime Lie l-algebra is presented.     

Subdirect decompositions and the radical of a generalized Boolean algebra extension of a lattice ordered group

The extension of a lattice ordered group A by a generalized Boolean algebra B will be denoted by A _B . In this paper we apply subdirect decompositions of A _B for dealing with a question proposed by Conrad and Darnel. Further, in the case when A is linearly ordered we investigate (i) the completely subdirect decompositions of A _B and those of B, and (ii) the values of elements of A _B and the radical R(A _B ).     

Special classes of <Emphasis Type="Italic">l</Emphasis>-rings and Anderson-Divinsky-Sulinski lemma

Special classes of lattice-ordered rings (l-rings) are studied and for special radicals of l-rings the Anderson-Divinsky-Sulinski lemma is proved, i.e., it is proved that if ρ is a special radical in the class of l-rings and I is an l-ideal of an l-ring R, then ρ(I) is an l-ideal of the l-ring R and ρ(I) = ρ(R) ∩ I.     

Weakly Special Classes of Semiprime Rings

This paper investigates closure properties possessed by certain classes of finite subdirect products of prime rings. If ℳ is a special class of prime rings then the class ℳ_∞ of all finite subdirect products of rings in ℳ is shown to be weakly special. A ring S is said to be a right tight extension [resp. tight extension] of a subring R if every nonzero right ideal [resp. right ideal and left ideal] of S meets R nontrivially. Every hereditary class of semiprime rings closed under tight extensions is weakly special. Each of the following conditions imposed on a semiprime ring yields a hereditary class closed under right tight extensions: ACC on right annihilators; finite right Goldie dimension; right Goldie. The class of all finite subdirect products of uniformly strongly prime rings is shown to be closed under tight extensions, answering a published question. This revised version was published online in June 2006 with corrections to the Cover Date.     

Archimedean orderpotent rings

The following paper shows the algebraic structure of Archimedean lattice-ordered rings in which a product of any n elements is comparable with zero (or n-orderpotent rings). It is shown that such rings are necessarily subdirect products of nilpotent e-rings and totally-ordered ones. If a given ring is also an f-ring, then it is a direct cardinal product of an Archimedean zero-ring and a subring of the reals.     

Serial rings and subdirect products

A basic Artinian serial ring can be realized as the subdirect product of factor rings of (S, M)-upper triangular matrix rings with S a local Artinian ring and M the maximal ideal of S. As an application the serial subdirect product of (S, M)-rings is shown to have self-duality.     

Finitely 1-convex f-rings

This paper investigates f-rings that can be constructed in a finite number of steps where every step consists of taking the fibre product of two f-rings, both being either a 1-convex f-ring or a fibre product obtained in an earlier step of the construction. These are the f-rings that satisfy the algebraic property that rings of continuous functions possess when the underlying topological space is finitely an F-space (i.e. has a Stone-?ech compactification that is a finite union of compact F-spaces). These f-rings are shown to be SV f-rings with bounded inversion and finite rank and, when constructed from semisimple f-rings, their maximal ideal space under the hull-kernel topology contains a dense open set of maximal ideals containing a unique minimal prime ideal. For a large class of these rings, the sum of prime, semiprime, primary and z-ideals are shown to be prime, semiprime, primary and z-ideals respectively.     

Associativity of the <Emphasis Type="Italic">∇</Emphasis>-operation on bands in rings

Given a multiplicative band of idempotents S in a ring R, for all e,f∈S the ∇-product e ∇ f=e+f+fe−efe−fef is an idempotent that lies roughly above e and f in R just as ef and fe lie roughly below e and f. In this paper we study ∇-bands in rings, that is, bands in rings that are closed under ∇, giving various criteria for ∇ to be associative, thus making the band a skew lattice. We also consider when a given band S in R generates a ∇-band.     

F*-Rings Are O*

O ^*-rings were introduced by Fuchs and recently characterized by Steinberg. A ring R is called O ^* if every partial order on R extends to a total order. We weaken the condition on the ordering of the ring by requiring that every partial order on R extends to an f-order. We call those rings F ^*-rings. We show that the two classes of rings coincide.     

On regular rings and V-rings

In this note, certain generalisations of strongly regular rings are considered in connection with regular rings andV-rings. The result that strongly regular rings are left (and right)V-rings [11] is extended. A condition for prime leftV-rings to be primitive with non-zero socle is given (this is related to a question ofFisher [7, Problem 3]. IfA is an ALD (almost left duo) ring, then (1) a simple leftA-module is injective iff it isp-injective; (2)A is von Neumann regular iff every maximal essential right ideal ofA isf-injective. Characterisations of semi-simple Artinian and simple Artinian rings are given in terms of regular andV-rings.     

Division closed lattice-ordered rings and commutative L*-rings

Abstract

The paper continues the study of division closed lattice-ordered rings and commutative L*-rings. More interesting properties of division closed lattice-ordered rings are presented and it is shown that under certain conditions such rings are f-rings. The main result on L*-rings is that for a commutative semilocal ring with the identity, it is L* if and only if it is O*.