ON RADICAL THEORY OF NON-ZEROSYMMETRIC NEAR-RINGS

Abstract

This paper is based on an expository talk given at the International Conference on Radicals and Rings ICOR 97. Our aim is not to give a comprehensive survey of the subject. We rather wish to present some selected results and on their basis to analyze the development of radical theory of near-rings, emphasizing the role of non-zerosymmetric near-rings. Besides we present some new results on Jacobson type radicals of non-zerosymmetric near-rings.     

A BROWN-McCOY RADICAL FOR T-RINGS

Abstract

A notion of G-regularity is introduced for a Γ-ring M, and from this notion a Brown-McCoy radical, B(M) is defined. B(M) is shown to be a radical in the sense of Kurosh and Amitsur, and analogies of various well-known results on the Brown-McCoy radical of a ring are proved. A “right” Brown McCoy radical, B'(M) can also be defined. In general, B(M) ? B'(M).     

Fixed zeros in the model matching problem for systems over semirings

This paper studies “fixed zeros” of solutions to the model matching problem for systems over semirings. Such systems have been used to model queueing systems, communication networks, and manufacturing systems. The main contribution of this paper is the discovery of two fixed zero structures, which possess a connection with the extended zero semimodules of solutions to the model matching problem. Intuitively, the fixed zeros provides an essential component that is obtained from the solutions to the model matching problem. For discrete event dynamic systems modeled in max-plus algebra, a common Petri net component constructed from the solutions to the model matching problem can be discovered from the fixed zero structure.     

SUBDIRECT IRREDUCIBILITY AND RADICALS

Abstract

We prove lemmas in Andrunakievich s-varieties on the transitivity of the relation “is an ideal of” and concerning subdirectly irreducible factor rings. Applying these lemmas we show that a Plotkin radical introduced in [8] has the ADS-property and is ideal hereditary. These lemmas are applicable in proving a subdirect decomposition for rings having an ideal with 0 antisimple radical. For Jordan algebras and near-rings (they do not form Andrunakievich varieties) we can prove a similar subdirect decomposition concerning ideals with 0 Brown-McCoy radical.     

On nilpotent subsemigroups of the matrix semigroup over an antiring

In this paper, nilpotent subsemigroups in the matrix semigroup over a commutative antiring are discussed. Some basic properties and characterizations for the nilpotent subsemigroups are given, and some equivalent conditions for the matrix semigroup over a commutative antiring to have a maximal nilpotent subsemigroup are obtained. Also, the maximal nilpotent subsemigroups in the matrix semigroup are described.     

Maximum order-index of matrices over commutative inclines: an answer to an open problem

This paper proves that the maximum order-index of n × n matrices over an arbitrary commutative incline equals (n − 1)² + 1. This is an answer to an open problem “Compute the maximum order-index of a member of M_n(L)”, proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984, where M_n(L) is the set of all n × n matrices over an incline L.     

Inequalities for Gondran-Minoux rank and idempotent semirings

The rank-sum, rank-product, and rank-union inequalities for Gondran-Minoux rank of matrices over idempotent semirings are considered. We prove these inequalities for matrices over quasi-selective semirings without zero divisors, which include matrices over the max-plus semiring. Moreover, it is shown that the inequalities provide the linear algebraic characterization for the class of quasi-selective semirings. Namely, it is proven that the inequalities hold for matrices over an idempotent semiring S without zero divisors if and only if S is quasi-selective. For any idempotent semiring which is not quasi-selective it is shown that the rank-sum, rank-product, and rank-union inequalities do not hold in general. Also, we provide an example of a selective semiring with zero divisors such that the rank-sum, rank-product, and rank-union inequalities do not hold in general.     

ON AN ALGEBRAIZATION OF MEASURE THEORY. ABSTRACT SEMIRINGS

Abstract

The concept of a semiring is introduced as a semilattice analogue of a semiring of sets. A detailed study of its embeddability into r-complete Boolean algebras is carried out. Some of its properties which are important from the standpoint of Measure Theory are elucidated. A representation of a semiring by a Semiring of sets is derived. A relation to implicative BCK-algebras is also mentioned.     

ON THE GROENEWALD-HEYMAN STRONGLY PRIME RADICAL

Strongly prime rings were introduced by Handelman and Lawrence [6], and in a recent paper [5] Groenewald and Heyman investigated the upper radical determined by the class of all strongly prime rings. In this paper we continue this investigation. Section 1 provides some alternative characterizations of the radical and in section 2 we discuss general properties of the radical and compare it with other well-known radicals. Finally, combinatorial results on polynomial identities are presented which, combined with our results in section 2. yield some new comnutativity theorems.

All rings considered are associative, but do not necessarily have an identity. As usual, I Δ A means that I is an ideal of the ring A. The notation <x₁,x₂,…> and (x_l,x₂,…) will stand for the subring and ideal, respectively, generated by the elements x₁,x₂,…. The rignt annihirator of a subset S of a ring A will be denoted by ann_A(S).

This work was supported in part by NSERC grants A-8775 and A-8789. and was completed while the first and third authors were visiting Dalhousie University. These authors would like to thank the Department of Mathematics, Statistics and Computing Science at Oalhousie University for its generous hospitality.     

Invariant forms of the elementary unitary Lie superalgebra

In this paper we describe the invariant forms of toral K-graded Lie superalgebras and, in particular, of the elementary unitary Lie superalgebra over a superring K containing .     

A NOTE ON THE g-PRIME RADICAL IN GAMMA RINGS

ABSTRACT

The g-prime radical of a Γ-ring M is equal to either the zero ideal or the prime radical of M. If the prime radical of M is a non-zero ideal, then the following three conditions are equivalent; (i) g-prime radical of M is equal to the prime radical of M; (ii) every g-prime ideal is a prime ideal; and (iii) every g-semiprime ideal is a semiprime ideal.     

A class of commutative semirings with stable range 2

Abstract

The notion and some properties of B-rings, in a natural way, are extended to B- and B_J -semirings which is somewhat similar to the notion of rings having stable range 2. Besides some examples and other results, it is shown that a Gelfand semiring [in particular, a simple semiring] has stable range one and is also a B-semiring (consequently, a B_J -semiring). Finally, we study a class of B_J -semirings and as a counterexample to the case of finite rings, construct two finite semirings such that neither of them is a B_J -semiring.     

Regular Fréchet-Lie groups of invertible elements in some inverse limits of unital involutive Banach algebras

We consider a wide class of unital involutive topological algebras provided with aC ^*-norm and which are inverse limits of sequences of unital involutive Banach algebras; these algebra sare taking a prominent position in noncommutative differential geometry, where they are often called unital smooth algebras. In this paper we prove that the group of invertible elements of such a unital solution smooth algebra and the subgroup of its unitary elements are regular analytic Fréchet-Lie groups of Campbell-Baker-Hausdorff type and fulfill a nice infinite-dimensional version of Lie's second fundamental theorem.     

Semilattice modes I: the associated semiring

We examine idempotent, entropic algebras (modes) which have a semilattice term. We are able to show that any variety of semilattice modes has the congruence extension property and is residually small. We refine the proof of residual smallness by showing that any variety of semilattice modes of finite type is residually countable. To each variety of semilattice modes we associate a commutative semiring satisfying 1 +r=1 whose structure determines many of the properties of the variety. This semiring is used to describe subdirectly irreducible members, clones, subvariety lattices, and free spectra of varieties of semilattice modes.Presented by J. Berman.Part of this paper was written while the author was supported by a fellowship from the Alexander von Humboldt Stiftung.     

Invertible and nilpotent matrices over antirings

In this paper, we characterize invertible matrices over an arbitrary commutative antiring S with 1 and find the structure of GL_n(S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent n×n matrix over an entire antiring can be written as a sum of ⌈log₂n⌉ square-zero matrices and also find the necessary number of square-zero summands for an arbitrary trace-zero matrix to be expressible as their sum.     

MOMENT OF LEvY MEASURE OF OPERATOR-STABLE LAW

In 1979, Jurek gave a characterization of the moment of a full operator-stable μ by eigenvalues of exponent matrix of μ. Here, a characterization of the moment of Lévy measure (restricted on a neighbor of 0) of a full operator-stable μ by eigenvalues of exponent matrix of μ is given.     

Torsion theories induced from commutative subalgebras

We begin a study of torsion theories for representations of finitely generated algebras U over a field containing a finitely generated commutative Harish-Chandra subalgebra Γ. This is an important class of associative algebras, which includes all finite W-algebras of type A over an algebraically closed field of characteristic zero, in particular, the universal enveloping algebra of (or ) for all n. We show that any Γ-torsion theory defined by the coheight of the prime ideals of Γ is liftable to U. Moreover, for any simple U-module M, all associated prime ideals of M in have the same coheight. Hence, the coheight of these associated prime ideals is an invariant of a given simple U-module. This implies the stratification of the category of U-modules controlled by the coheight of the associated prime ideals of Γ. Our approach can be viewed as a generalization of the classical paper by Block (1981) [4]; it allows, in particular, to study representations of beyond the classical category of weight or generalized weight modules.     

On sets of integers with prescribed gaps

For a fixed setI of positive integers we consider the set of paths (p _o,...,p _k ) of arbitrary length satisfyingp _l –p _l–1I forl=2,...,k andp ₀=1,p _k =n. Equipping it with the uniform distribution, the random path lengthT _n is studied. Asymptotic expansions of the moments ofT _n are derived and its asymptotic normality is proved. The step lengthsp _l –p _l–1 are seen to follow asymptotically a restricted geometrical distribution. Analogous results are given for the free boundary case in which the values ofp ₀ andp _k are not specified. In the special caseI={m+1,m+2,...} (for some fixed m) we derive the exact distribution of a random m-gap subset of {1,...,n} and exhibit some connections to the theory of representations of natural numbers. A simple mechanism for generating a randomm-gap subset is also presented.