PRIME AND SEMIPRIME BI-IDEALS

Abstract

Prime and semiprime bi-ideals in associative rings are defined. This provides a setting for a generalization of the well-known theorem that a commutative ring is Von Neumann regular iff every ideal is semiprime.     

A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

Generalized weakly symmetric rings

A ring R is defined to be GWS   if abc=0$abc=0$ implies bac⊆N(R)$bac\subseteq N(R)$ for a,b,c∈R$a,b,c\in R$, where N(R)$N(R)$ stands for the set of nilpotent elements of R. Since reduced rings and central symmetric rings are GWS, we study sufficient conditions for GWS rings to be reduced and central symmetric. We prove that a ring R is GWS   if and only if the n×n$n\times n$ upper triangular matrices ring _Un(R,R)$U_n(R,R)$ is GWS for any positive integer n. It is proven that GWS rings are directly finite and left min-abel. For a GWS ring R, R is a strongly regular ring if and only if R is a von Neumann regular ring if and only if R is a left SF   ring and J(R)=0$J(R)=0$; R is an exchange ring if and only if R is a clean ring. Finally, we show that GWS exchange rings have stable range 1 and a GWS semiperiodic ring R   with N(R)≠J(R)$N(R)\ne J(R)$ is commutative.     

Quasi-duo rings and stable range descent

In a recent paper, the first author introduced a general theory of corner rings in noncommutative rings that generalized the classical theory of Peirce decompositions. This theory is applied here to the study of the stable range of rings upon descent to corner rings. A ring is called quasi-duo if every maximal 1-sided ideal is 2-sided. Various new characterizations are obtained for such rings. Using some of these characterizations, we prove that, if a quasi-duo ring R has stable range ?n, the same is true for any semisplit corner ring of R. This contrasts with earlier results of Vaserstein and Warfield, which showed that the stable range can increase unboundedly upon descent to (even) Peirce corner rings.     

On Faith's correspondence theorem

A simple proof of an extension of Faith's correspondence theorem for projective modules is given for a Morita context (R, V, W, S) in whichVWV=V andWVW=W.Research supported by N.S.E.R.C. (Canada), Grant No. A 8075.     

Cartan determinants for gentle algebras

The determinant of the Cartan matrix of a finite dimensional algebra is an invariant of the derived category and can be very helpful for derived equivalence classifications. In this paper we determine the determinants of the Cartan matrices for all gentle algebras. This is a class of algebras of tame representation type which occurs naturally in various places in representation theory. The definition of these algebras is of a purely combinatorial nature, and so are our formulae for the Cartan determinants.Received: 29 October 2004     

Annihilator Conditions on Polynomials II

In this paper, we continue the study of various annihilator conditions which were used by Rickart and Kaplansky to abstract the algebraic properties of von Neumann algebras. In our main results, we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of polynomials. These results are somewhat surprising since, in contrast to the polynomial ring case, the nearring of polynomials has substitution for its multiplication operation. Moreover they indicate connections between the ring and nearring structures on polynomials. Examples are provided to illustrate and delimit our results.This revised version was published online in October 2004 with a corrected Received date.The second author was partially supported by the National Science Council of the Republic of China, Taiwan under grant number NSC 90-2115-M-143-001.     

Simultaneous Approximation for Generalized Baskakov-Durrmeyer-Type Operators

The present paper deals with the study of a Durrmeyer-type integral modification of certain modified Baskakov operators. Here we study simultaneous approximation properties for these operators by using the iterative combinations. We obtain an asymptotic formula and an error estimation in terms of higher order modulus of continuity for these operators.      

Ein kurzer elementarer Beweis eines Satzes von O. Steinfeld über die einstufig nichtprimen Ringe

Ohne Zusammenfassung

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Dem fünfzigsten Geburtstag von Professor Dr. Otto Steinfeld gewidmet     

Double ore extensions

A double Ore extension is a natural generalization of the Ore extension. We prove that a connected graded double Ore extension of an Artin-Schelter regular algebra is Artin-Schelter regular. Some other basic properties such as the determinant of the DE-data are studied. Using the double Ore extension, we construct 26 families of Artin-Schelter regular algebras of global dimension four in a sequel paper.     

Hopf structures on ambiskew polynomial rings

We derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type. This construction generalizes many known Hopf algebras, for example U(sl₂), U_q(sl₂) and the enveloping algebra of the three-dimensional Heisenberg Lie algebra. In a torsion-free case we describe the finite-dimensional simple modules, in particular their dimensions, and prove a Clebsch-Gordan decomposition theorem for the tensor product of two simple modules. We construct a Casimir type operator and prove that any finite-dimensional weight module is semisimple.     

Stability of differential equations with piecewise constant arguments of generalized type

In this paper we continue to consider differential equations with piecewise constant argument of generalized type (EPCAG) [M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type, Nonlinear Anal. TMA 66 (2007) 367–383]. A deviating function of a new form is introduced. The linear and quasilinear systems are under discussion. The structure of the sets of solutions is specified. Necessary and sufficient conditions for stability of the zero solution are obtained. Our approach can be fruitfully applied to the investigation of stability, oscillations, controllability and many other problems of EPCAG. Some of the results were announced at The International Conference on Hybrid Systems and Applications, University of Louisiana, Lafayette, 2006.     

Algèbre Pré-Gerstenhaber à homotopie près

This paper is concerned by the concept of algebra up to homotopy for a structure defined by two operations . and $[,]$. An important example of such a structure is the Gerstenhaber algebra (i.e. commutatitve structure with degree 0 and Lie structure with degree ?1). The notion of Gerstenhaber algebra up to homotopy ($G_{\infty }$ algebra) is known: it is a codifferential bicogebra.Here, we give a definition of pre-Gerstenhaber algebra (pre-commutative and pre-Lie) allowing a similar construction for a $\text{pre}{G}_{\infty }$ algebra.Given a structure of pre-commutative (Zinbiel) and pre-Lie algebra and working over the corresponding Koszul dual operads, we will give an explicit construction of the associated pre-Gerstenhaber algebra up to homotopy: it is a bicogebra (Leibniz and permutative) equipped with a codifferential which is a coderivation for the two coproducts.     

A note on interpolation in the Lumer's Nevanlinna class

A sufficient condition for free interpolation in the Lumer's Nevanlinna class in the unit ball of ⁿ is given in term of existence of pluriharmonic majorants.     

Strongly regular rings and rational identities of division rings

We look at division rings in the variety of strongly regular rings and show a connection to the study of rational identities on division rings.Research partially supported by a Grant from NSERC.     

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.     

Torsion-free groups and modules with the involution property

An Abelian group or module is said to have the involution property if every endomorphism is the sum of two automorphisms, one of which is an involution. We investigate this property for completely decomposable torsion-free Abelian groups and modules over the ring of p-adic integers.