An alternative approach to the structure theory of PI-rings

We expose a rather simple and direct approach to the structure theory of prime PI-rings (“Posner’s theorem”), based on fundamental properties of the extended centroid of a prime ring.     

Algebraic prime subalgebras in simple Artinian algebras

We prove a Wedderburn-Artin type theorem for algebraic prime subalgebras in simple Artinian algebras, giving a generalized version of Yahaghi’s theorem [B.R. Yahaghi, On F-algebras of algebraic matrices over a subfield F of the center of a division ring, Linear Algebra Appl. 418 (2006) 599-613]. We also show that every semiprime left algebraic subring in a semiprime right Goldie ring must be a semiprime Artinian ring.     

Special radicals of near-rings and Γ-near-rings

It was previously shown that every special radical classR of rings induces a special radical class R of -rings. Amongst the special radical classes of near-rings, there are some, called the -special radical classes, which induce, special radical classes of -near-rings by the same procedure as used in the ring case. The -special radicals of near-rings possess very strong hereditary properties. In particular, this leads to some new results for the equiprime andI ₃ radicals.     

CLOSED SUBMODULES OF NORMALIZING BIMODULES OVER SEMIPRIME RINGS

In this paper we study closed sub-bimodules of normalizing bimodules over semiprime rings. We extend the main results which are known for centred bimodules and several other results which are new even for centred bimodules are also obtained. In particular, we prove that the theorem on one-to-one correspondence between closed submodules obtained in former papers for centred bimodules is also true for normalizing bimodules. Finally, we give some applications of the main results.

     

A note on the splitting principle

We offer a new perspective on the splitting principle. We give an easy proof that applies to all classical types of vector bundles and in fact to G-bundles for any compact connected Lie group G. The perspective gives precise calculational information and directly ties the splitting principle to the specification of characteristic classes in terms of classifying spaces.     

Commutativity conditions for rings: 1950–2005

During the last 55 years there have been many results concerning conditions that force a ring to be commutative. These results were stimulated by Jacobson's famous result and were extensively developed by Herstein. This paper will survey the area by organizing the results according to whether they come from variations on Herstein's conditions, depend on general polynomial conditions, depend on the presence of a derivation, or whether a ring has special properties that make commutativity more easily accessible. Finally, the most recent conditions concern product sets and lead to results in a new area of inquiry.     

TORSION PRERADICALS—A TOOL FOR ANALYSING RINGS

Abstract

This paper surveys a selection of results in the literature on torsion preradicals; these are left exact preradical functors on the category of unital right modules over an associative ring with identity. Various well known classes of rings such as semisimple, artinian, perfect and strongly prime are characterized in terms of torsion preradicals. A classification of prime rings using torsion preradicals is also exhibited. Rings all of whose torsion preradicals are radicals and rings whose torsion preradicals commute, are investigated. An application of the latter condition to Jacobson's Conjecture is presented.     

GENERALISED IDEALS OF RINGS

Abstract

Using a general definition of a regularity for rings, F- and F- qausi-ideals of a ring are defined. These concepts are shown to be generalizations of ideals or one-sided ideals of a ring. An F-semi prime F—(F-quasi-) ideal of a ring R is also defined. F-regular rings are characterized in terms of F-semi prime F- (F-quasi-) ideals for a large class of polynomial regularities including some well known regularities. A more general characterization of the prime radical β(R) of a ring are given in terms of F—(F-quasi-) ideals.     

UNIFORMLY STRONGLY PRIME RADICALS

Abstract

In this article different characterizations for a uniformly strongly prime ring are given as well as a way of constructing a uniformly strongly prime ring. Uniformly strongly prime rings of bound one as well as the upper radical determined by this special class of rings are also investigated.     

Topological invariance of intersection homology without sheaves

In (1) Goresky and MacPherson defined intersection homology groups for triangulable pseudomanifolds and showed they were PL invariants. Then in [2] they generalized these groups to any pseudomanifold and showed they were topological invariants. These groups have generated a great deal of interest. However, [2] is difficult for many mathematicians (including this author) because it requires a familiarity with a great deal of hefty sheaf-theoretic machinery. This is too bad, because the basic ideas behind intersection homology (elucidated in [1]) are very geometric.In this paper we give a sheafless definition of intersection homology groups for an arbitrary stratified set and we give an elementary sheafless proof that they are topological invariants, i.e. independent of the stratification.In doing so, we find some new perversities whose intersection homology groups are topological invariants. Unfortunately, these new perverse intersection homology classes do not seem to intersect with anything (which is probably why they were ignored by Goresky and MacPherson). But in any case these groups are invariants of singular spaces which might be of some interest.     

Quasi-Armendariz rings relative to a monoid

For a monoid M, we introduce M-quasi-Armendariz rings which are a generalization of quasi-Armendariz rings, and investigate their properties. The M-quasi-Armendariz condition is a Morita invariant property. The class of M-quasi-Armendariz rings is closed under some kinds of upper triangular matrix rings. Every semiprime ring is M-quasi-Armendariz for any unique product monoid and any strictly totally ordered monoid M. Moreover, we study the relationship between the quasi-Baer property of a ring R and those of the monoid ring R[M]. Every quasi-Baer ring is M-quasi-Armendariz for any unique product monoid and any strictly totally ordered monoid M.     

On Jordan Triple Derivations and Related Mappings

In this article we study certain functional equations and systems of functional equations related to (generalized) derivations on semiprime rings. In particular, we prove that any generalized Jordan triple derivation on a 2- torsion free semiprime ring is a generalized derivation. We also prove that any (generalized) Jordan triple *-derivation on a 2-torsion free semiprime *-ring is a (generalized) Jordan *-derivation. The second author was supported in part by the Ministry of Science, Education and Sports of the Republic of Croatia (project No. 037-0372784-2757).     

Second-order differential invariants of the rotation group O(n) and of its extensions: E(n), P(1, n), G(1, n)

Functional bases of second-order differential invariants of the Euclid, Poincaré, Galilei, conformal, and projective algebras are constructed. The results obtained allow us to describe new classes of nonlinear many-dimensional invariant equations.     

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.     

Right distributive quasigroups on algebraic varieties

In this paper we develop a structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes generating algebraic Fischer groups (in the sense of [6]) such that the mappingx x ^–1 ax, fora , is an automorphism of (as variety). We also give examples of algebraic Fischer groups where this does not happen. It becomes clear that the class of algebraic right distributive quasigroups has nice properties concerning subquasigroups, normal subquasigroups and direct product.We give a complete classification of one- and two-dimensional as well as of minimal algebraic right distributive quasigroups.     

Homological characterization of lean algebras

Certain classes of lean quasi-hereditary algebras play a central role in the representation theory of semisimple complex Lie algebras and algebraic groups. The concept of a lean semiprimary ring, introduced recently in [1] is given here a homological characterization in terms of the surjectivity of certain induced maps between Ext¹-groups. A stronger condition requiring the surjectivity of the induced maps between Ext ^k -groups for allk≥1, which appears in the recent work of Cline, Parshall and Scott on Kazhdan-Lusztig theory, is shown to hold for a large class of lean quasi-hereditary algebras. Research partially supported by NSERC of Canada and by Hungarian National Foundation for Scientific Research grant no. 1903 Research partially supported by NSERC of Canada     

A class of strongly homotopy Lie algebras with simplified sh-Lie structures

Given a complex that is a differential graded vector space, it is known that a single mapping defined on a space of it where the homology is non-trivial extends to a strongly homotopy Lie algebra (on the graded space) when that mapping satisfies two conditions. This strongly homotopy Lie algebra is non-trivial (it is not a Lie algebra); however we show that one can obtain an sh-Lie algebra where the only non-zero mappings defining it are the lower order mappings. This structure applies to a significant class of examples. Moreover in this case the graded space can be replaced by another graded space, with only three non-zero terms, on which the same sh-Lie structure exists.