R. Wiegandt

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 and (xl,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 annA(S). This work was supported in part by NSERC grants A-8775 and A-8789. and was completed while the first and ...

Essential covers and complements of radicals

We show that a radical has a semisimple essential cover if and only if it is hereditary and has a complement in the lattice of hereditary radicals. In 1971 Snider gave a full description of supernilpotent radicals which have a complement. Recently Beidar, Fong, Ke, and Shum have determined radicals with semisimple essential covers. Using their results, we are able to provide a lower radical representation of complemented subidempotent radicals. This completes Sniders description of hereditary complemented radicals. In the context of radical theory the usefulness of the essential cover operator £ has been known from Armendariz [2] and Rjabuhin [8], who showed that a semisimple class is closed under essential extension if and only if the corresponding radical class is hereditary. In 1970, Stewart [10] characterised semisimple radical classes in terms of subdirect sums of a finite set of finite fields. In 1983, Loi [7] showed that a radical class is semisimple if and only if it is closed under essential extensions (also see Gardner [6]). The last two results naturally lead one to consider the classification of the essential covers of radicals in terms of semisimplicity. In 1994, Birkenmeier [4] showed that the essential cover £p of a supernilpotent radical p is nearly a semisimple class: it is hereditary, closed under extensions, finite subdirect sums, arbitrary direct sums and products. Hence Ep only lacks the requirement of being closed under arbitrary subdirect sums to become a semisimple class. Thus the question arises in [4]: which supernilpotent classes have semisimple essential covers? Imposing this seemingly mild extra condition on the essential cover Sp has turned out to be very restrictive: none of the classical radicals have semisimple essential covers [5]. Recently, Beidar, Fong, Ke, and Shum [3] have fully described radicals having semisimple essential covers. Their description is reminiscent of Stewarts characterisation of radical semisimple classes [10].

The hereditariness of the upper radical

SummaryStarting from a regular classM, one can construct the upper radicalUM of the classM in a category which is like that of associative, alternative or not necessarily associative rings, or that of Lie rings. It turns out that in quite a few cases the upper radical is hereditary. (cf.Suliński [7], Rjabuhin [6], Armendariz [2], Szász—Wiegandt [8]).W. G. Leavitt has suggested the problem: Give a necessary and sufficient condition to be satisfied by the regular classM so that the upper radical classUM ofM is hereditary. In the present paper we shall give such a necessary and sufficient condition. If the classM satisfies an even stronger condition, then theUM-semisimple objects are subdirectly embeddable in a (direct) product ofM-objects. Also a necessary and sufficient condition is given which assures that eachUM-semisimple object can be subdirectly embedded in a (direct) product ofM-objects.

A NOTE ON SPECIAL RADICALS AND PARTITIONS OF SIMPLE RINGS

ABSTRACT For any partition of simple prime rings the lower special radical L determined by is properly contained in the upper radical U of all subdirectly irreducible rings with heart in . Under certain constraints on the class it can be achieved that a polynomial ring is in U but not in L.

RINGS DISTINCTIVE IN RADICAL THEORY

Abstract The purpose of this survey is two-fold, primarily to compile a selection of rings and ring constructions which distinguish radical theoretical properties of rings. This will be achieved mainly by the secondary aim which is to localize the position of most of the known radicals, in particular that J ϕ ≠ B (the existence of a simple primitive ring without non-zero idempotent), K ≠ K p (the existence of a ring A with zero total such that for every prime ideal P(≠ A) the total of A/P is not zero) and that (Veldsmans left superprime radical is properly contained in Olsons uniformly strongly prime radical).

Involutions on Universal Algebras

The concept of an involution in the category of rings is extended to universal algebras and is further generalized in that setting. This approach yields four distinct types of involutions on algebras with two binary operations and two distinct types on the category of rings. Subdirectly irreducible universal algebras with involution are considered in detail. A subdirectly irreducible universal algebra with involution is either subdirectly irreducible as an algebra without involution or it is the subdirect product of two subdirectly irreducible algebras and the involution is the exchange involution. An example from the category of rings is given to illustrate that this result is sharp: no direct sum decomposition can be achieved in general. Focus then turns to algebras with two binary operations, particularly near-rings and rings. Subdirectly irreducible objects in the categories of distributive near-rings and of rings are characterized in greater detail, with close attention given to their additive structure.

A CONDITION IN GENERAL RADICAL THEORY AND ITS MEANING FOR RINGS, TOPOLOGICAL SPACES AND GRAPHS

Dedicated to my teacher, Professor L. Rddei on his 75 th birthday In this note we establish a necessary and sufficient condition for the hereditariness and cohereditariness of a so-called C-class and D-class, respectively. In the ring theory a C-class and a D-class corresponds to a tad/ca1 class and a semisimple class, respectively. As far as hereditary radicals are very common, homomorphically closed semisimple classes are very rare. This fact shows that one of the conditions is a much stronger requirement than the other, ttiough they are categorically dual. Investigating homomorphically closed semisimple classes we shall get a condition involving that a class of rings contains almost all simple rings (Corollary 1). For topological spaces and graphs C-classes and D-classes yield connectednesses and disconnectednesses, respectively. It will turn out that hereditariness (cohereditariness) is almost incompatible with connectedness (disconnectedness). These considerations yield characterizations of the class of all T0-spaces (Corol~ary 6) and of the class of all graphs having at most one loop but with other edges (Corollary 8).

On Radicals with Amitsur Property

Abstract A radical γ has the Amitsur property, if γ(A[x]) = (γ(A[x]) ∩ A)[x] for every ring A. To any radical γ with Amitsur property we construct the smallest radical γ x which coincides with γ on polynomial rings. Distinct special radicals with Amitsur property are given which coincide on simple rings and on polynomial rings, answering thus a stronger version of M. Ferreros problem. Radicals γ with Amitsur property are characterized which satisfy A[x, y] ∈ γ whenever A[x] ∈ γ.

Radical theory in varieties of near-rings in which the constants form

Not all the good properties of the Kurosh-Amitsur radical theory in the variety of associative rings are preserved in the bigger variety of near-rings. In the smaller and better behaved variety of O-symmetric near-rings the theory is much more satisfactory. In this note we show that many of the results of the 0-symmetric near-ring case can be extended to a much bigger variety of near-rings which, amongst others, includes all the O-symmetric as well as the constant near-rings. The varieties we shall consider are varieties of near-rings, called Fuchs varieties, in which the constants form an ideal. The good arithmetic of such varieties makes it possible to derive more explicit conditions. (i) for the subvariety of constant near-rings to be a semisimple class (or equivalently, to have attainable identities), (ii) for semisimple classes to be hereditary. We shall prove that the subvariety of 0-symmetric near-rings has attainable identities in a Fuchs variety, and extend the theory of overnilpotent radicals of...

Characterizations of the Brown-McCoy radical

We characterize the Brown--McCoy radical in an arbitrary universal class of not necessarily associative rings or near-rings, as a lower radical. We also establish the corresponding upper radical representation with respect to an arbitrary universal class of alternative rings or near-rings, as well as the intersection property with respect to the upper class of this representation. Finally we exhibit a natural module counterpart for this ring radical.