Class Of Groups Research Articles

On subidempotent radical classes

Several papers (see [5], [6], [7] and [8]) gave necessary and sufficient conditions for UM to be a hereditary radical class if M is a class of rings satisfying various conditions. Analogous to the concepts of special classes and special radical classes in the case of supernilpotent radical classes, we will use some of the results from the above—mentioned papers to define the concepts of super classes and super radical classes in the case of subidempotent radical classes.

Quast m-rings and their radicals

The concept, of quasi M-rings is introduced if N is a class of rings satisfying certain conditions. This concept enables us to characterize certain radical classes and also to construct new N-radicals.

A NOTE ON RADICALS FOR HEMIRINGS

Abstract The concept of upper radicals for hemirings is developed and used to establish results similar to those in ring theory for special class of hemirings. The class of all nil hemirings is shown to be a radical class.

On a radical class of lattice ordered groups

On a radical class of lattice ordered groups

General Heredity and Strength for Radical Classes

An H-relation, as introduced by Rossa and Tangeman [4], is a relation σ on the class of associative rings with their subrings satisfying the following conditions:(1) IσR implies that I is a subring of R;(2) if IσR and f is a homomorphism of R, then (If)σ(Rf);(3) if IσR and J is an ideal of R, then (I∩J)σJ.Puczylowski [3] imposes also the condition(4) if J is an ideal of R, then JσR.A further condition satisfied by many familiar H-relations is the following:(5) if f is a homomorphism from a ring R onto a ring S and BσS, then there exists a subring A of R such that AσR and Af = B.

Radical classes of finite groups with the lockett condition

Radical classes of finite groups with the lockett condition

SUBIDEMPOTENT RADICAL CLASSES

Abstract We show that a semisimple class of rings M satisfies condition (β) (i.e. I ◃ A ◃ M and A2 = 0 implies A/I e M) if and only if the corresponding radical class is hypersolvable or hypo-idempotent. Any radical class R which satisfies condition (F) (i.e. J ◃ I ◃ A and I/J e R implies J ◃ A) must by hypo-idempotent. If the radical class is regular, the converse is also true. We also give characterizations of the semisimple classes of hypo-idempotent and subidempotent radical classes.

MODULO-CONSTANT IDEAL-HEREDITARY RADICALS OF NEAR-KINGS

ABSTRACT It is known that no “good” radical of (not necessarily o-symmetric) near-rings can be ideal-hereditary. Using the results of the o-symmetric case, we show that the situation is not as bad as on first appearances and we give several examples of (Kurosh-Amitsur) radicals of near-rings for which the semisimple class is hereditary and the radical class is hereditary on left invariant ideals. We also extend some recent results on left strong radicals from the o-symmetric case to the general case.

Closure operators on the lattice of radical classes of lattice ordered groups

Closure operators on the lattice of radical classes of lattice ordered groups

A uniformly strongly prime radical

AbstractThe class of all uniformly strongly prime rings is shown to be a special class of rings which generates a radical class which properly contains both the right and left strongly prime radicals and which is independent of the Jacobson and Brown-McCoy radicals.

Closure operators on radical classes of lattice-ordered groups

Closure operators on radical classes of lattice-ordered groups

Nil and s-prime Ω-groups

AbstractThe concepts nilpotent element, s-prime ideal and s-semi-prime ideal are defined for Ω-groups. The class {G|G is a nil Ω-group} is a Kurosh-Amitsur radical class. The nil radical of an Ω-group coincides with the intersection of all the s-prime ideals. Furthermore an ideal P of G is an s-semi-prime ideal if and only if G/P has no non-zero nil ideals.

Book reviews

BRITISH INDUSTRIAL RELATIONS Gill Palmer SOCIAL INEQUALITY AND CLASS RADICALISM IN FRANCE AND BRITAIN Duncan Gallie

Necessary conditions for the coperiodicity of quotients of direct products of Abelian groups

The problem of describing the structure of the group P---Xli~zAi/~iAi has attracted the attention of specialists in the theory of Abelian groups. The task of investigating the structure of such a group was proposed in [I] (see Problem 24). It was proved in [2, 3] that the group P is algebraically compact whenever the set of indices is countable. This result had a considerable influence on the development of the theory of narrow groups, groups of extensions, radical classes, etc. The above-mentioned result was generalized in [4] to the case of K-direct sums. The search for necessary conditions for the group P to be algebraically compact was begun in [5, 6], and such a necessary condition was found there for quotients of a power of a single group. Finally, a necessary condition was found in [7] for algebraical compactness of an arbitrary group ~eIAi/H. It follows from [2, 3] that the for H=~,i~A~ , this condition is also sufficient. In view of the special importance of the class of coperiodic groups, the question of the coperiodicity of P is a very natural one. The main result of our article is Theorem i, which gives a necessary condition for the group OA~/H to be coperiodic. This result immediately implies a necessary condition for K the coperiodicity of I],=xAi/H (see Theorem 2). If K is a o-ideal of the Boolean algebra B (f), H = @Ai, where K I C K, then our necessary condition turns out to be sufficient as KI well (see Theorem 3). In Theorem 4, we give a necessary and sufficient condition for P to be coperiodic. In Theorem 5, we prove that there exists a maximal ideal KCB (I), such that the group @Ai/@A i is coperiodic. K KI All the groups considered below are assumed to be Abelian. The following notation will be adhered to throughout the article: i) B(I) is the Boolean algebra of all subsets of the set I.

Radical classes, connectednesses and torsion theories

The equality or non-equality of radical classes, torsion classes and connectednesses is investigated in a category that could be any of the concrete categories in which a radical theory, connectednesses theory or a torsion theory have been developed. Among others, it is shown that every connectedness is a radical class. The converse is not true. If, however, a radical class is strict, then it is a connectedness. Furthermore it is proved that every torsion class is a radical class and the converse is not necessarily true. Every radical class with the ADS-property is a torsion class.

A categorical approach to largest and smallest connectednesses and disconnectednesses

Motivated by the existence of largest and smallest proper connectednesses and disconnectednesses in the categories of topological spaces (Top) and undirected graphs with loops (Graph), we look for necessary and sufficient conditions for the existence of such classes in a general categorical setting. This leads us to consider certain classes of simple objects. We start by giving the necessary theory of radical classes (connectednesses) and semisimple classes (disconnectednesses). In the next section we define certain subclasses of the class of all simple objects. These classes are then used in the third section to obtain largest and smallest radical and semisimple classes. Lastly we generalize the results of Wiegandt [8] concerning the hereditariness and cohereditariness of connectednesses and disconnectednesses. K will always be an arbitrary category and E and M will denote subclasses of the class of K-epimorphisms and K-monomorphisms respectively. Throughout this paper we shall assume that both E and M are closed under composition with isomorphisms and that the class of all K-identities is contained in EflM. Note that these two assumptions imply that the class of all K-isomorphisms is contained in E flM. A~BCE will mean there is a morphism ~: A-~B in Kwith c~EE. A ~ B E M will have a similar meaning. Let T be the class of all trivial objects in K (cf. [2]), i.e. T is that class of objects which satisfies the following conditions:

ON THE EXISTENCE AND NON-EXISTENCE OF COMPLEMENTARY RADICAL AND SEMISIMPLE CLASSES

ABSTRACT In certain categories of mathematical structures, non-trivial complementary radical classes (torsion classes or connectednesses) can be found. The question is why this is true for some but not for all categories. The answer depends on the embedding of trivial objects into nontrivial objects and is given by our main result: Any ‘reasonable’ category has no non-trivial complementary radical and semisimple classes if and only if for every trivial object T and every non-trivial object A there is a morphism T → A. Roughly, a ‘reasonable’ category in our sense is one with at least one object into which a terminal object can be embedded and has finite products, coproducts or lexicographic products.

On radical and coradical classes of I-groups

On radical and coradical classes of I-groups

On Brown–McCoy radical classes in categories

What does a simple ring with unity, a topological T0-space and a graph that has at most one loop but may possess edges, have in common? In this note we show that they all are Brown–McCoy semisimple. Suliński has generalised the well-known Brown–McCoy radical class of associative rings (cf. [1]) to a category which satisfies certain conditions. In [3] he defines a simple object, a modular class of objects and the Brown–McCoy radical class as the upper radical class determined by a modular class in a category which, among others, has a zero object and kernels. To include categories like that of topological spaces and graphs, we use the concepts of a trivial object and a fibre. We then follow Suliński and define a simple object, a modular class of objects and then the Brown–McCoy radical class as the upper radical class determined by a modular class.

Upper radicals of regular classes

The class M is called regular if MC= SUM, or equivalently if UM is the largest radical class relative to which every REM is semisimple. A radical class UM is said to have the intersection property relative to a class N if for every ring R the radical UM(R)= A(I I R/IEN). A radical class is called hereditary if I-~ R E UM implies IE UM, and in a recent paper several criteria were derived equivalent to UM hereditary [5; Theorem 4]. In the case that M has the property