Class Of Groups Research Articles

Characterizations of semisimple classes

AbstractTwo characterizations of semisimple classes of associative and alternative rings (and semigroups with 0) are given:(i) A class is a semisimple class if and only if it is hereditary, closed under extensions and subdirect sums;(ii) A class is a semisimple class if and only if it is hereditary, closed under extensions, and has the co-inductive property.The first characterization sharpens Armendariz's (1968) result proved for associative rings, the second one is categorically dual to a characterization of radical classes due to Amitsur (1954).

On Relations among radical properties

In this paper we work in the class of associative rings. The fundamental definitions and properties of radicals may be found in Divinsky [1].If α is a radical class of rings we shall denote the class of semi-simple rings by the radical of a ring R we shall denote by α(R).

Reflected radical classes

1. Introduction. Our categorical terminology is selected from the books of FR~YD [5] and MITCHELL [11 ]. We shall not distinguish notationally between a subobject and a representative morphism. On the basis of ~UL'GEn~ER'S work in [13] we say that a category is suitable for radical theory if it has a zero object, kernels and conormal epimorphic images, if unions of ascending chains of normal subobjeets exist and are normal and if images under conormal epimorphisms of normal subobjects are normal subobjects. Throughout this paper "category" will mean a category suitable for radical theory in which the class of normal subobjects of each object is a set. Let ~ be a non-empty, isomorphism closed class of objects in a category ~. A subobjects S ~ A is a ~-subobject if S ~ ~. The class ~ is a radical class if (see [13, Theorem 3.1]) R1. If .4 ~ B is a conormal epimorphism and .4 E ~, then B E ~. R2. An object A is in & if, for every non-zero conormal epimorphism .4 ~ B, B has a non-zero normal ~-subobject. If ~ is a radical class then each object .4 has a unique largest ~-subobject ~(.4) ~ A, and `4 is called ~ semi-simple if ~(A) = 0. Suppose that ~ also satisfies: for each normal subobject N ~ B and conormal epimorphism `4 ~ B the pullback

Strong Radical Classes and Idempotents

All rings are associative but do not necessarily have identities. Definitions and basic results about radical classes can be found in [2]. A radical class is strong [3] if for every ring A, (A) contains all left and right -ideals of A.

Semi-simple radical classes of algebras and attainability of identities

This paper is concerned with the investigation of some interplay between the theories of (Kurosh-Amitsur) radicals and varieties of (not necessarily associative) algebras. Specifically, it is shown that a variety is a radical class if and only if it is closed under extensions, while a radical class which is also a semi-simple class is the same thing as a variety with attainable identities in the sense of T. Tamura (J. Algebra 3 (1966), 261-276). In certain instances it is shown that the two properties of varieties are equivalent.

On semi-simple radical classes

It has been incorrectly asserted that each non-trivial semi-simple radical class of associative rings is a variety defined by an equation of the form xn = x. In this paper we give, for each non-trivial semi-simple radical class of associative rings, a set of equations which does define that class as a variety.

The radical equation P(An) = (P(A))n

The purpose of this paper is to impose conditions on a radical class P so that the P-radical of the ring of n × n-matrices over a ring A is equal to the ring of n×n-matrices over the ring P(A). In (1), Amitsur gave such conditions, but with the stipulation that the radical class P contained all zero-rings (rings in which all products are zero). In what follows, we shall be working within the class of associative rings.

A simple ring separating certain radicals

All rings considered will be associative. For a class M of rings let UM be the class of all rings having no non-zero homomorphic image in M. A hereditary class M of prime rings is called a “special class” [see 1, p. 191] if it has the property that when I ∈ M with I an ideal of a ring R, then R/I* ∈ Mwhere I* is the annihilator of I in R, and the corresponding radical class UM is then a “special radical”. Let S be the class of all subdirectly irreducible rings with simple heart.

Sub-prime radical classes determined by zerorings

It is shown that the correspondence which associates with each radical class τ of abelian groups the (radical) class of prime radical rings with additive groups in τ gives a complete classification of those radical classes of rings which are determined (as lower radicals) by zerorings.

Some radical constructions for associative rings

Dickson's construction [1] of radical and semi-simple classes for certain abelian categories is a rather straightforward procedure in comparison with the methods traditionally used in more general situations. In §2 of the present paper we use a well-known characterization of the lower radical class to obtain, via consideration of maps with accessible images, a similar “homomorphic orthogonality” characterization of radical and semi-simple classes of associative rings. By substituting certain other subring properties for accessibility, we are then able to obtain simple constructions of various types of radical classes, including those which are strict in the sense first used by Kurosh [3] for groups.

Radical Classes need not have a Unique Maximal R0-Closed Subclass

It was shown in [1] that certain classes of groups which are closed under quotients, and extensions contain a unique maximal inclosed subclass. These results prompted the question whether there exists a class of groups which is closed under quotients and extensions and yet does not have a unique maximal R0-closed subclass. This note provides an example of such a class.

Crystal symmetry groups and their representations II orbits, strata and crystallographic classes of space groups

Crystal symmetry groups and their representations II orbits, strata and crystallographic classes of space groups

The Internal Stratification of the Working Class: System Involvements of Auto Workers in Four Countries

WILLIAM H. FORM University of Illinois, Urbana-Champaign American Sociological Review 1973, Vol. 38 (December):697-711 The relevance of structural differentiation for social stratification has rarely been studied empirically. This research compares automobile workers of different skill levels for their involvement in the family, work group, union, neighborhood, community, and nation in four countries varying in industrial development. In the more industrialized countries,. compared to other employees, skilled workers participate more in work-related social systems and in systems which extend beyond family and neighborhood. The increasing differences in participation within the working class which accompany industrialization and the increasing social power of the skilled workers in the community partly result from their greater ability to link other systems to their solidary work groups. Industrialization and structural differentiation of society may not homogenize industrial workers but rather create internal cleavages which increase the social power of skilled workers and decrease the possibility of a unified or radical working class movement.

On classes of groups defined by systems of subgroups

On classes of groups defined by systems of subgroups

Varieties and section closed classes of groups

Varieties and section closed classes of groups

A supernilpotent non-special radical class

Let F be the upper radical determined by all fields. The supernilpotent radical classes which are not special have thus far always contained F properly. The purpose of this note is to construct a countably infinite number of supernilpotent radical classes which are not special and each of which is properly contained in F. The construction involves a ring due to Leavitt which is interesting in its own right and is not generally known. All rings considered are associative.

Strongly Hereditary Complements of Radical Classes

Strongly Hereditary Complements of Radical Classes

Strict Radical Classes of Associative Rings

Strict Radical Classes of Associative Rings

Stable automorphisms of certain classes of groups

Stable automorphisms of certain classes of groups

Strict radical classes of associative rings

A class of rings is strongly hereditary if it is closed under taking subrings. Strict radical classes (that is, radical classes for which the class of semisimple rings is strongly hereditary) are studied, strongly hereditary strict radical classes are classified, and it is shown that the category of associative rings has no proper localizing subcategories in the sense of Šul’geĭfer.