Radical Classes Of Rings Research Articles

Algebraic Structure of Supernilpotent Radical Class Constructed from a Topology Thychonoff Space

A radical class of rings is called a supernilpotent radicals if it is hereditary and it contains the class for some positive integer In this paper, we start by exploring the concept of Tychonoff space to build a supernilpotent radical. Let be a Tychonoff space that does not contain any isolated point. The set of all continuous real-valued functions defined on is a prime essential ring. Finally, we can show that the class of rings is a supernilpotent radical class containing the matrix ring .

Topological connectednesses and congruences

It is known that the connectednesses of topological spaces in the sense of Preuß is the topological analogue of the Kurosh-Amistsur radicals of algebraic structures in a categorical sense. Here this connection is further explored. As in universal algebra, a congruence on a topological space has been defined. It is shown that a connectedness can be characterized in terms of conditions on congruences which are the precise topological analogues of those conditions that characterize the radical classes of rings in terms of ideals.

FURTHER REMARKS ON ELEMENTARY RADICALS AND ASSOCIATED FILTERS OF IDEALS

AbstractRyabukhin showed that there is a correspondence between elementary radical classes of rings and certain filters of ideals of the free ring on one generator, analogous to the Gabriel correspondence between torsion classes of left unital modules and certain filters of left ideals of the coefficient ring. This correspondence is further explored here. All possibilities for the intersection of the ideals in a filter are catalogued, and the connections between filters and other ways of describing elementary radical classes are investigated. Some generalisations to nonassociative rings and groups are also presented.

Radicalizers

ABSTRACTFor a Kurosh–Amitsur radical class of rings, we investigate the existence, for a radical subring S of a ring A, of a largest subring T of A for which S is the radical. When T exists, it is called the radicalizer of S. There are no radical classes of associative rings for which every radical subring of every ring has a radicalizer. If a subring is the radical of its idealizer, then the idealizer is a radicalizer. We examine radical classes for which each radical subring is contained in one which is the radical of its own idealizer.

ESSENTIAL COVERS OF RADICAL CLASSES

This paper is primarily a survey of the results on essential covers of radical classes of rings. Some problems are posed, and some new results are presented.

REGULARITIES AND SUBDIRECT PRODUCTS OF RINGS

Abstract Structure theorems are obtained for certain radical classes of rings (including the Brown-McCoy radical class, the class of λ-rings, the class of E 5-rings, the class of E 6-rings and the class of f-regular rings) by generalizing the concept of a prime ideal.

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).