Elementary radical classes

B J Gardner

doi:10.24330/ieja.373634

Abstract

A radical class R of rings is elementary if it contains precisely those rings whose singly generated subrings are in R. Many examples of ele- mentary radical classes are presented, and all those which are either contained in the Jacobson radical class or disjoint from it are described. Attention is given to those elementary radical classes which are de nable by composition subsemigroups of the free ring on one generator. Whether every elementary radical class is of this form remains an open question.

Highlights

A radical class R is elementary if it satisfies the condition A ∈ R if and only if every singly generated subring of A is in R
Such classes have been considered by several authors under several names. They were called elementary by Stewart [25], semi-strictly hereditary by Ryabukhin [23] and 1-radical classes by the author [9]
R+ denotes the additive group of a ring R, G0 or R0 the zeroring on an abelian group G or the additive group of a ring R respectively

Summary

Introduction

A radical class R is elementary if it satisfies the condition A ∈ R if and only if every singly generated subring of A is in R. Such classes have been considered by several authors under several names. Elementary radical classes are examples of locally equational classes as defined by Hu [15] The latter are generalizations of varieties, so since Mal’tsev-Neumann products of varieties are varieties, there may be some interest in the result that products of locally equational classes need not be locally equational. This follows from results on products of radical classes in [11]

Examples of elementary radical classes defined by semigroups

Are all elementary radical classes defined by semigroups?

Closing remarks