Abstract

We consider the construction of associated group of a ring with identity element. The characterization of rings with a periodic, FC-group, or nilpotent associated group are given. It is shown that if the adjoint group R∘ of a semiperfect ring R with some finiteness conditions is an Engel group then it is nilpotent and R is a Lie nilpotent ring.

Introduction

Let R be an associative ring with an identity element. The set of all elements of R forms a semigroup with the identity element 0 ∈ R under the operation a ∘ b = a + b + ab for all a and b of R. The group of all invertible elements of this semigroup is called the adjoint group of R and is denoted by R∘. Clearly, if R has the identity 1, then 1 + R∘ coincides with the group of units U(R) of the ring R and the map a → 1 + a with a ∈ R is an isomorphism from R∘ onto U(R).

Many authors have studied rings with prescribed adjoint groups (or equivalently, groups of units) (see, for example, [1-16]).

This paper is concerned with the question of how properties of associated group influence some characteristics of ring structure. The idea of associated group was introduced in [1] for radical rings. We extend this construction to arbitary associative rings with identity element.