Published online by Cambridge University Press: 11 January 2010
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.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.