The Sufficient and Necessary Conditions on Group G So that U is Locally Finite and Finitely Generated

Abstract

Let  G  be a  group and  A be a ring . Let U be the units group in the group ring A[G].  Many recent studies in algebra deal with the structure of the group  U  knowing the structure of the group  G  and vice versa.  In this paper we tried to prove the following problems :

1. What are the necessary and sufficient  conditions on G so that  U  is                                         locally finite?
2. What are the necessary and sufficient  conditions on G so that  U  is  finitely generated ?