On the Unit Groups of a Class of Total Quotient Rings of Characteristic p k with k ≥ 3

Abstract

Let R be a commutative completely primary finite ring. The struc tures of the groups of units for certain classes of R have been determined. It is well known that completely primary finite rings play a crucial role in the endeavors towards the classification of finite rings. Let G be an arbitrary finite group. The classification of all finite rings Ri so that U(Ri) ∼= G is still an open problem. In this paper, we consider S ⊂ R to be a saturated multiplicative subset of R and construct a total quotient ring RS whose group of units is characterized, when char RS = p k , k ≥ 3. It is observed that U(R) ∼= U(RS), since R ∼= RS. The cases when char RS = p k , k = 1, 2 have been studied in a related work.