Maximal quotient rings of group rings

Abstract

The two which have received the greatest attention are theclassical (Ore) quotient ring and the maximal (Utumi) quotient ring.The classical quotient ring has a relatively straightforward description,but it is only defined for rings which satisfy the so-called Ore con-dition. In contrast the maximal quotient ring is less easy to describebut is defined for all rings. In both cases there are distinct notionsof left and right quotient rings and we will always consider leftquotient rings.For group rings the classical quotient ring has been studied byHerstein and Small [2], Passman [5, 6], M. Smith [7], and P. F. Smith[81, and the maximal quotient ring has been studied by Burgess [1].This paper investigates the relationship of the maximal quotientrings of group rings, subgroup rings, and the centers of group rings.The object is to obtain for the maximal quotient ring analogues oftheorems of Passman and M. Smith on the classical quotient ring.Their techniques are used for the group ring arguments while thequotient ring arguments reflect the formalism of the maximal quotientring.