The Units of Group-rings

when addition and multiplication are defined in the obvious way, form a ring, the group-ring of G over K, which will be denoted by R (G, K). Henceforward, we suppose that K has the modulus 1, and we denote the identity in G by e0. Then R(G,K) has the modulus l.e0. Since no confusion can arise thereby, the element 1. e in R(G, K) will be written as e, and whenever it is convenient, the elements e0 in G and re0 in R(G, K) as 1 and r respectively. The symbol e, with or without subscripts, will always denote an element in G. If the elements Ex, E2 in R(G, K) satisfy JE1E2 = 1, Ex will be said to be a left unit, and E9 a right unit in R(G, K). If 77 is a left (or right) unit in K, then rje is a left (or right) unit in R(G, K). Such a unit will be described as trivial. The units in R(G, K) form a group if and only if every right unit is also a left unit. This is so, for instance, if both G and K are Abelian. It is also true if G is a finite group, and K is any ring of complex numbers, for then the regular representation| of G can be extended to give an isomorphism of R(G, K) in the ring of ordinary matrices. The first object of this paper is to study units in R(G,C), where C is the ring of rational integers. In § 2 we take G to be a finite Abelian group,