Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

KIRÁLY, B., On the powers of the augmentation ideal of a group ring

On the powers of the augmentation ideal oi a group ring BERTALAN KIRÁLY* Abstract. In this paper we give necessary and sufficient conditions for groups which have finite terminais with respect to a commutative ring with unity of non-zero characteristic. 1. Introduction Let R be a commutative ring with unity, G a group and RG its group ring and let A(RG) dénoté the augmentation ideal of RG

, that is the kernel of the ring homomorphism 0 : RG —» R which maps the group elements to 1. It is easy to see that an E-module A(RG) is a free module with the elements g — 1 (g G G) as a basis. It is clear that A(RG) is the ideal genereted by ail elements of the form g — 1, <7 G G. The powers A

X(RG) of A(RG) are defined inductivelyiA^G) = A

1 (RG), A

X+

1(RG) = A

X(RG) • A(RG), if A is not a limit ordinal, and otherwise A\RG) = P| A

v(RG). I/< A It is easy to see that the right ideal A

X(RG) is a two-sided ideal of RG for ail ordinals À > 1. Evidently there exists a least ordinal r = T

R(G) such that A

R(RG) = A

T+

1(RG). In [2] r was called the augmentation terminal (or terminal for simple when it is obvious from the context what ring R we are working with) of G with respect to R. We shall use this terminology, and also we shall write oo A

U(RG) = P| A

n(RG) 71= 1 for the first limit ordinal U. ff G = (1) we put T

r(G) = 1. In generál, the question of the classification of groups in regarding to values of the terminais and also of the computation of these terminais, is far from being simple ( see [2]). * Research supported by the Hungárián National Foundation for Scientific Research Grant, No T014279.