Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

KlRALY, B., Residual Lie nilpotence of the augmentation ideal

88 Bertalan Király (1) G is discriminated by the class VQ or (2) G is discriminated by the class V

p. Proof. If G' is with no generalized torsion elements (with respect to the lower central series of G), then by Theorem A A^(Z

PG) = 0 if and only if G is discriminated by the class V

0

. Let us consider the case when G' contains a generalized torsion element. Let A^(Z

PG) = 0. By Theorem C there exists a non-empty subset 0 of the set of primes, such that r\

q^J

q(Z

p) =• 0. It is known that J

p(Z

p

) = 0 and for a prime q ^ p, J

q(Z

p) = Z

p. Therefore p G ÍÍ. If O = {p}, then by the last theorem G is discriminated by V

p. If Í2 contains a prime q ^ p, then we choose A C ft such that ft \ A = {P}. Then V[

q^\J

q(Z

v) / 0 and by Theorem C G is discriminated by the class V

p. Conversely. If G is discriminated by the class V

p

, we put Q = {p}, and the proof follows from Theorem C. From Theorem A and C we also get the results of I. Musson and A. Weiss ([2], Theorem A). References [1] KIRALY B., The residual nilpotency of the augmentation ideal, Publ. Math. Debrecen., 45 (1994), 133-144. [2] MUSSON I., WEISS A., Integral group rings with residually nilpotent unit groups', Arch. Math., 38 (1982), 514-530. [3] PARMENTER, M. M., PASSI, I. B. S. and SEHGAL, S. K., Polynomial ideals in group rings, Canad. J. Math., 25 (1973), 1174-1182. [4] PASSI, I. B. S., Group ring and their augmentation ideals, Lecture notes in Math., 715, SpringerVerlag, Berlin-Heidelberg-New York, 1979. [5] PASSI, I. B. S., PASSMAN D. S. and Sehgal S. K., Lie solvable group rings, Canad. J. Math. 25 (1973), 748-757. [6] SEHGAL S. K., Topics in group rings, Marcel-Dekker Inc., New YorkBasel, 1978. BERTALAN KIRÁLY ESZTERHÁZY KÁROLY TEACHERS' TRAINING COLLEGE DEPARTMENT OF MATHEMATICS LEÁNYKA U. 4. 3301 EGER, PF. 43. HUNGARY E-mail: kiraly@gemini.ektf.hu