Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1995-1996. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 23)
KlRALY, B., The Lie augmentation terminals of group
The Lie augmentation terminals of groups 69 We remind that in the proof of this part we obtained the following inequalitions: from (7) we have that T R[G] > R P[G] (9) and from (8) we obtain that T r[G] > R°[A(RG/G P}. (10) Conversely, let G/G P is a nilpotent group whose derived group is a finite p-group. Then by Theorem 2.1 A [K ](RG/G P) = 0 for the Lie nilpotency index R°[A(RG)] = k of A(RG/G P). It follows that AW(RG) C I{G P). Hence, by Lemma 3.2, we obtain that A^(RG) C A^(RG). The inverse inclusion, of course, is trivial. Therefore A^(RG) — A^(RG). Consequently, T R[G ] is finite and T r[G) < R°[A(RG/GP)]. (11) The proof of the theorem is complete. Theorem 3.2. Let R be a commutative ring of characteristic p s and the Lie augmentation terminal of G is finite. Then T r[G] = T°[A{RG /Gp)] > R P[G\. The proof of this theorem follows from statements 1) and 2) of Theorem 3.1 and from (9), (10) and (11). References [1] PARMENTER, M. M., PASSI, I. B. S. and SEHGAL, S. K., Polynomial ideals in group rings, Canad. J. Math., 25 (1973), 1174-1182. [2] PASSI, I. B., Group ring and their augmentation ideals, Lecture notes in Math., 715, SpringerVerlag, Berlin-Heidelberg-New York, 1979. [3] SANDLING, R., The dimension subgroup problem, J. Algebra , 21 (1972), 216-231.
