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
Residual Lie nilpotence of the augmentation ideal 81 r) n n n 7 1 Applying identity (3) repeatedly to [h[ h?, • • • h^ — 1) from the previous congruence it follows that 771 m P n / n\ h~i = Ysihf -m = )( h< -^ ( mod Alk ](xQ): i-1 i=l 1 ^ ^ ' where b{ E RG. Because h l E G' ~ 72(C), from Lemma 2.5. (cases 1 and 3) we obtain that (/i; - I)7 E (ÄG) for every i and j. If n > sk, then p s divides [ p. ) for every j — 1, 2,. .., k — 1. Therefore m m k — 1 h- ie -^ f s E E - ^ i=i j^i j=i = p sX(k,h) (mod ^(ÄG)), where X(M) = E?=i Ef=k ~ WE ÄG\ p sd 3 = (*"). The Lemma is proved. It is easy to show that if # E G' and E D^(RG) then (4) P m{g~ 1) E for a large enough m. Lemma 3.2. ([1], Lemma 3.6.) Let K, be a class oi groups and {CajaG/ a family of normal subgroups of G such that for all a (a £ I) the conditions (1) G/G a E K (2) G a is torsion-free hold. If G is not discriminated by /C then there exists a finite set of distinct elements gi , g 2,..., g s from G such that the non-zero element y = {g\ 1)(<72 — 1) • • • (<7s — 1) lies in the ideal n a e//(G a). The torsion subgroup T(R +) of the additive group R + of a ring R is the direct sum of its p-primary components S P(R +). Let II be the set of those primes for which the p-primary components S P(R +) of T(R +) are non-zero. An element a of an additive Abelian group A is called an element of infinite p-height for a prime p, if the equation p nx — a has a solution in A for every natural number n. Proposition 3.3. ([1], Theorem 3.3.) LetT(R +) ^ 0, and suppose that for some p E II group T(R +) has no element of infinite p-height. Further
