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 77 is a ring homomorphism induced by a group homomorphism <j>:G —> H and a ring homomorphism ifi: R —• K

, then e(A

f[RG)) C Af(KH). (It is assumed here that IP(1R) = 1 /<-, where and 1/c are identities of rings R and K respectively.) For every natural number n A^(RG) is a polynomial ideal (see in particular [4], Corollary 1.9., page 6.) and by Lemma 2.1. 4>{AW(RG)) C AW{RG/L) for every n. From this inclusion it can be obtained easily that (l) 4>{A^\RG)) c AM(RG/L). If /C denotes a class of groups we define the class RX of residually-/C groups by letting G G R/C if and only if: whenever 1 / g G G, there exists a normal subgroup H

g of the group G such that G/H

g G /C and g ^ H

g. It is easy to see that G G RA^ if and only if there exists a family {

HÍ

}

Í

^

I of normal subgroups G such that GjE{ G /C for every i G I and D

Í

^

I

H

Í = (1). A group G is said to be discriminated by /C if for every finite set gi,g2, • • • ,g

n of distinct elements of G, there exists a group H £ IC and a homomorphism 4>:G —H such that 4>{gi) ^ 4>(gj) if i j

1 j, (1 < i,j < Lemma 2.2. Let a class of groups /C be closed with respect to forming subgroups and finite direct products and let G be a residually-/C group. Then G is discriminated by /C. The proof can be obtained easily. It is easy to show that if G is discriminated by a class of groups /C and if £ is a non-zero element of RG

, then there exists a group H G /C and a •homomorphism <j) of RG to RH such that 4>{x) / 0. From this fact and from inclusion (1) we have Lemma 2.3. If G is discriminated by a class of groups K, and for each H £JC the equation A^(RH) = 0 holds, then A^(RG) = 0. We use the following notations for standard group classes: VQ — the class of those nilpotent groups whose derived groups are torsionfree. V

p — the class of nilpotent groups whose derived groups are p-groups of bounded exponent.