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

Ori the powers of the augmentation ideal of a group ring 81 PROOF. Let T

r(G) = k. It is obvious that T

r(G/G

P

) is finite and also the inequality T

r(G) > T

R(G/G

P) holds. By Theorem 3.1 GJG

V is finite pgroup. Keeping in mind the previous inequalities, by Theorem 2.1, we have that A

K(RG/G

P) = 0. Consequently, R°(A(RG/G

P)) < R

R(G). Now we show that T°(A(RG/G

p)) = T

r(G). If this équation is not true we can choose a nonnegative integer i < k such that A

L(RG/G

P

) = 0. Hence we have that A

L{RG) Ç I(G

P). By Lemma 3.2, I(G

P) = A

U(RG). Therefore A*(RG) Ç A"(RG) and A

L(RG) = A

I+

L(RG) which contradicts to the équation T

r(G) = k. Consequently, T

r(G) = T°(A(RG/G

P)). From T

r(G) = k it follows that A

K(RG) = A

U

J(RG) and, by Lemmas 3.1 and 3.2, I(G

PI

K) Ç A

K(RG) = A

U(RG) = I{G

P). Then G

P,

K Ç G

P and by (1) we obtain that G

P

TK = G

P

I that is r

P(G) < T

r(G). This complétés the proof of the theorem. Let n(n) dénoté the set of prime divisors of a natural number n. Theorem 3.3. Let R be a, commutative ring of non-zero characteristic n. Then the augmentation terminal of G with respect to R is finite if and only if G/G

p is finite p-group for ail p £ II(n). PROOF. Let n — P™

1P™

2 • • •pbe the prime power décomposition of the natural number n. We shah write R

P

i = R/riiR for n

l = p™'

, where Pi e n(ra) = {p

1,p

2,.

..,Pt}

. Let T

r(G) be finite. It follows that R

R

P (G) is also finite and (5) r

f

i(G)>r

Äp

i(G) for ail pi G n(n). Then, by Threorem 3.1, G/G

P

i is a finite p

{-group for ail Pi e n(n). We notice that from (5) the inequality (6) T

R(G)> max {T

RJG)} p,en(n) follows. Conversely. Let G/G

P

I be finite p^-group for ail G II(n). Then by Theorem 3.1, the augmentation terminal R

R

P (G) is finite for ail G II(n). Let k= max {r

R (G)}. PiGll(n)