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

82 Bertalan Király Then (7) ... D A

k-\R

P

iG) D A

k{R

V

iG) = A

k+

1(R

P

iG) = ... = A

u(R

P

iG) for ail pi G n(n). From the isomorphism A

k(R

P

iG) = (A

k(RG

) + n

xR • RG)ln

xR • RG and from (7) it follows that for every pi G II(n), an arbitrary element x of A

k(RG) can be written as (8) x = xi + riidi, where x

{ G A

k+

l (RG), a; G RG and i = 1,2,

..., t. If n

i — n/rii

, then rí; is non-zero and = 0 in R. Then n^x = riiX

t for ail i = 1,2,

. .., t and from (8) we have that t t (^Tï

{)x = ^n.Xi. i=i i=i It is easy to see that rfj and n; are coprimes and also n^ divides ríj for all t t i ^ j. Therefore the numbers and n are coprimes. Hence ^T^ni is i=

1 i=

1 invertible in R

, because the characteristic of Ä equals to n. Then from the previous équation we obtain that t x = a y^njXj, i=i t where a^^fîi — 1 £ R* and R* is the unit group of R. Therefore x G i=i A

k+

1(RG) and hence we conclude that A

k(RG) Ç A

k+

l(RG). The inverse inclusion is trivial. Consequently, A

k(RG) =.A

k+

1(RG) and (9) M«) < max{r„

p(G)}, p

t€ll(n) that is the augmentation terminal T

R(G

) of G in regarding to R is finite which was to be proved.