TY - JOUR
AU - Smith, P. F.
AB - By P. F. SMITH [Received 4 January 1971—Revised 10 November 1971] Throughout this paper 'ring' will mean 'associative ring with identity 1^0 ' and 'module' will mean 'unitary right module'. Let G be a group having a series G=:ff 2(? 2...2ff 2^ = {l} 1 l B 1 of subgroups such that, for each i e (1,2, ...,n}, G is a normal subgroup i+1 of Gi and the factor group GJG is either finite or cyclic. Let h(G) denote iJrX the number of groups G /G (1 ^ i ^ n) which are infinite. Then we i i+1 prove in § 2 that for any right noetherian ring A the Krull dimension of the group ring AG is the sum of the Krull dimension of A and the integer h(G). Now suppose that A is a commutative noetherian ring. Then in §4 we show that AG satisfies the descending chain condition on prime ideals. Moreover if, in addition, each subgroup G is normal in G then the Krull dimension of AG is the supremum of the lengths of chains of prime ideals In § 5 it is shown that if Z is the ring of integers and G is 
TI - On the Dimension of Group Rings
JF - Proceedings of the London Mathematical Society
DO - 10.1112/plms/s3-25.2.288
DA - 1972-08-01
UR - https://www.deepdyve.com/lp/wiley/on-the-dimension-of-group-rings-Oea10jyfSx
SP - 288
EP - 302
VL - s3-25
IS - 2
DP - DeepDyve
ER -