On the Dimension of Group Rings
On the Dimension of Group Rings
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论群环的维数
DOI：
10.1112/plms/s3-25.2.288
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发表时间：
1972
影响因子：
1.8
通讯作者：
P. F. Smith
中科院分区：
文献类型：
--
作者：
P. F. Smith
G=：ff 1 2（？l 2... 2ff B 2^ 1={1}，使得对于每个i e（1，2，.，n}，Gi+ 1是Gi的正规子群，且因子群GJGiJrX是有限的或循环的。设h（G）表示无限群Gi/Gi+ 1（1^ i^ n）的个数.在§ 2中我们证明了对任意右Noether环A，群环AG的Krull维数是A的Krull维数与整数h（G）之和.现在假设A是一个交换的诺特环。然后在§ 4中我们证明了AG满足关于素理想的降链条件。此外，如果每个子群Gi在G中正规，则AG的Krull维数是素理想链长度的上确界。在§ 5中，证明了如果Z是整数环，G是可解群，则群环ZG有有限Krull维数当且仅当G满足子群的极大条件。我感谢裁判大大简化了定理2.5的证明，并提出了其他一些有益的建议。
G=: ff 1 2 (? l 2... 2ff B 2^ 1={l} of subgroups such that, for each i e (1, 2,..., n}, Gi+ 1 is a normal subgroup of Gi and the factor group GJGiJrX is either finite or cyclic. Let h (G) denote the number of groups Gi/Gi+ 1 (1^ i^ n) which are infinite. Then we 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 Gi is normal in G then the Krull dimension of AG is the supremum of the lengths of chains of prime idealsIn § 5 it is shown that if Z is the ring of integers and G is a soluble group then the group ring ZG has finite Krull dimension if and only if G satisfies the maximal condition for subgroups. I am indebted to the referee for greatly simplifying the proof of Theorem 2.5 and for making several other helpful suggestions.