an:00792873
Zbl 0830.16018
Martin, R.
Skew group rings and maximal orders
EN
Glasg. Math. J. 37, No. 2, 249-263 (1995).
00026924
1995
j
16S35 16H05 16P40 16D25
prime Noetherian rings; finite groups; \(X\)-outer automorphisms; skew group rings; reflexive height-1 \(G\)-prime ideals; \(G\)-maximal orders; prime maximal orders; height-1 prime ideals; strongly graded rings; tame orders
Let \(S\) be a prime Noetherian ring and let \(G\) be a finite group acting on \(S\) such that \(G\) is \(X\)-outer. Let \(T=S*G\) be the skew group ring and let \(\Omega_0\) be the set of reflexive height-1 \(G\)-prime ideals of \(S\). In the main theorem it is proved that if (a) \(S\) is a \(G\)-maximal order (i.e. an order which is not properly contained in any \(G\)-invariant order to which it is equivalent), and (b) \(p_0 T\) is a prime ideal of \(T\) for all \(p_0\) in \(\Omega_0\), then \(T\) is a prime maximal order. Conversely, if \(T\) is a (prime) maximal order and the order of \(G\) is a unit in \(S\) then (a) and (b) both hold. An example is given to show that the restriction on the order of \(G\) is necessary. In order to prove this theorem the author develops a theory of \(G\)-maximal orders analogous to that for maximal orders. Now let \(S\) be commutative and for each \(1\neq g\in G\) define \(I(G)\) to be the ideal of \(S\) generated by the set \(s-s^g\) (\(s\in S\)). Then it is proved that \(T\) is a prime maximal order if and only if \(S\) is integrally closed and there does not exist \(1 \neq g\in G\) and a height-1 prime ideal \(p\) in \(S\) such that \(I(g)\subseteq p\). \textit{E. Nauwelaerts} and \textit{F. Van Oystaeyen} [J. Algebra 101, 61-68 (1986; Zbl 0588.16002)] have given sufficient conditions for a ring \(R\) strongly graded by a finite group \(G\), with the order of \(G\) a unit in \(R\), to be a tame order.
P.F.Smith (Glasgow)
Zbl 0588.16002