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
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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