Skew group rings and maximal orders.

*(English)*Zbl 0830.16018Let \(S\) 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 \(I(g)\subseteq p\). E. Nauwelaerts and 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.

Reviewer: P.F.Smith (Glasgow)

##### MSC:

16S35 | Twisted and skew group rings, crossed products |

16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |

16P40 | Noetherian rings and modules (associative rings and algebras) |

16D25 | Ideals in associative algebras |

##### Keywords:

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
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DOI

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