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Some correspondences for a center Galois extension. (English) Zbl 1003.16017

Summary: Let \(B\) be a ring with \(1\), \(G\) a finite automorphism group of \(B\), \(C\) the center of \(B\), \(B^G\) the set of elements in \(B\) fixed under each element in \(G\). \(B*G\) a skew group ring in which the multiplication is given by \(gb=g(b)g\) for \(b\in B\) and \(g\in G\). Assume \(C\) is a Galois algebra with Galois group \(G|_C\cong G\). Two correspondence theorems are given between some sets of separable extensions in the skew group ring \(B*G\). Moreover, a necessary and sufficient condition is given for a subring \(S\) of \(B\) over \(B^G\) such that \(S=B^K\) for some subgroup \(K\) of \(G\). Consequently, a correspondence is established between the set of subgroups of \(G\) and a set of subrings of \(B\) over \(B^G\).

MSC:

16S35 Twisted and skew group rings, crossed products
16W20 Automorphisms and endomorphisms
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
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