On crossed product orders.
Let \(K\) be the quotient field of a complete local Dedekind domain \(R\), and \(A\) be a central simple algebra over \(K\). The authors have considered an embedding problem of the hereditary \(R\)-order \({\mathcal O}_ Z\) of a simple subalgebra \(Z\) of \(A\) over \(K\) into the hereditary \(R\)-order \({\mathcal O}_ A\) of \(A\) [\textit{H. Benz}, J. Reine Angew. Math. 225, 30-75 (1967; Zbl 0155.09802) and \textit{H. Zassenhaus}, Number Theory and Algebra, 363-393 (1977; Zbl 0375.16007)]. In those considerations, crossed product orders played an important role. In this paper, they investigate the arithmetic of crossed products and prove the uniqueness of existence of an hereditary \(R\)-order \({\mathcal O}_ A\) of \(A\) satisfying some conditions for each crossed product order \({\mathcal O}\) and the maximal one of all crossed product orders contained in \({\mathcal O}_ A\).
scientific article; zbMATH DE number 3952984