Integral semihereditary orders inside a Bézout maximal order
This paper is a continuation of the author's article reviewed above [see Zbl 0871.16009]. For a valuation domain \(V\) with quotient field \(F\), semi-hereditary \(V\)-orders \(H\) in a central simple \(F\)-algebra \(Q\) are considered which admit a Bézout overorder. If, in addition, this overorder \(B_0\) is a Dubrovin valuation ring of \(Q\), the author shows that extremal \(V\)-orders \(R\subset B_0\) are semi-hereditary. (The converse does hold in general and has been proved in the above mentioned previous article.) If \(B_0\) is just Bézout, there are only finitely many maximal overorders of \(H\), each of them being a Bézout order. Moreover, if \(\text{Rad }V\) is principal, the intersection of all Bézout overorders coincides with \(H\). Otherwise, it is shown that \(B_0\) is the unique maximal overorder of \(H\).
scientific article; zbMATH DE number 1013985
scientific article