Bézout orders and Henselization
Let \(V\) be a valuation ring of a field \(F\), and \(B\) a Bézout maximal \(V\)-order in some central simple \(F\)-algebra \(S\). If \(F_h\), \(V_h\) denotes Henselization, the authors show that \(B_h=V_h\otimes_V B\) is a semi-hereditary maximal \(V_h\)-order in \(S_h=F_h\otimes_F S\). Moreover, they prove that \(B_h\) is Bézout if and only if \(B\) is a Dubrovin valuation ring in \(S\), i.e. if \(B/\text{Rad }B\) is simple artinian, and for any \(s\in S\setminus B\) there are \(b,c\in B\) with \(bs,sc\in B\setminus\text{Rad }B\). Let \(n\), \(t\) be the sizes of the matrix rings \(S_h\) and \(B /\text{Rad }B\), respectively. The authors use the above theorem in order to give an alternative proof of Gräter's result that \(n/t\) coincides with the extension number of \(V\) in \(S\).
scientific article; zbMATH DE number 763951