On semi-hereditary orders integral over a commutative valuation ring. (Q1886817)

Let \(V\) be a (commutative) valuation ring with quotient field \(K\). For a central simple \(K\)-algebra \(Q\), the author considers semi-hereditary \(V\)-orders \(R\) in \(Q\), i.e., semi-hereditary subrings with \(KR=Q\) which are integral over their center \(V\). Like in the case of hereditary orders, there is a bijection between \(V\)-overorders and finitely generated idempotent ideals of \(R\), given by the left conductor and the right order, respectively. More explicitly, the set \({\mathcal I}(R)\) of idempotent ideals \(I\) of \(R\) with \(_RI\) finitely generated, is described in terms of modified cycles, generalizing the corresponding theory for hereditary orders. In this way, the cardinality of \({\mathcal I}(R)\) is calculated. Accordingly, the number of \(V\)-overorders of \(R\) is determined. Using the structure of maximal ideals of \(R\), the author provides a criterion for \(\text{rad\,}R\) to be principal. Some further results, concerning the nilpotency of \(\text{rad\,}R\) modulo \(R(\text{rad\,}V)\), are obtained.