On semi-hereditary orders integral over a commutative valuation ring

In this paper, we study semi-hereditary V-orders in a simple Artinian ring Q with finite dimension over its center, where V is a commutative valuation ring. Let S be a minimal V-overring of a semi-hereditary V-order R. In Section 1, we investigate some relations between all maximal ideals of S and R, and characterize the commutativities of idempotent maximal ideals of R in terms of orders of ideals. In Section 2, we show that there is a bijection between the set of all V-overrings of R and the set of all idempotent ideals which are finitely generated as left ideals. Any element in the latter set is characterized by four different types of cycles. In Section 3, we discuss the principalness of the Jacobson radical J(R). Some results in Sections 1–3 are used to derive the exact numbers of all semi-hereditary maximal V-orders containing R and of all V-overrings of R, and to study the nilpotency of J(R) modulo J(V)R. Some invariant properties of semi-hereditary V-orders are also given.