Integral Semihereditary Orders, Extremality, and Henselization Original Research Article

In this paper, we study integral semihereditary orders over a valuation ring in a finite-dimensional simple Artinian ring. In the first section we prove that such orders are extremal. Consequently, in a central division algebra admitting a total valuation ring, the intersection of all the conjugates of the total valuation ring is the unique integral semihereditary order over the center of the total valuation ring. In the second section we characterize, up to conjugacy, integral semihereditary orders over a Henselian valuation ring. In the last section we show that an integral orderRover an arbitrary valuation ringVis semihereditary iff its Henselization,R circle times operator VVh, whereVhis the Henselization ofV, is a semihereditaryVh-order. In this case, there is an inclusion preserving bijective correspondence between semihereditaryV-orders insideRand semihereditaryVh-orders insideR circle times operator VVh.