Value Functions on Simple Artinian Rings

Abstract

In this paper we introduce the notion of a value function ω on a simple Artinian ring Q whose restriction to the center of Q is a Krull valuation. We then study the structure of a valuation ring Bω corresponding to ω and its ideals. It is shown that Bω is a prime Goldie ring and the set of two-sided ideals of Bω is totally ordered by inclusion with the maximal ideal J(BW). Furthermore, it is proved that if Q is a central simple algebra, then the valuation ring Bω is a Dubrovin valuation ring which is integral over its center.