We deal in this paper with Bezout orders R in semisimple artinian rings Q such that R / J ( R ) is semisimple artinian and call these orders valuation rings. This name can be justified if one considers on the one hand the various notions of valuations and valuation rings used in Number Theory, in Algebraic Geometry, in the theory of Quadratic Forms, division algebras and simple artinian rings, and on the other hand considers the properties, in particular the extension properties, of these rings. Several authors have previously dealt with some aspects of these orders, see for example [D4], [R2], [Wl], and [W2]. W. KRULL in [K] extended S. Kurschak's real valued, i.e., rank one valuations of (commutative) fields to allow commutative ordered groups of arbitrary rank as value sets; this concept corresponds to Krull valuation rings, where a subring V of a field F is in this class if and only if x in F \ V implies 2-l in V. A Krull valuation ring of a field F can also be characterized as a Bezout order V in F with V / J ( V ) a field, where J ( V ) is the Jacobson radical of V. By replacing field by skew field in the last sentence, one obtains the definition of a total valuation ring B of a skew field D, which is equivalent to the condition that for all x in D either x or x -l is in B. SCHILLING in [S2] considers invariant total valuation subrings which correspond to valuations of skew fields with ordered groups as value sets. A more general concept of valuation ring was introduced by N. I. DUBROVIN in [D2] where he not only shows that Bezout orders B in simple artinian rings with B / J ( B ) simple artinian share many properties with total valuation rings, but that this class of rings, now called Dubrovin valuation rings, is not only closed under Morita equivalence, but also has much better extension properties