A classification of prime segments in simple artinian rings

Let A be a simple artinian ring. A valuation ring of A is a Bezout order R of A so that R/J(R) is simple artinian, a Goldie prime is a prime ideal P o f R so that R/P is Goldie, and a prime segment of A is a pair of neighbouri ng Goldie primes of R. A prime segment P-1 superset of P-2 is archimedean i f K(P-1) = {a is an element of P-1\P(1)aP(1) subset of P-1} is equal to P-1 ; it is simple if K(P-1) = P-2 and it is exceptional if P-1 superset of K(P -1) superset of P-2. In this last case, K(P-1) is a prime ideal of R so tha t R/K(P-1) is not Goldie. Using the group of divisorial ideals, these resul ts are applied to classify rank one valuation rings according to the struct ure of their ideal lattices. The exceptional case splits further into infin itely many cases depending on the minimal n so that K(P-1)(n) is not diviso rial for n greater than or equal to 2.