A classification of prime segments in simple artinian rings

Abstract

Let A A be a simple artinian ring. A valuation ring of A A is a Bézout order R R of A A so that R / J ( R ) R/J(R) is simple artinian, a Goldie prime is a prime ideal P P of R R so that R / P R/P is Goldie, and a prime segment of A A is a pair of neighbouring Goldie primes of R . R. A prime segment P 1 ⊃ P 2 P_{1}\supset P_{2} is archimedean if K ( P 1 ) = { a ∈ P 1 | P 1 a P 1 ⊂ P 1 } K(P_{1})=\{a\in P_{1}\vert P_{1} aP_{1}\subset P_{1}\} is equal to P 1 , P_{1}, it is simple if K ( P 1 ) = P 2 K(P_{1})=P_{2} and it is exceptional if P 1 ⊃ K ( P 1 ) ⊃ P 2 . P_{1}\supset K(P_{1})\supset P_{2}. In this last case, K ( P 1 ) K(P_{1}) is a prime ideal of R R so that R / K ( P 1 ) R/K(P_{1}) is not Goldie. Using the group of divisorial ideals, these results are applied to classify rank one valuation rings according to the structure of their ideal lattices. The exceptional case splits further into infinitely many cases depending on the minimal n n so that K ( P 1 ) n K(P_{1})^{n} is not divisorial for n ≥ 2. n\ge 2.