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A classification of prime segments in simple artinian rings

Authors:	H. H. Brungs H. Marubayashi E. Osmanagic

Affiliation:	Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 ; Department of Mathematics, Naruto University of Education, Naruto, Japan ; Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Abstract:	Let be a simple artinian ring. A valuation ring of is a Bézout order of so that is simple artinian, a Goldie prime is a prime ideal of so that is Goldie, and a prime segment of is a pair of neighbouring Goldie primes of A prime segment $P_{1}supset P_{2}$ is archimedean if $K(P_{1})={ain P_{1}vert P_{1} aP_{1}subset P_{1}}$ is equal to $P_{1},$ it is simple if $K(P_{1})=P_{2}$ and it is exceptional if $P_{1}supset K(P_{1})supset P_{2}.$ In this last case, $K(P_{1})$ is a prime ideal of so that $R/K(P_{1})$ 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 so that $K(P_{1})^{n}$ is not divisorial for

Keywords:	Dubrovin valuation ring, local Bé zout order, total valuation ring, Goldie prime, localizable prime, divisor group

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