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 of R so that R/P is Goldie, and a prime segment of A is a pair of neighbouring Goldie primes of R. A prime segment P1 ⊃ P2 is archimedean if K(P1) = {a ∈ P1|P1aP1 ⊂ P1} is equal to P1, it is simple if K(P1) = P2 and it is exceptional if P1 ⊃ K(P1) ⊃ P2. In this last case, K(P1) is a prime ideal of R so that R/K(P1) 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 so that K(P1) is not divisorial for n ≥ 2.

/pdf/a-classification-of-prime-segments-in-simple-artinian-rings-4nh1iijer5.pdf

A classification of prime segments in simple artinian rings

Ideal Theory of Right Cones and Associated Rings

Valuation Rings in Ore Extensions

This volume was sparked by the fact that Alan Schoenfeld and Gunter Torner were both celebrating their 65th birthdays in July 2012. The book started out as part of a Festschrift to celebrate that event. Although the volume was not ready in time for their birthdays, the result is a belated celebration in print to recognize their contributions to the field of mathematics education.

Proficiency and beliefs in learning and teaching mathematics : learning from Alan Schoenfeld and Günter Törner

A right chain ordered semigroup is an ordered semigroup whose right ideals form a chain. In this paper we study the ideal theory of right chain ordered semigroups in terms of prime ideals, completely prime ideals and prime segments, extending to these semigroups results on right chain semigroups proved in Ferrero et al. (J Algebra 292:574–584, 2005).

/pdf/on-right-chain-ordered-semigroups-4ry3ulavi7.pdf

On right chain ordered semigroups

An additive subgroup P of a skew field F is called a prime of F if P does not contain the identity, but if the product xy of two elements x and y in F is contained in P, then x or y is in P. A prime segment of F is given by two neighbouring primes P 1 ⊃ P 2; such a segment is invariant, simple, or exceptional depending on whether A(P 1) = {a ∈ P 1 | P 1 aP 1 ⊂ P 1} equals P 1, P 2 or lies properly between P 1 and P 2. The set T(F) of all primes of F together with the containment relation is a tree if |T(F)| is finite, and 1 < |T(F)| < ∞ is possible if F is not commutative. In this paper we construct skew fields with prescribed types of sequences of prime segments as skew fields F of fractions of group rings of certain right ordered groups. In particular, groups G of affine transformations on ordered vector spaces V are considered, and the relationship between properties of Dedekind cuts of V, certain right orders on G, and chains of prime segments of F is investigated. A general result in Section 4 describing the possible orders on vector spaces over ordered fields may be of independent interest.

/pdf/prime-segments-of-skew-fields-3ov6ljo7fd.pdf

Prime Segments of Skew Fields

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Prime Segments of Skew Fields