Semiorderings and localizations of planar ternary rings

We continue our study of quadratic semiorderings in planar ternary rings started in [11]. Following PRESTEL [16] and LAM [14], we establish the notion of compatibility between semiorderings and places of planar ternary rings and transfer a Baer-Krull like theorem for semiorderings to our setting. As in the classic case, any archimedean semiordering turns out to be an ordering. Finally, we show that over planar ternary rings with rational prime field each semiordering gives rise to a natural place into the reals.     

The holomorphy rings of planar ternary rings with rational prime field

By a deep, recent result of Hartmann and Priess-Crampe any ordered planar ternary ring with rational prime field admits a natural order-compatible place into the reals. This allows us to extend the classical machinery about real places and spaces of orderings, the notion of real and relative holomorphy rings, the Kadison-Dubois representation for these rings, and the Brown-Marshall inequality to arbitrary planar ternary rings with rational prime field.     

The isostrophisms of planar ternary rings Part II

In Part I the family of ternary rings determined with respect to the base points A,B,C,D of a projective plane was considered, when the different members of the family are determined by permutations of A,B,C,D. Some of the algebraic relationships between the different rings were studied. In Part II these results are applied to problems of collineations. There is a large body of results relating the properties of ternary rings to collineations, and relating the existence of one set of collineations to the existence of others. Once several basic theorems are established many theorems involving collineations follow quite easily. The usual method of determining what a given collineation implies about a ternary ring is to compute the image points and image lines and see what results. That is not necessary for the collineations presented here using the techniques of isostrophisms.This research is included in the author's doctoral dissertation submitted to Temple University. The author would like to express his appreciation to Professor R. Artzy for his assistance in completing this work.     

The isostrophisms of planar ternary rings part I

It is well known that a projective plane nay be coordinatized by picking four arbitrary, ordered points such that no three of them are collinear. This gives rise to an algebraic structure called the planar ternary ring T with respect to the base points A,B,C,D. The existence of certain collineations in the plane will be reflected by algebraic properties in T, and conversely. In this paper the interrelationships between certain algebraic properties in T, and the linearity of those ternary rings which are obtained from T and various permutations of A,B,C,D are considered. Many well known types of algebraic structures, such as quasifields and alternative division rings will occur if the ternary rings generated by suitable permutations of A,B,C,D are linear.This research is included in the author's doctoral dissertation submitted to Temple University. The author would like to express his appreciation to Professor R. Artzy for his assistance in completing this work.     

Orderings and valuations on twisted polynomial rings

The real spectra of certain "twisted" polynomial algebras A over R are examined. In certain cases, including Weyl algebras and universal enveloping algebras of solvable Lie algebras, the stability indices of the residue spaces of Sper(A) are estimated. The results show that the Bröcker-Scheiderer theory of minimal generation of constructible sets applies to Sper(A) in these cases. An attempt is made to compute all orderings on the Weyl algebra over R. Several open problems are mentioned.     

Trivialization of fans in planar ternary rings with rational prime field

Summary Making use of the intimate relations between real places and orderings, we continue studying the spaces of orderings of planar ternary rings (PTRs) with rational prime field. As in the classical case, given a preorderingS of such a PTRT, then the productA ^S of all natural place ringsA _p associated to the orderingsP X/S is itself a place ring ofT. In particular, ifS is a nontrivial fan ofT thenA ^s T. Thus L. Bröcker's celebrated theorem on the trivialization of fans also applies to our setting: For any fan Sof a PTRT with rational prime field there exists a place µ: T T {} such that the push down S:= (S)\{0,} of S is a trivial fan of T. Further, Bröcker's global stability formula and, by means of an approximation theorem, some classical, valuation theoretic characterizations of SAP-preorderings and fans also extend to PTRs with rational prime field.     

Saturation for valuations on two-dimensional regular local rings

We study valuations centered in a regualar local ring of dimension two. We define the notion of saturation with respect to such a valuation, extending the classical definitions. The invariants associated in a natural way to the valuation are related with the saturated ring and som geometric properties are deduced. Received February 16, 1998; in final form February 9, 1999     

Embedding of near-fields with real valuations into planar near-fields

In [7] Wefelscheid studied all examples of nonplanar near-fields known so far. It was shown that there exists a valuation on each of these near-fields F, such that F together with the topology induced by the valuation is a topological near-field which can be completed relative to its additive uniform structure. The completion $hat F$ of F always turned out to be planar. The aim of the note presented here is to generalize these results to a class of near-fields which contains the examples mentioned above.     

Construction of finite ternary rings

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 10, pp. 1313–1318, October, 1989.