Fields with two incomparable henselian valuation rings

Let K be a field and C, C' be two incomparable valuation rings of the separable closure of K, Theorem 1.2 states that the intersection of the decomposition groups of C, C', with respect to K, is precisely the inertia group of the composition ring C·C'. We apply this theorem in the study of two special cases of valued fields (L,B). In the first case, B is henselian and there is a subfield K of L such that L|K is a normal extension and B K is not henselian. The second case is that in which B has exactly two prolongations in the separable closure of L. We call these rings semihenselian rings, and they are characterized through Theorems 2.6 and 2.12.This paper is part of author's doctoral dissertation. Financial support for this research was provided by CNP_q (National Research Council) and by Universidade Estadual de Campinas.     

Ascent of properties for absolutely flat homomorphisms

We show that if A – B is an absolutely flat homomorphism of consultative rings and A is arithmetical, then B is such, thus generalising a previous result (see [9]) on valutation rings.

To prove the above theorem we show first that it is true when A is a local ring and B is a local ind-étale homomorphism (in par ticular if B is a strict henselization of A ) , and we apply the following general fact:a local property which ascends to strict henselization and descends by faithful flatness ascends also by ab solutely flat homomorphlsms. This last result also applies to other properties, such as locally noetherian, geometrically unibranche, R_n,S_n, Cohen-Macaulay.     

Order-topological separable complete modular ortholattices admit order continuous faithful valuations

We prove that on every separable complete atomic modular ortholattice (i.e.order topological) there exists an order continuous faithful valuation. We also give a construction of the existing order continuous faithful valuation. For separable atomic modular ortholattices we give a necessary and sufficient condition to admit an order continuous faithful valuation and we show that it is equivalent with the condition to have a modular MacNeille completion. We improve one statement on complete metric lattices from Birkhoff's Lattice Theory.

     

Completions of Convexly Ordered Valuation Rings

We prove that every convexly ordered valuation ring has a unique completion as a uniform space, which furthermore is a convexly ordered valuation ring. In addition, we give a model theoretic characterisation of complete convexly ordered valuation rings, and give a necessary and sufficient condition for the completion of a convexly ordered valuation ring to be a real closed ring. Mathematics Subject Classification : 03C60, 13L05, 54E15.     

An imbedding property of K-groups

For a pair of rings A⊂B satisfying a certain condition (?) we get general imbedding results for Karoubi-Villamayor, homotopy and Quillen K-theories. One important case is when A and B are respectively the henselization and completion of a regular ring of finite type over a field at a prime ideal.     

Ramification of Valuations and Local Rings in Positive Characteristic

In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension.     

Structure des algebres de valuation discrete

We define a topology τ_e, on anyC-algebra of discrete valuation, generalizing the topology of coefficientwise convergence on C[[X]] studied by G. R. Allan. We give a necessary and sufficient condition for τ_e to be complete and prove that the completion provides an algebra of discrete valuation. We also prove that if aC-algebra of discrete valuation is Fréchet andm-convex for τ_e then it is isomorphic to (C[[X]], τ_e) and then τ_e is the uniqueF-algebra topology in A. We prove that a commutative, unital Fréchet l.m.c.a. that is aC-algebra of valuation is in fact aC-algebra of discrete valuation and so is embeddable in (C[[X]], τ_e). Whence a result of H. Bouloussa.     

A Valuation Theory for Nonassociative Quaternion Algebras

A nonassociative quaternion algebra over a field F is a 4-dimensional F-algebra A whose nucleus is a separable quadratic extension field of F. We define the notion of a valuation ring for A, and we also define a value function on A with values from a totally ordered group. We determine the structure of the set on which the function assumes non-negative values and we prove that, given a valuation ring of A, there is a value function associated to it if and only if the valuation ring is integral and invariant under proper F-automorphisms of A.     

The Merkuriev-Suslin theorem for any semi-local ring

We introduce here a method which uses étale neighborhoods to extend results from smooth semi-local rings to arbitrary semi-local rings A by passing to the henselization of a smooth presentation of A. The technique is used to show that étale cohomology of A agrees with Galois cohomology, to show that the Merkuriev-Suslin theorem holds for A, and to describe torsion in K₂(A).     

Spectral norms on valued fields

Let be a perfect valued field, be an algebraic closure of be an extension of to and be the G-spectral norm on Let be an algebraic extension of K and be the completion of L relative to We associate to any element a real number and prove that if for all x in , then and is a zero-dimensional regular ring. We show that and prove that is algebraic over (with some additional conditions on K and L). We give a Galois type correspondence between the set of all closed K-subalgebras of and the subfields of L. We prove that is an algebraic closed and zero-dimensional regular ring. Received: 3 March 1999; in final form: 21 February 2000 / Published online: 4 May 2001     

A combinatorial principle and endomorphism rings I: Onp-groups

Two lines of research are involved here. One is a combinatorial principle, proved in ZFC for many cardinals (e.g., any λ = λ^ℵ ₀) enabling us to prove things which have been proven using the diamond or for strong limit cardinals of uncountable cofinality. The other direction is building abelian groups with few endomorphisms and/or prescribed rings of endomorphisms. We prove that for a ringR, whose additive group is thep-adic completion of a freep-adic module,R is isomorphic to the endomorphism ring of some separable abelianp-groupG divided by the ideal of small endomorphisms, withG of power λ for any λ = λ^ℵ ₀≧|R|. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research.     

Different and trace

Let E/F be a finite separable field extension and suppose given a discrete valuation on F that has a unique extension to E. We prove the relation d = e − 1 + c, where d is the differential exponent, e the ramification and c the trace contraction.     

The Structure of Lattice-Subspaces

In Polyrakis (1983; Math. Proc. Cambridge Phil. Soc. 94, 519) it is proved that each infinite-dimensional, closed lattice-subspace of . . .₁ is order-isomorphic to . . .₁ and in Polyrakis (1987; Math. Anal. Appl. 184, 1) that each separable Banach lattice is order isomorphic to a closed lattice-subspace of C[0,1]. Therefore . . .₁ contains only one lattice-subspace but C[0,1] contains all the separable Banach lattices. In the first section of this article we study the kind of the order embeddability of a separable Banach lattice in C[0,1]. We show that the AM spaces have the ``best' behavior and the AL-spaces the ``worst'. In the second section we prove that the closure of a lattice-subspace is not necessarily a lattice-subspace and in the least one we study lattice-subspaces with positive bases.     

On locally finite alternative division rings with valuation

We prove that a locally finite complete alternative division ring with valuation is a field, without using the celebrated theorem of Bruck and Kleinfeld, which classifies all division rings with IP.The author's research is supported by the National Fund for Scientific research (Belgium).     

On some notions of good reduction for endomorphisms of the projective line

Let Φ be an endomorphism of ${\mathbb{P}^1_{\overline{\mathbb{Q}}}}$ , the projective line over the algebraic closure of ${\mathbb{Q}}$ , of degree ≥ 2 defined over a number field K. Let v be a non-archimedean valuation of K. We say that Φ has critically good reduction at v if any pair of distinct ramification points of Φ do not collide under reduction modulo v and the same holds also for any pair of branch points. We say that Φ has simple good reduction at v if the map Φ _v , the reduction of Φ modulo v, has the same degree of Φ. We prove that if Φ has critically good reduction at v and the reduction map Φ _v is separable, then Φ has simple good reduction at v.     

Some New Results on Optimal Codes Over F5

We present some results on almost maximum distance separable (AMDS) codes and Griesmer codes of dimension 4 over over the field of order 5. We prove that no AMDS code of length 13 and minimum distance 5 exists, and we give a classification of some AMDS codes. Moreover, we classify the projective strongly optimal Griesmer codes over F₅ of dimension 4 for some values of the minimum distance.     

Real Closed Valuation Rings

The real closed valuation rings, i.e., convex subrings of real closed fields, form a proper subclass of the class of real closed domains. It is shown how one can recognize whether a real closed domain is a valuation ring. This leads to a characterization of the totally ordered domains whose real closure is a valuation ring. Real closures of totally ordered factor rings of coordinate rings of real algebraic varieties are very frequently valuation rings. In particular, the real closure of the coordinate ring of a curve is an SV-ring (i.e., the factor rings modulo prime ideals are valuation rings). Real closed valuation rings play a role in the definition of real closed rings, as well as in the construction of real closures of rings and porings. They can also be used for the study of univariate differentiable semi-algebraic functions. This leads to the notion of differentiablility of semi-algebraic functions along half branches of curves.     

Model companion and model completion of theories of rings

Extending the language of rings to include predicates for Jacobson radical relations, we show that the theory of regular rings defined by Carson, Lipshitz and Saracino is the model completion of the theory of semisimple rings. Removing the requirement on the Jacobson radical (reduced to {0}), we prove that the theory of rings with no nilpotents does not admit a model companion relative to this augmented language.      

On strongly prime rings and ideals

Strongly prime rings may be defined as prime rings with simple central closure. This paper is concerned with further investigation of such rings. Various characterizations, particularly in terms of symmetric zero divisors, are given. We prove that the central closure of a strongly (semi-)prime ring may be obtained by a certain symmetric perfect one sided localization. Complements of strongly prime ideals are described in terms of strongly multiplicative sets of rings. Moreover, some relations between a ring and its multiplication ring are examined.     

Global dimension two orders are quasi-hereditary

Motivated by results of Cline, Parshall and Scott on quasi-hereditary algebras, in [8] the concept of a quasi-hereditary order is introduced in integral representation theory. In this note we show that the results of Dlab and Ringel on quasi-hereditary semiprimary rings and hereditary artinian rings presented in [6] have integral analogues in the theory of orders. In particular, we prove as our main result the followingTheorem: An order of global dimension at most two over a complete Dedekind domain R in a separable algebra over the quotient field of R is quasi-hereditary.