Hilbert's 17th Problem for Real Closed Rings

We recall the characterisation of positive definite polynomial functions over a real closed ring due to Dickmann, and give a new proof of this result, based upon ideas of Abraham Robinson. In addition we isolate the class of convexly ordered valuation rings for which this characterisation holds. Mathematics Subject Classification: 03C60, 13L05, 12D15.     

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.     

Irreducibility in the Total Ring of Quotients

Let R be a ring whose total ring of quotients Q is von Neumann regular. We investigate the structure of R when it admits an ideal that is irreducible as a submodule of the total ring of quotients. We characterize those rings which contain a maximal ideal that is irreducible in its total ring of quotients Q. An integral domain has a Q-irreducible ideal which is a maximal ideal if and only if R is a valuation domain. We show that when the total ring of quotients of R is von Neumann regular, then having a maximal ideal that is Q-irreducible is equivalently to certain valuation like properties, including the property that the regular ideals are linearly ordered.     

Dubrovin赋值斜群环

本文中对一个斜群环为Dubrovin赋值环给出了一系列等价刻画,并且刻画了一个Dubrovin赋值斜群环的所有素理想.     

Another proof of the almost purity theorem for perfectoid valuation rings

The almost purity theorem is central to the geometry of perfectoid spaces and has numerous applications in algebra and geometry. This result is known to have several different proofs in the case that the base ring is a perfectoid valuation ring. We give a new proof by exploiting the behavior of Faltings' normalized length under the Frobenius map.     

Integral domains having a unique Kronecker function ring

We study the class of integrally closed domains having a unique Kronecker function ring, or equivalently, domains in which the completion (or b-operation) is the only e.a.b star operation of finite type. Such domains are a generalization of Prüfer domains and have fairly simple sets of valuation overrings. We give characterizations by studying valuation overrings and integral closure of finitely generated ideals. We provide new examples of such domains and show that for several well-known classes of integral domains the property of having a unique Kronecker function ring makes them fall into the class of Prüfer domains.     

Dubrovin Valuation Skew Group Rings

Some equivalent characterizations for a skew group ring to be a Dubrovin valuation ring are given. Among them all the prime ideals of a Dubrovin valuation skew group ring are characterised. Received June 8, 1999, Revised June 4, 2001, Accepted July 20, 2001     

On Von Neumann Regular Rings of Skew Generalized Power Series

In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S?{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]].     

关于p除环结构的几个定理

本文讨论了强ｐ除环的结构与赋值，并且给出了ｐ除环是广义四元数体的一个子体的充要条件。     

Uniserial Noetherian centrally essential rings

Abstract

It is proved that a ring R is a right uniserial, right Noetherian centrally essential ring if and only if R is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings.     

Valuations and rank of ordered abelian groups

It is shown that there exists an ordered abelian group that has no smallest positive element and that has no sequence of nonzero elements converging to zero. Some formulae for the rank of ordered abelian groups have been derived and a necessary condition for an order type to be rank of an ordered abelian group has been discussed. These facts have been translated to the spectrum of a valuation ring using some well-known results in valuation theory.

     

Growth of Rank 1 Valuation Semigroups

We consider semigroup S of a rank 1 valuation ? centered on a local ring R. We show that the Hilbert polynomial of R gives a bound on the growth of the valuation semigroup S. This allows us to give a very simple example of a well ordered subsemigroup of Q+, which is not a value semigroup of a local domain. We also consider the rates of growth which are possible for S. We show that quite exotic behavior can occur. In the final section, we consider the general question of characterizing rank 1 value semigroups.     

Diagonal F-thresholds and F-pure Thresholds of Hibi Rings

Hibi rings are a kind of graded toric ring on a finite distributive lattice D = J(P), where P is a partially ordered set. In this paper, we compute diagonal F-thresholds and F-pure thresholds of Hibi rings and give a characterization of Hibi rings which satisfy the equality between these invariants in terms of its trivialness in the sense of Herzog–Hibi–Restuccia.     

Formal power series with cyclically ordered exponents

We define and study a notion of ring of formal power series with exponents in a cyclically ordered group. Such a ring is a quotient of various subrings of classical formal power series rings. It carries a two variable valuation function. In the particular case where the cyclically ordered group is actually totally ordered, our notion of formal power series is equivalent to the classical one in a language enriched with a predicate interpreted by the set of all monomials.Received: 24 February 2003     

On Valuation Rings

In this article, we provide necessary and sufficient conditions for R = A ∝ E to be a valuation ring where E is a non-torsion or finitely generated A-module. Also, we investigate the (n, d) property of the valuation ring.     

Value Functions on Simple Artinian Rings

In this paper we introduce the notion of a value function ω on a simple Artinian ring Q whose restriction to the center of Q is a Krull valuation. We then study the structure of a valuation ring Bω corresponding to ω and its ideals. It is shown that Bω is a prime Goldie ring and the set of two-sided ideals of Bω is totally ordered by inclusion with the maximal ideal J(BW). Furthermore, it is proved that if Q is a central simple algebra, then the valuation ring Bω is a Dubrovin valuation ring which is integral over its center.     

On VC-minimal fields and dp-smallness

In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.     

Indecomposable Modules and Gelfand Rings

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R _P is an Artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings. It is shown that each commutative ring R with a Hausdorff and totally disconnected maximal spectrum is local-global. Moreover, if R is arithmetic, then R is an elementary divisor ring.     

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.