Formally real fields,pythagorean fields,C-fields andW-groups

Supported in part by the National Science Foundation and the Natural Sciences and Engineering Research Council of Canada     

Strongly real hilbert rings and the real jacobson semisimplicity

In this paper, several false results in reference [1] related to rela Hilbert rings and the ral Jacobson semisimplicity are negated by a counterexample. By introducing the notion of a strongly real Hilbert ring, we characterize those rings of which every finitely generated real extension is real jacobson semisimple. Moreover, the so-called strictly real Hilbert rings are considered.     

Representation of real commutative rings

During the last 10 years there have been several new results on the representation of real polynomials, positive on some semi-algebraic subset of . These results started with a solution of the moment problem by Schmüdgen for compact semi-algebraic sets. Later, Wörmann realized that the same results could be obtained by the so-called “Kadison–Dubois” Representation Theorem.The aim of our paper is to present this representation theorem together with its history, and to discuss its implication to the representation of positive polynomials. Also recent improvements of both topics by T. Jacobi and the author will be included.     

On the algebraic closure in rings

The ``algebraic closure" of a subset of a ring is an algebraic analogue of topological closure.

     

Gabriel filters in real closed rings

Real closed rings arise in semi-algebraic geometry and topology as well as in the investigation of partially ordered rings. It is shown that localizations of real closed rings with respect to Gabriel filters, or more generally: multiplicative filters, are again real closed. Thus, real closedness is preserved under a large number of important ring theoretic constructions. For a few particularly simple cases the multiplicative filters are classified and the localizations are determined. Received: August 26, 1996     

Sums of squares in real rings

Let be an excellent ring. We show that if the real dimension of is at least three then has infinite Pythagoras number, and there exists a positive semidefinite element in which is not a sum of squares in .

     

Tight closure and linkage classes in Gorenstein rings

We study the relationship between the tight closure of an ideal and the sum of all ideals in its linkage class I thank Mel Hochster and Craig Huneke for their support and encouragement and for many helpful discussions.The author was partially supported by the NSF.     

On polynomial rings over nil rings in several variables and the central closure of prime nil rings

We prove that the ring of polynomials in several commuting indeterminates over a nil ring cannot be homomorphically mapped onto a ring with identity, i.e. it is Brown-McCoy radical. It answers a question posed by Puczy?owski and Smoktunowicz. We also show that the central closure of a prime nil ring cannot be a simple ring with identity, solving a problem due to Beidar.     

Localization of tight closure in two-dimensional rings

It is shown that tight closure commutes with localization in any two-dimensional ringR of prime characteristic if eitherR is a Nagata ring orR possesses a weak test element. Moreover, it is proved that tight closure commutes with localization at height one prime ideals in any ring of prime characteristic.     

Sums of squares in real analytic rings

Let be an analytic ring. We show: (1) has finite Pythagoras number if and only if its real dimension is , and (2) if every positive semidefinite element of is a sum of squares, then is real and has real dimension .

     

Reflexive rings and their extensions

A right ideal I is reflexive if xRy ∈ I implies yRx ∈ I for x, y ∈ R. We shall call a ring R a reflexive ring if aRb = 0 implies bRa = 0 for a, b ∈ R. We study the properties of reflexive rings and related concepts. We first consider basic extensions of reflexive rings. For a reduced iedal I of a ring R, if R/I is reflexive, we show that R is reflexive. We next discuss the reflexivity of some kinds of polynomial rings. For a quasi-Armendariz ring R, it is proved that R is reflexive if and only if R[x] is reflexive if and only if R[x; x ^?1] is reflexive. For a right Ore ring R with Q its classical right quotient ring, we show that if R is a reflexive ring then Q is also reflexive. Moreover, we characterize weakly reflexive rings which is a weak form of reflexive rings and investigate its properties. Examples are given to show that weakly reflexive rings need not be semicommutative. It is shown that if R is a semicommutative ring, then R[x] is weakly reflexive.