Embedding Ordered Valued Domains into Division Rings

Klep, Igor

doi:10.1007/s10468-007-9062-5

Algebr Represent Theor

2007

DOI: 10.1007/s10468-007-9062-5

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Embedding Ordered Valued Domains into Division Rings

Igor Klep

Abstract: We study some classes of ordered domains that are embeddable in division rings. We prove the ordered version of the Cohn-Lichtman embedding theorem for valued domains. A question of Glass is answered in the negative. Furthermore, we prove that universal enveloping algebras of Lie algebras over formally real fields can be embedded into ordered division rings.

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Cited by 2 publications

References 17 publications

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Generalized orderings and rings of fractions

Cimpric

Klep

2006

Algebra univers.

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ABSTRACT. Every total ordering of a commutative domain can be extended uniquely to its field of fractions. This result is extended in two directions. Firstly, the notion of a total ordering is generalized so that a nonzero element can have more than two signs (in fact, these signs form a group). Secondly, commutative domains are replaced by noncommutative ones and we consider the following types of rings of fractions: Ore extensions, maximal (right or two-sided) rings of fractions, division hulls of free algebras and epic fields. Throughout the paper several examples are given to illustrate the theory.It is well-known that every total ordering of a commutative domain extends uniquely to a total ordering of its field of fractions. There are several generalizations of this result to (noncommutative) associative domains. While there is only one definition of a total ordering of an associative domain, several nonequivalent definitions of a ring of fractions exist in the literature. Existence and uniqueness of extensions of total orderings depend on the definition chosen. This paper is organized as follows. In Section 1 we introduce the notion of a G-ordering of an associative unital ring which generalizes the notion of an ordering of higher level. We are motivated by the work of T. Kanzaki [Ka] on signatures. In Sections 2-5 we discuss the extension theory of G-orderings. Each section is devoted to one variant of rings of fractions and in each section we recall the corresponding results for total orderings.The authors would like to thank the referee for the time and effort he took with the paper. His comments improved the paper a great deal in both form and content.

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Generalized orderings and rings of fractions

Cimpric

Klep

2006

Algebra univers.

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Embedding ∗-ordered domains into skew fields

Klep¹

2006

Journal of Algebra

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We study the problem of embedding domains with * -orderings into skew fields. Assuming that the natural valuation associated to a * -ordered domain satisfies an Ore-type condition, we prove that the domain embeds in an order-preserving way into a * -ordered skew field. We call this the * -ordered version of the Dauns embedding theorem for domains with valuations. A number of concrete examples, where this result can be applied, is given. Moreover, a question of Murray Marshall regarding * -orderable groups is answered in the affirmative.Unlike in the commutative case, not every domain (i.e., an associative unital ring without zero divisors) is embeddable in a skew field. Even if such an embedding exists, it need not be unique or canonical. So one looks for necessary and sufficient conditions to ensure the existence and perhaps additional properties of an embedding. These have been given by P.M. Cohn, see, e.g., [Co2] for a nice exposition. The situation remains the same for orderable domains. There exist orderable domains not embeddable in skew fields and embeddings, if they exist, need not be unique or canonical. For results in this regard we refer to [Co2,CK]. On the other hand, the case of * -orderable domains is still open. It is unknown whether such domains always admit an embedding in a skew field (cf. [Ma1, 1.6.Note (2)] or [Ci, Section 3]). Our aim is to discuss ✩

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Embedding Ordered Valued Domains into Division Rings

Cited by 2 publications

References 17 publications

Generalized orderings and rings of fractions

Generalized orderings and rings of fractions

Embedding ∗-ordered domains into skew fields

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