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no. S1
On f-rings that are not formally real
James J. Madden1
1 Louisiana State University, Baton Rouge
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 143-157.

Henriksen and Isbell showed in 1962 that some commutative rings admit total orderings that violate equational laws (in the language of lattice-ordered rings) that are satisfied by all totally-ordered fields. In this paper, we review the work of Henriksen and Isbell on this topic, construct and classify some examples that illustrate this phenomenon using the valuation theory of Hion (in the process, answering a question posed in [E]) and, finally, prove that a base for the equational theory of totally-ordered fields consists of the f-ring identities of the form 0=0∨(f 1 ∧⋯∧f n ), n=1,2,..., where {f 1 ,...,f n }⊆ℤ[X 1 ,X 2 ,...] is not a subset of any positive cone.

Henriksen et Isbell ont montré en 1962 que certains anneaux commutatifs admettent des ordres totaux qui ne vérifient pas les lois equationnelles (dans le language des anneaux réticulés) vérifiées par tous les corps totalement ordonnés. Dans cet article, nous revisitons le travail de Henriksen et Isbell sur ce sujet. En suite nous construisons et classifions quelques exemples qui testifient à ce phenomène utilisant la théorie des valuations de Hion (ce que nous permet, en particulier, de répondre á la question posée dans [E]). Finalement, nous montrons qu’une base pour la théorie equationnelle des corps totalement ordonnés consiste des identités dans les f-anneaux de la forme 0=0∨(f 1 ∧⋯∧f n ), n=1,2,..., où {f 1 ,...,f n }⊆ℤ[X 1 ,X 2 ,...] n’est contenu dans aucun cône positif.

DOI: 10.5802/afst.1279

James J. Madden 1

1 Louisiana State University, Baton Rouge 
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     title = {On $f$-rings that are not formally real},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {143--157},
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James J. Madden. On $f$-rings that are not formally real. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 143-157. doi : 10.5802/afst.1279. https://afst.centre-mersenne.org/articles/10.5802/afst.1279/

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