On f -rings that are not formally real

James J. Madden[1]

  • [1] Louisiana State University, Baton Rouge

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

  • Volume: 19, Issue: S1, page 143-157
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Madden, James J.. "On $f$-rings that are not formally real." Annales de la faculté des sciences de Toulouse Mathématiques 19.S1 (2010): 143-157. <http://eudml.org/doc/115894>.

@article{Madden2010,
abstract = {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 $0=0\vee (f_1\wedge \cdots \wedge f_n)$, $n=1,2,\ldots $, where $\lbrace \,f_1,\ldots , f_n\,\rbrace \subseteq \mathbb\{Z\}[X_1,X_2,\ldots ]$ is not a subset of any positive cone.},
affiliation = {Louisiana State University, Baton Rouge},
author = {Madden, James J.},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {total orderings; lattice-ordered rings; valuation theory; -rings},
language = {eng},
month = {4},
number = {S1},
pages = {143-157},
publisher = {Université Paul Sabatier, Toulouse},
title = {On $f$-rings that are not formally real},
url = {http://eudml.org/doc/115894},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Madden, James J.
TI - On $f$-rings that are not formally real
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2010/4//
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - S1
SP - 143
EP - 157
AB - 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 $0=0\vee (f_1\wedge \cdots \wedge f_n)$, $n=1,2,\ldots $, where $\lbrace \,f_1,\ldots , f_n\,\rbrace \subseteq \mathbb{Z}[X_1,X_2,\ldots ]$ is not a subset of any positive cone.
LA - eng
KW - total orderings; lattice-ordered rings; valuation theory; -rings
UR - http://eudml.org/doc/115894
ER -

References

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  7. M. Henriksen and J. Isbell, Lattice ordered rings and function rings, Pacific J. Math. 12 (1962), 533–66. Zbl0111.04302MR153709
  8. Ya. V. Hion, Rings normed by the aid of semigroups. Izv. Akad. Nauk SSSR. Ser. Mat.21(1957), 311–328. (Russian) MR 19 (1958), p. 530. MR88502
  9. J. Isbell, Notes on ordered rings, Algeb ra Universalis1 (1972), 393–399. Zbl0238.06013MR295994
  10. J. Madden, Pierce-Birkhoff rings. Arch. der Math. 53 (1989), 565–70. Zbl0691.14012MR1023972
  11. N. Schwartz and J. Madden, Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings. Lecture Notes in Mathematics, 1712. Springer-Verlag, Berlin, 1999. Zbl0967.14038MR1719673
  12. B. Sturmfels, Gröbner bases and convex polytopes, AMS, Providence, RI, 1996. Zbl0856.13020MR1363949

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