Doi: http://dx.doi.org/10.15446/recolma.v49n1.54177

¹Universidad de Costa Rica, San José, Costa Rica. Email: jorge.guier@ucr.ac.cr

The purpose of this paper is to characterize the lattice-ordered convex subrings of von Neumann regular f-rings. They turn out to be the reduced projectable f-rings satisfying the convexity property, i.e.: for all a, b, if 0 < a < b then b divides a. A real closed version of this result can also be stated.

Key words: Lattice-ordered ring, projectable f-ring, von Neumann regular ring, convex subring, first convexity property, real closed ring, ring of quotients, valuation ring.

El propósito de este artículo es caracterizar los subanillos convexos de los f-anillos von Neumann regulares. Estos son los f-anillos reducidos, proyectables y que satisfacen la propiedad de convexidad, i.e.: para todo a, b, si 0 < a < b entonces b divide a. También se da una versión real cerrada de este resultado.

Palabras clave: Anillo reticulado, f-anillo proyectable, anillo von Neumann regular, subanillo convexo, primera propiedad de convexidad, anillo real cerrado, anillo de cocientes, anillo de valuación.

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References

[1] M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, 1969.         [ Links ]

[2] A. Bigard, K. Keimel, and S. Wolfenstein, Groupes et anneaux réticulés, `Lectures Notes in Mathematics´, (1977), Vol. 608, Springer, Berlin, Germany.         [ Links ]

[3] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer, Berlin, Germany, 1981.         [ Links ]

[4] S. Burris and H. Werner, `Sheaf Constructions and Their Elementary Properties´, Trans. Amer. Math. Soc. 248, (1979), 269-309.         [ Links ]

[5] G. Cherlin and M. A. Dickmann, `Real Closed Rings II. Model Theory´, Ann. Pure Appl. Logic 25, (1983), 213-231.         [ Links ]

[6] M. A. Dickmann, `Elimination of Quantifiers for Ordered Valuation Rings´, J. Symbolic Logic 52, (1987), 116-128.         [ Links ]

[7] J. I. Guier, `Boolean Products of Real Closed Rings and Fields´, Ann. Pure Appl. Logic 112, (2001), 119-150.         [ Links ]

[8] K. Keimel, The Representation of Lattice-Ordered Groups and Rings by Sections of Sheaves, `Lectures Notes in Mathematics´, (1971), Vol. 248, Springer, Berlin, Germany, p. 1-98.         [ Links ]

[9] M. Knebusch, `Positivity and Convexity in Rings of Fractions´, Positivity 11, (2007), 639-686.         [ Links ]

[10] J. Lambek, Lectures on Rings and Modules, Chelsea, New York, USA, 1976.         [ Links ]

[11] S. Larson, `Convexity Conditions on f-Rings´, Canad. J. Math. 38, (1986), 48-64.         [ Links ]

[12] J. Martínez, `The Maximal Ring of Quotient f-Ring´, Algebra Universalis 33, (1995), 355-369.         [ Links ]

[13] N. Schwartz, Real Closed Rings - Examples and Applications, `Séminaire de Structures Algébriques Ordonnées´, 1997, Vol. 61, 1995-96 (Delon, Dickmann, Gondard eds), Paris VII-CNRS Logique, Prépublications, Paris, France.         [ Links ]

[14] N. Schwartz, `Epimorphic Extensions and Prüfer Extensions of Partially Ordered Rings´, Manuscripta Math. 102, (2000), 347-381.         [ Links ]

[15] N. Schwartz, `Convex Subrings of Partially Ordered Rings´, Math. Nachr. 283, (2010), 758-774.         [ Links ]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

Doi: http://dx.doi.org/ @ARTICLE{RCMv49n1a08,
    AUTHOR  = {Guier, Jorge I.},
    TITLE   = {{Convex Lattice-Ordered Subrings of von Neumann Regular \boldsymbol{f}-Rings}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2015},
    volume  = {49},
    number  = {1},
    pages   = {161--170}
}