Abstract
In pointfree topology, F-frames have been defined by Ball and Walters-Wayland by means of a frame-theoretic translation of the topological characterization of F-spaces as those whose cozero-sets are C*-embedded. This is a departure from the way in which F-spaces were defined by Gillman and Henriksen as those spaces X for which the ring C(X) is Bézout, meaning that every finitely generated ideal is principal. In this note, we show that, as in the case of spaces, a frame L is an F-frame precisely when the ring \({\mathcal{R}L}\) of continuous real-valued functions on L is Bézout. A commutative ring with identity is called almost weak Baer if the annihilator of each element is generated by idempotents. We establish that \({\mathcal{R}L}\) is almost weak Baer iff L is a strongly zero-dimensional F-frame.
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References
Ball, R.N., Walters-Wayland, J.: C- and C*-quotients in pointfree topology. Dissertationes Mathematicae (Rozprawy Mat.), vol. 412 (2002)
Banaschewski, B.: The real numbers in pointfree topology. Textos de Matemática Série B, no. 12. Departamento de Matemática da Universidade de Coimbra (1997)
Banaschewski B.: A uniform view of localic realcompactness. J. Pure Appl. Algebra 143, 49–68 (1999)
Banaschewski B.: On the function ring functor in pointfree topology. Appl. Categ. Structures 13, 305–328 (2005)
Banaschewski B., Brümmer G.C.L.: Functorial uniformities on strongly zero-dimensional frames. Kyungpook Math. J. 41, 179–190 (2002)
Banaschewski B., Gilmour C.: Pseudocompactness and the cozero part of a frame. Comment. Math. Univ. Carolin. 37, 577–587 (1996)
Banaschewski B., Pultr A.: Samuel compactification of uniform frames. Math. Proc. Camb. Phil. Soc. 108, 63–78 (1990)
Blair R.L., Hager A.W.: Extensions of zero-sets and real-valued functions. Math. Z. 136, 41–52 (1974)
Dube T.: Notes on C- and C*-quotients of frames. Order 25, 369–375 (2008)
Dube T.: Some ring-theoretic properties of almost P-frames. Algebra Universalis 60, 145–162 (2009)
Dube, T.: Notes on pointfree disconnectivity with a ring-theoretic slant. Appl. Categ. Struct. DOI10.1007/s10485-008-9162-3
Dube, T.: Concerning P-, essential P- and strongly zero-dimensional frames. Algebra Universalis (to appear)
Dube T., Walters-Wayland J.: Coz-onto frame maps and some applications. Appl. Categ. Structures 15, 119–133 (2007)
Ghosh S.K.: Intersections of minimal prime ideals in the rings of continuous functions. Comment. Math. Univ. Carolin. 47, 623–632 (2006)
Gillman L., Henriksen M.: Rings of continuous functions in which every finitely generated ideal is principal. Trans. Amer. Math. Soc. 82, 366–391 (1956)
Gillman, L., Jerison, M.: Rings of Continuous Functions. Van Nostrand, Princeton (1960)
Henriksen, M.: Rings of continuous functions from an algebraic point of view. In: Ordered algebraic structures (Curaao, 1988). Math. Appl., vol. 55, pp. 143–174. Kluwer Acad. Publ., Dordrecht (1989)
Johnstone P.T.: Stone Spaces. Cambridge Univ. Press, Cambridge (1982)
Madden J., Vermeer H.: Lindelöf locales and realcompactness. Math. Proc. Camb. Phil. Soc. 99, 473–480 (1986)
Mandelker M.: F′-spaces and z-embedded subspaces. Pacific J. Math. 28, 615–621 (1969)
Martinez J., Woodward S.: Bézout and Prüfer rings. Comm. Algebra 20, 2975–2989 (1992)
Niefield S.B., Rosenthal K.I.: Sheaves of integral domains on Stone spaces. J. Pure Appl. Algebra 47, 173–179 (1987)
Rudd D.: On two sums theorem for ideals of C(X). Michigan Math. J. 17, 139–141 (1970)
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Presented by J. Martinez.
In memory of Paul F. Conrad
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Dube, T. Some algebraic characterizations of F-frames. Algebra Univers. 62, 273–288 (2009). https://doi.org/10.1007/s00012-010-0054-7
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DOI: https://doi.org/10.1007/s00012-010-0054-7