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Fraction-Dense Algebras and Spaces

Published online by Cambridge University Press:  20 November 2018

Anthony W. Hager  and
Jorge Martinez
Anthony W. Hager
Affiliation:
Department of Mathematics Wesleyan University Middletown, Connecticut 06457 U.S.A.
Jorge Martinez
Affiliation:
Department of Mathematics University of Florida Gainesville, Florida 32611 U.S.A.
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Abstract

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A fraction-dense (semi-prime) commutative ring A with 1 is one for which the classical quotient ring is rigid in its maximal quotient ring. The fractiondense ƒ- rings are characterized as those for which the space of minimal prime ideals is compact and extremally disconnected. For archimedean lattice-ordered groups with this property it is shown that the Dedekind and order completion coincide. Fractiondense spaces are defined as those for which C(X) is fraction-dense. If X is compact, then this notion is equivalent to the coincidence of the absolute of X and its quasi-F cover. R-embeddings of Tychonoff spaces are re-introduced and examined in the context of fraction-density.

Keywords

06F2554C40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 5 , 01 October 1993 , pp. 977 - 996
Copyright
Copyright © Canadian Mathematical Society 1993

References

[AC] Anderson, M. and R Conrad, The hulls ofC(X), Rocky Mountain J. (1) 12 Winter (1982), 7–22.Google Scholar
[AF] Anderson, M. and Feil, T., Lattice-Ordered Groups; an Introduction, Reidel Texts, Math. Sci., Kluwer, 1988.Google Scholar
[BH1] Ball, R.N. and Hager, A.W., Archimedean kernel-distinguishing extensions of archime de an l-groups with weakunit. Indian J. Math. (3) 29(1987), 351–368.Google Scholar
[BH2] Ball, R.N., Characterizations of epimorphisms in archimedean lattice-ordered groups and vector lattices, Ch. 8, Lattice-Ordered Groups, (eds. Glass, A.M.W. and Holland, W.C.), Kluwer Acad. Publishers, 1989.Google Scholar
[Ba] Banaschewski, B., Maximal rings of quotients of semi-simple commutative rings, Archiv Math. XVI (1965), 414–420.Google Scholar
[BKW] Bigard, A., Keimel, K. and Wolfenstein, S., Groupes et Anneaux Réticulés, Springer Lecture Notes 608, Springer Verlag, 1977.Google Scholar
[B] Blair, R.L., Spaces in which special sets are z-embedded, Canad. J. Math. 28(1976), 673–690.Google Scholar
[B1H] Blair, R.L. and Hager, A.W., Extensions of zero-sets and of real-valued functions, Math. Zeitsch. 136(1974), 41–57.Google Scholar
[Bl] Bleier, R., The orthocompletion of a lattice-ordered group, Proc. Kon. Ned. Akad. v. Wetensch. (A) 79(1976), 1–7.Google Scholar
[CHN] Comfort, W.W., Hindman, N. and Negrepontis, S., F'-spaces and their products with P-spaces, Pac. J. Math. 28(1969), 489–502.Google Scholar
[C] Conrad, P., The essential closure of an archimedean lattice-ordered group, Proc. London Math. Soc. 38 (1971), 151–160.Google Scholar
[CM1] Conrad, P. and Martinez, J., Complemented lattice-ordered groups, to appear; Proc. Kon. Ned. Akad. v. Wetensch. [CM2] , Complementing lattice-ordered groups, Order, to appear.Google Scholar
[DHH] Dashiell, F., Hager, A.W. and Henriksen, M., Order-Cauchy completions of rings and vector lattices of continuous functions, Canad. J. Math. 32(1980), 657–685.Google Scholar
[FGL] Fine, N., Gillman, L. and Lambek, J., Rings of Quotients of Rings of Functions, McGill University, 1965.Google Scholar
[Gl] Gleason, A., Projective topological spaces, Illinois J. Math. 2(1958), 482–489.Google Scholar
[GJ] Gillman, L. and Jerison, M., Rings of Continuous Functions, Grad. Texts in Math. 43, Springer Verlag, 1976.Google Scholar
[H] Hager, A.W., Minimal covers of Topological Spaces, Ann. NY Acad. Sci., Papers on Gen. Topol. and Rel. Cat. Th. and Top. Alg. 552(1989), 44–59.Google Scholar
[HaJ] Hager, A.W. and Johnson, D.G., A note on certain suhalgebras of C(X), Canad. J. Math. 20(1968), 389–393.Google Scholar
[HJ] Henriksen, M. and Jerison, M., The space of minimal prime ideals of a commutative ring, Trans. AMS 115(1965), 110–130.Google Scholar
[HVW1] Henriksen, M., Vermeer, J. and Woods, R.G., Quasi-F covers of Tychonoff spaces, Trans AMS (2) 303(1987), 779–803.Google Scholar
[HVW2] Henriksen, M., Vermeer, J. and Woods, R.G., Wallman covers of compact spaces, Diss. Math., to appear.Google Scholar
[HW] Henriksen, M. and Wilson, R.R., When is C(X)/ Pa valuation ring for every prime ideal R, Proc. Oxford Topol. Conf., to appear.Google Scholar
[HP] Huijsmans, C.B. and de Pagter, B., Maximal d-ideals in a Riesz space, Canad. J. Math. (6) XXXV(1983), 1010–1029.Google Scholar
[La] Lambek, J., Lectures on Rings and Modules, Ginn-Blaisdell, Waltham Mass., 1966.Google Scholar
[Lv] Levy, R., Almost P-spaces, Canad. J. Math. 29(1977), 284–288.Google Scholar
[LZ] W Luxemburg, A.J. and Zaanen, A.C., Riesz Spaces, Vol. I, North. Holland, 1971.Google Scholar
[M] Martinez, J., The maximal ring of quotients of an f-ring, submitted.Google Scholar
[Pa] Papangelou, F., Order convergence and topological completion of commutative lattice- g roups, Math. Ann. 155(1964), 81–107.Google Scholar
[PW] Porter, J.R. and Woods, R.G., Extensions and Absolutes of Hausdorff Spaces, Springer Verlag, 1988.Google Scholar
[Si] Sikorski, R., Boolean Algebras, Ergebnisse der Math, und ihrer Grenzgeb. 25, Springer Verlag, 1960.Google Scholar
[Sm] Smirnov, Y.M., On normally disposed sets of normal spaces, Mat. Sbor. (N.S.), (Russian) 29(1951), 173–176.Google Scholar
[VI] Vermeer, J., On perfect irreduciblepre-images, Topology Proc. 9(1984), 173–189.Google Scholar
[V2] Vermeer, J., The smallest basically disconnected preimage of a space, Topol. and its Appl. 17(1984), 217- 232.Google Scholar
[Wl] Walker, R., The Stone-Cech Compactification, Ergebnisse der Math, und ihre Grenzgeb. 83, Springer Verlag,(1974).Google Scholar
[ZK] Zaharov, V.K. and Koldunov, A.V., The sequential absolute and its characterizations, Sov. Math. Dokl. 22(1980), 70–74.Google Scholar