SV and related f -rings and spaces

Suzanne Larson[1]

  • [1] Loyola Marymount University Los Angeles, California 90045

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

  • Volume: 19, Issue: S1, page 111-141
  • ISSN: 0240-2963

Abstract

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An f -ring A is an SV f -ring if for every minimal prime ℓ -ideal P of A , A / P is a valuation domain. A topological space X is an SV space if C ( X ) is an SV f -ring. SV f -rings and spaces were introduced in [HW1], [HW2]. Since then a number of articles on SV f -rings and spaces and on related f -rings and spaces have appeared. This article surveys what is known about these f -rings and spaces and introduces a number of new results that help to clarify the relationship between SV f -rings and spaces and related f -rings and spaces.

How to cite

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Larson, Suzanne. "SV and related $f$-rings and spaces." Annales de la faculté des sciences de Toulouse Mathématiques 19.S1 (2010): 111-141. <http://eudml.org/doc/115892>.

@article{Larson2010,
abstract = {An $f$-ring $A$ is an SV $f$-ring if for every minimal prime $\ell $-ideal $P$ of $A$, $A/P$ is a valuation domain. A topological space $X$ is an SV space if $C(X)$ is an SV $f$-ring. SV $f$-rings and spaces were introduced in [HW1], [HW2]. Since then a number of articles on SV $f$-rings and spaces and on related $f$-rings and spaces have appeared. This article surveys what is known about these $f$-rings and spaces and introduces a number of new results that help to clarify the relationship between SV $f$-rings and spaces and related $f$-rings and spaces.},
affiliation = {Loyola Marymount University Los Angeles, California 90045},
author = {Larson, Suzanne},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {-rings; SV-rings; partially ordered rings},
language = {eng},
month = {4},
number = {S1},
pages = {111-141},
publisher = {Université Paul Sabatier, Toulouse},
title = {SV and related $f$-rings and spaces},
url = {http://eudml.org/doc/115892},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Larson, Suzanne
TI - SV and related $f$-rings and spaces
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2010/4//
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - S1
SP - 111
EP - 141
AB - An $f$-ring $A$ is an SV $f$-ring if for every minimal prime $\ell $-ideal $P$ of $A$, $A/P$ is a valuation domain. A topological space $X$ is an SV space if $C(X)$ is an SV $f$-ring. SV $f$-rings and spaces were introduced in [HW1], [HW2]. Since then a number of articles on SV $f$-rings and spaces and on related $f$-rings and spaces have appeared. This article surveys what is known about these $f$-rings and spaces and introduces a number of new results that help to clarify the relationship between SV $f$-rings and spaces and related $f$-rings and spaces.
LA - eng
KW - -rings; SV-rings; partially ordered rings
UR - http://eudml.org/doc/115892
ER -

References

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  1. Ali Rezaei Aliabad, Ahvaz. Pasting topological spaces at one point. Czech. Math. J. 2006, 56 (131), 1193 - 1206. Zbl1164.54338MR2280803
  2. Banerjee, B; Henriksen, M. Ways in which C ( X ) mod a prime ideal can be a valuation domain; something old and something new. Positivity (Trends in Mathematics), Birkhäuser Basel: Switzerland, 2007, 1- 25. Zbl1149.46042MR2382213
  3. Bigard, A.; Keimel, K.; Wolfenstein, S. Groupes et Anneaux Réticulés, Lecture Notes in Mathematics 608; Springer-Verlag: New York, 1977. Zbl0384.06022MR552653
  4. Cherlin, G.; Dickmann, M. Real-closed rings I. Fund. Math. 1986, 126, 147-183. Zbl0605.54014MR843243
  5. Darnel, Michael. Theory of Lattice-Ordered Groups; Marcel Dekker, Inc: New York, 1995. Zbl0810.06016MR1304052
  6. Dashiell, F.; Hager, A.; Henriksen, M. Order-Cauchy completions of rings and vector lattices of continuous functions. Canad. J. Math. 1980, 32(3), 657-685. Zbl0462.54009MR586984
  7. Gillman, L.; Jerison, M. Rings of Continuous Functions; D. Van Nostrand Publishing: New York, 1960. Zbl0093.30001MR116199
  8. Henriksen, M.; Larson, S. Semiprime f -rings that are subdirect products of valuation domains. Ordered Algebraic Structures (Gainesville, FL 1991), Kluwer Acad. Publishing: Dordrecht, 1993, 159-168. Zbl0799.06031MR1247304
  9. Henriksen, M.; Larson, S.; Martinez, J.; Woods, R. G. Lattice-ordered algebras that are subdirect products of valuation domains. Trans. Amer. Math. Soc. 1994, 345, 193-221. Zbl0817.06014MR1239640
  10. Henriksen, M.; Wilson, R. When is C ( X ) / P a valuation ring for every prime ideal P ? Topology and Applications 1992, 44, 175-180. Zbl0801.54014MR1173255
  11. Henriksen, M.; Wilson, R. Almost discrete SV-spaces. Topology and Applications 1992, 46, 89-97. Zbl0791.54049MR1184107
  12. Kuratowski, K. Topology, vol. 1; Academic Press; New York, 1966. Zbl0158.40802MR217751
  13. Larson, S. Convexity conditions on f -rings. Canad. J. Math. 1986, 38, 48-64. Zbl0588.06011MR835035
  14. Larson, S. f -Rings in which every maximal ideal contains finitely many minimal prime ideals. Comm. in Algebra 1997, 25 (12), 3859-3888. Zbl0952.06026MR1481572
  15. Larson, S. Constructing rings of continuous functions in which there are many maximal ideals with nontrivial rank. Comm. in Algebra 2003, 31 (5), 2183-2206. Zbl1024.54015MR1976272
  16. Larson, S. Rings of continuous functions on spaces of finite rank and the SV property. Comm. in Algebra 2007, 35 (8), 2611 - 2627. Zbl1146.54008MR2345805
  17. Larson, S. Images and open subspaces of SV spaces. Comm. in Algebra, 2008, 36, 1 - 13. Zbl1147.54008MR2387527
  18. Larson, S. Functions that map cozerosets to cozerosets. Commentationes Mathematicae Universitatis Carolinae 2007, 48 (3), 507-521. Zbl1199.54099MR2374130
  19. Martinez, J; Woodward, S. Bezout and Prüfer f -Rings. Comm. in Algebra 1992, 20, 2975 - 2989. Zbl0766.06018MR1179272
  20. Schwartz, N. The Basic Theory of Real Closed Spaces, AMS Memoirs 397; Amer. Math. Soc.: Providence, RI, 1989. Zbl0697.14015MR953224
  21. Schwartz, N. Rings of continuous functions as real closed rings. Ordered Algebraic Structures (Curaçao 1995), Kluwer Acad. Publishing: Dordrecht, 1997, 277-313. Zbl0885.46024MR1445117

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