# 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

## Access Full Article

top## Abstract

top## How to cite

topLarson, 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

top- Ali Rezaei Aliabad, Ahvaz. Pasting topological spaces at one point. Czech. Math. J. 2006, 56 (131), 1193 - 1206. Zbl1164.54338MR2280803
- Banerjee, B; Henriksen, M. Ways in which $C\left(X\right)$ 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
- Bigard, A.; Keimel, K.; Wolfenstein, S. Groupes et Anneaux Réticulés, Lecture Notes in Mathematics 608; Springer-Verlag: New York, 1977. Zbl0384.06022MR552653
- Cherlin, G.; Dickmann, M. Real-closed rings I. Fund. Math. 1986, 126, 147-183. Zbl0605.54014MR843243
- Darnel, Michael. Theory of Lattice-Ordered Groups; Marcel Dekker, Inc: New York, 1995. Zbl0810.06016MR1304052
- 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
- Gillman, L.; Jerison, M. Rings of Continuous Functions; D. Van Nostrand Publishing: New York, 1960. Zbl0093.30001MR116199
- 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
- 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
- Henriksen, M.; Wilson, R. When is $C\left(X\right)/P$ a valuation ring for every prime ideal $P$? Topology and Applications 1992, 44, 175-180. Zbl0801.54014MR1173255
- Henriksen, M.; Wilson, R. Almost discrete SV-spaces. Topology and Applications 1992, 46, 89-97. Zbl0791.54049MR1184107
- Kuratowski, K. Topology, vol. 1; Academic Press; New York, 1966. Zbl0158.40802MR217751
- Larson, S. Convexity conditions on $f$-rings. Canad. J. Math. 1986, 38, 48-64. Zbl0588.06011MR835035
- 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
- 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
- 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
- Larson, S. Images and open subspaces of SV spaces. Comm. in Algebra, 2008, 36, 1 - 13. Zbl1147.54008MR2387527
- Larson, S. Functions that map cozerosets to cozerosets. Commentationes Mathematicae Universitatis Carolinae 2007, 48 (3), 507-521. Zbl1199.54099MR2374130
- Martinez, J; Woodward, S. Bezout and Prüfer $f$-Rings. Comm. in Algebra 1992, 20, 2975 - 2989. Zbl0766.06018MR1179272
- Schwartz, N. The Basic Theory of Real Closed Spaces, AMS Memoirs 397; Amer. Math. Soc.: Providence, RI, 1989. Zbl0697.14015MR953224
- Schwartz, N. Rings of continuous functions as real closed rings. Ordered Algebraic Structures (Curaçao 1995), Kluwer Acad. Publishing: Dordrecht, 1997, 277-313. Zbl0885.46024MR1445117

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.