Equations for the set of overrings of normal rings and related ring extensions

Mabrouk Ben Nasr; Ali Jaballah

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 3, page 921-935
  • ISSN: 0011-4642

Abstract

top
We establish several finiteness characterizations and equations for the cardinality and the length of the set of overrings of rings with nontrivial zero divisors and integrally closed in their total ring of fractions. Similar properties are also obtained for related extensions of commutative rings that are not necessarily integral domains. Numerical characterizations are obtained for rings with some finiteness conditions afterwards.

How to cite

top

Ben Nasr, Mabrouk, and Jaballah, Ali. "Equations for the set of overrings of normal rings and related ring extensions." Czechoslovak Mathematical Journal 73.3 (2023): 921-935. <http://eudml.org/doc/299099>.

@article{BenNasr2023,
abstract = {We establish several finiteness characterizations and equations for the cardinality and the length of the set of overrings of rings with nontrivial zero divisors and integrally closed in their total ring of fractions. Similar properties are also obtained for related extensions of commutative rings that are not necessarily integral domains. Numerical characterizations are obtained for rings with some finiteness conditions afterwards.},
author = {Ben Nasr, Mabrouk, Jaballah, Ali},
journal = {Czechoslovak Mathematical Journal},
keywords = {total ring of fractions; ring extension; intermediate ring; overring; finite direct product; FIP extension; FCP extension; integrally closed; integral domain; Prüfer domain; valuation domain; normal pair; normal ring; length of ring extension; number of intermediate ring; number of overring},
language = {eng},
number = {3},
pages = {921-935},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Equations for the set of overrings of normal rings and related ring extensions},
url = {http://eudml.org/doc/299099},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Ben Nasr, Mabrouk
AU - Jaballah, Ali
TI - Equations for the set of overrings of normal rings and related ring extensions
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 3
SP - 921
EP - 935
AB - We establish several finiteness characterizations and equations for the cardinality and the length of the set of overrings of rings with nontrivial zero divisors and integrally closed in their total ring of fractions. Similar properties are also obtained for related extensions of commutative rings that are not necessarily integral domains. Numerical characterizations are obtained for rings with some finiteness conditions afterwards.
LA - eng
KW - total ring of fractions; ring extension; intermediate ring; overring; finite direct product; FIP extension; FCP extension; integrally closed; integral domain; Prüfer domain; valuation domain; normal pair; normal ring; length of ring extension; number of intermediate ring; number of overring
UR - http://eudml.org/doc/299099
ER -

References

top
  1. Anderson, D. D., Dobbs, D. E., Mullins, B., The primitive element theorem for commutative algebras, Houston J. Math. 25 (1999), 603-623 corrigendum ibid 28 2002 217-219. (1999) Zbl0999.13003MR1829123
  2. Ayache, A., Jaballah, A., 10.1007/PL00004598, Math. Z. 225 (1997), 49-65. (1997) Zbl0868.13007MR1451331DOI10.1007/PL00004598
  3. Badawi, A., Jaballah, A., Some finiteness conditions on the set of overrings of a φ -ring, Houston J. Math. 34 (2008), 397-408. (2008) Zbl1143.13010MR2417400
  4. Bastida, E., Gilmer, R., 10.1307/mmj/1029001014, Mich. Math. J. 20 (1973), 79-95. (1973) Zbl0239.13001MR0323782DOI10.1307/mmj/1029001014
  5. Nasr, M. Ben, 10.1007/s00605-008-0090-y, Monatsh. Math. 158 (2009), 97-102. (2009) Zbl1180.13008MR2525924DOI10.1007/s00605-008-0090-y
  6. Nasr, M. Ben, 10.1142/S0219498816500225, J. Algebra Appl. 15 (2016), Article ID 1650022, 8 pages. (2016) Zbl1335.13006MR3479800DOI10.1142/S0219498816500225
  7. Nasr, M. Ben, Jaballah, A., 10.1016/j.exmath.2007.09.002, Expo. Math. 26 (2008), 163-175. (2008) Zbl1142.13004MR2413833DOI10.1016/j.exmath.2007.09.002
  8. Nasr, M. Ben, Jaballah, A., 10.1142/S0219498820501716, J. Algebra Appl. 19 (2020), Article ID 2050171, 12 pages. (2020) Zbl1451.13023MR4136742DOI10.1142/S0219498820501716
  9. Nasr, M. Ben, Jarboui, N., 10.1007/s11587-013-0169-1, Ric. Mat. 63 (2014), 149-155. (2014) Zbl1301.13008MR3211064DOI10.1007/s11587-013-0169-1
  10. Nasr, M. Ben, Zeidi, N., 10.1142/S0219498817501857, J. Algebra Appl. 16 (2017), Articles ID 1750185, 11 pages. (2017) Zbl1390.13028MR3703540DOI10.1142/S0219498817501857
  11. Nasr, M. Ben, Zeidi, N., 10.1017/S0004972716000721, Bull. Aust. Math. Soc. 95 (2017), 14-21. (2017) Zbl1365.13011MR3592540DOI10.1017/S0004972716000721
  12. Davis, E. D., 10.1090/S0002-9947-1973-0325599-3, Trans. Am. Math. Soc. 182 (1973), 175-185. (1973) Zbl0272.13004MR0325599DOI10.1090/S0002-9947-1973-0325599-3
  13. Dobbs, D. E., Mullins, B., Picavet, G., Picavet-L'Hermitte, M., 10.1081/AGB-200066123, Commun. Algebra 33 (2005), 3091-3119. (2005) Zbl1120.13009MR2175382DOI10.1081/AGB-200066123
  14. Dobbs, D. E., Picavet, G., Picavet-L'Hermitte, M., 10.1016/j.jalgebra.2012.07.055, J. Algebra 371 (2012), 391-429. (2012) Zbl1271.13022MR2975403DOI10.1016/j.jalgebra.2012.07.055
  15. Dobbs, D. E., Shapiro, J., 10.1142/S0219498811004628, J. Algebra Appl. 10 (2011), 335-356. (2011) Zbl1221.13012MR2795742DOI10.1142/S0219498811004628
  16. Gaur, A., Kumar, R., 10.1080/00927872.2021.1986517, Commun. Algebra 50 (2022), 1613-1631. (2022) Zbl1482.13015MR4391512DOI10.1080/00927872.2021.1986517
  17. Gilmer, R., Multiplicative Ideal Theory, Pure and Applied Mathematics 12. Marcel Dekker, New York (1972). (1972) Zbl0248.13001MR0427289
  18. Gilmer, R., 10.1090/S0002-9939-02-06816-8, Proc. Am. Math. Soc. 131 (2003), 2337-2346. (2003) Zbl1017.13009MR1974630DOI10.1090/S0002-9939-02-06816-8
  19. Grothendieck, A., 10.1007/BF02684778, Publ. Math., Inst. Hautes Étud. Sci. 4 (1960), 1-228 French. (1960) Zbl0118.36206MR0217083DOI10.1007/BF02684778
  20. Jaballah, A., Subrings of Q , J. Sci. Technology 2 (1997), 1-13. (1997) 
  21. Jaballah, A., 10.1080/00927879908826495, Commun. Algebra 27 (1999), 1307-1311. (1999) Zbl0972.13008MR1669083DOI10.1080/00927879908826495
  22. Jaballah, A., Finiteness of the set of intermediary rings in normal pairs, Saitama Math. J. 17 (1999), 59-61. (1999) Zbl1073.13500MR1740247
  23. Jaballah, A., 10.1016/j.exmath.2005.02.003, Expo. Math. 23 (2005), 353-360. (2005) Zbl1100.13008MR2186740DOI10.1016/j.exmath.2005.02.003
  24. Jaballah, A., 10.1007/s10587-010-0002-x, Czech. Math. J. 60 (2010), 117-124. (2010) Zbl1224.13011MR2595076DOI10.1007/s10587-010-0002-x
  25. Jaballah, A., 10.1007/s00605-010-0205-0, Monatsh. Math. 164 (2011), 171-181. (2011) Zbl1228.13023MR2837113DOI10.1007/s00605-010-0205-0
  26. Jaballah, A., 10.1142/S0219498811005658, J. Algebra Appl. 11 (2012), Article ID 1250041, 18 pages. (2012) Zbl1259.13004MR2983173DOI10.1142/S0219498811005658
  27. Jaballah, A., 10.1007/s13366-012-0101-y, Beitr. Algebra Geom. 54 (2013), 111-120. (2013) Zbl1267.13013MR3027669DOI10.1007/s13366-012-0101-y
  28. Jaballah, A., Integral domains whose overrings are discrete valuation rings, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 62 (2016), 361-369. (2016) Zbl1389.13022MR3680213
  29. Jaballah, A., 10.1007/s13366-022-00677-5, (to appear) in Beitr. Algebra Geom. DOI10.1007/s13366-022-00677-5
  30. Jaballah, A., Jarboui, N., 10.1017/S0004972720000015, Bull. Aust. Math. Soc. 102 (2020), 15-20. (2020) Zbl1443.05187MR4120745DOI10.1017/S0004972720000015
  31. Matsumura, H., 10.1017/CBO9781139171762, Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge (1989). (1989) Zbl0666.13002MR1011461DOI10.1017/CBO9781139171762
  32. Picavet, G., Picavet-L'Hermitte, M., FIP and FCP products of ring morphisms, Palest. J. Math. 5 (2016), 63-80. (2016) Zbl1346.13013MR3477616
  33. Stacks Project. Part 1: Preliminaries. Chapter 10: Commutative Algebra. Section 10.37: Normal rings. Lemma 10.37.16, Available at https://stacks.math.columbia.edu/tag/030C. 
  34. Stacks Project. Part 1: Preliminaries. Chapter 10: Commutative Algebra. Section 10.23: Glueing propertiesLemma 10.23.1, Available at https://stacks.math.columbia.edu/tag/00HN. 

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.