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
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topBen 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- 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
- Ayache, A., Jaballah, A., 10.1007/PL00004598, Math. Z. 225 (1997), 49-65. (1997) Zbl0868.13007MR1451331DOI10.1007/PL00004598
- Badawi, A., Jaballah, A., Some finiteness conditions on the set of overrings of a -ring, Houston J. Math. 34 (2008), 397-408. (2008) Zbl1143.13010MR2417400
- Bastida, E., Gilmer, R., 10.1307/mmj/1029001014, Mich. Math. J. 20 (1973), 79-95. (1973) Zbl0239.13001MR0323782DOI10.1307/mmj/1029001014
- Nasr, M. Ben, 10.1007/s00605-008-0090-y, Monatsh. Math. 158 (2009), 97-102. (2009) Zbl1180.13008MR2525924DOI10.1007/s00605-008-0090-y
- Nasr, M. Ben, 10.1142/S0219498816500225, J. Algebra Appl. 15 (2016), Article ID 1650022, 8 pages. (2016) Zbl1335.13006MR3479800DOI10.1142/S0219498816500225
- 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
- Nasr, M. Ben, Jaballah, A., 10.1142/S0219498820501716, J. Algebra Appl. 19 (2020), Article ID 2050171, 12 pages. (2020) Zbl1451.13023MR4136742DOI10.1142/S0219498820501716
- 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
- Nasr, M. Ben, Zeidi, N., 10.1142/S0219498817501857, J. Algebra Appl. 16 (2017), Articles ID 1750185, 11 pages. (2017) Zbl1390.13028MR3703540DOI10.1142/S0219498817501857
- Nasr, M. Ben, Zeidi, N., 10.1017/S0004972716000721, Bull. Aust. Math. Soc. 95 (2017), 14-21. (2017) Zbl1365.13011MR3592540DOI10.1017/S0004972716000721
- 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
- 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
- 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
- Dobbs, D. E., Shapiro, J., 10.1142/S0219498811004628, J. Algebra Appl. 10 (2011), 335-356. (2011) Zbl1221.13012MR2795742DOI10.1142/S0219498811004628
- Gaur, A., Kumar, R., 10.1080/00927872.2021.1986517, Commun. Algebra 50 (2022), 1613-1631. (2022) Zbl1482.13015MR4391512DOI10.1080/00927872.2021.1986517
- Gilmer, R., Multiplicative Ideal Theory, Pure and Applied Mathematics 12. Marcel Dekker, New York (1972). (1972) Zbl0248.13001MR0427289
- 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
- Grothendieck, A., 10.1007/BF02684778, Publ. Math., Inst. Hautes Étud. Sci. 4 (1960), 1-228 French. (1960) Zbl0118.36206MR0217083DOI10.1007/BF02684778
- Jaballah, A., Subrings of , J. Sci. Technology 2 (1997), 1-13. (1997)
- Jaballah, A., 10.1080/00927879908826495, Commun. Algebra 27 (1999), 1307-1311. (1999) Zbl0972.13008MR1669083DOI10.1080/00927879908826495
- Jaballah, A., Finiteness of the set of intermediary rings in normal pairs, Saitama Math. J. 17 (1999), 59-61. (1999) Zbl1073.13500MR1740247
- Jaballah, A., 10.1016/j.exmath.2005.02.003, Expo. Math. 23 (2005), 353-360. (2005) Zbl1100.13008MR2186740DOI10.1016/j.exmath.2005.02.003
- Jaballah, A., 10.1007/s10587-010-0002-x, Czech. Math. J. 60 (2010), 117-124. (2010) Zbl1224.13011MR2595076DOI10.1007/s10587-010-0002-x
- Jaballah, A., 10.1007/s00605-010-0205-0, Monatsh. Math. 164 (2011), 171-181. (2011) Zbl1228.13023MR2837113DOI10.1007/s00605-010-0205-0
- Jaballah, A., 10.1142/S0219498811005658, J. Algebra Appl. 11 (2012), Article ID 1250041, 18 pages. (2012) Zbl1259.13004MR2983173DOI10.1142/S0219498811005658
- Jaballah, A., 10.1007/s13366-012-0101-y, Beitr. Algebra Geom. 54 (2013), 111-120. (2013) Zbl1267.13013MR3027669DOI10.1007/s13366-012-0101-y
- 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
- Jaballah, A., 10.1007/s13366-022-00677-5, (to appear) in Beitr. Algebra Geom. DOI10.1007/s13366-022-00677-5
- Jaballah, A., Jarboui, N., 10.1017/S0004972720000015, Bull. Aust. Math. Soc. 102 (2020), 15-20. (2020) Zbl1443.05187MR4120745DOI10.1017/S0004972720000015
- Matsumura, H., 10.1017/CBO9781139171762, Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge (1989). (1989) Zbl0666.13002MR1011461DOI10.1017/CBO9781139171762
- Picavet, G., Picavet-L'Hermitte, M., FIP and FCP products of ring morphisms, Palest. J. Math. 5 (2016), 63-80. (2016) Zbl1346.13013MR3477616
- 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.
- 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.
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