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

Ben Nasr, Mabrouk ; Jaballah, Ali

Czechoslovak Mathematical Journal, Tome 73 (2023) no. 3, pp. 921-935

Abstract (VO)

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.

Détail

DOI : 10.21136/CMJ.2023.0358-22

Classification : 13B02, 13B22, 13B30, 13E15, 13E99, 13F05, 13G05
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

Bibliographie

[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. | MR | JFM

[2] Ayache, A., Jaballah, A.: Residually algebraic pairs of rings. Math. Z. 225 (1997), 49-65. | DOI | MR | JFM

[3] Badawi, A., Jaballah, A.: Some finiteness conditions on the set of overrings of a $\phi$-ring. Houston J. Math. 34 (2008), 397-408. | MR | JFM

[4] Bastida, E., Gilmer, R.: Overrings and divisorial ideals of rings of the form $D+M$. Mich. Math. J. 20 (1973), 79-95. | DOI | MR | JFM

[5] Nasr, M. Ben: On finiteness of chains of intermediate rings. Monatsh. Math. 158 (2009), 97-102. | DOI | MR | JFM

[6] Nasr, M. Ben: An answer to a problem about the number of overrings. J. Algebra Appl. 15 (2016), Article ID 1650022, 8 pages. | DOI | MR | JFM

[7] Nasr, M. Ben, Jaballah, A.: Counting intermediate rings in normal pairs. Expo. Math. 26 (2008), 163-175. | DOI | MR | JFM

[8] Nasr, M. Ben, Jaballah, A.: The number of intermediate rings in FIP extension of integral domains. J. Algebra Appl. 19 (2020), Article ID 2050171, 12 pages. | DOI | MR | JFM

[9] Nasr, M. Ben, Jarboui, N.: New results about normal pairs of rings with zero-divisors. Ric. Mat. 63 (2014), 149-155. | DOI | MR | JFM

[10] Nasr, M. Ben, Zeidi, N.: A special chain theorem in the set of intermediate rings. J. Algebra Appl. 16 (2017), Articles ID 1750185, 11 pages. | DOI | MR | JFM

[11] Nasr, M. Ben, Zeidi, N.: When is the integral closure comparable to all intermediate rings. Bull. Aust. Math. Soc. 95 (2017), 14-21. | DOI | MR | JFM

[12] Davis, E. D.: Overrings of commutative rings. III: Normal pairs. Trans. Am. Math. Soc. 182 (1973), 175-185. | DOI | MR | JFM

[13] Dobbs, D. E., Mullins, B., Picavet, G., Picavet-L'Hermitte, M.: On the FIP property for extensions of commutative rings. Commun. Algebra 33 (2005), 3091-3119. | DOI | MR | JFM

[14] Dobbs, D. E., Picavet, G., Picavet-L'Hermitte, M.: Characterizing the ring extensions that satisfy FIP or FCP. J. Algebra 371 (2012), 391-429. | DOI | MR | JFM

[15] Dobbs, D. E., Shapiro, J.: Normal pairs with zero-divisors. J. Algebra Appl. 10 (2011), 335-356. | DOI | MR | JFM

[16] Gaur, A., Kumar, R.: Maximal non-Prüfer and maximal non-$\phi$-Prüfer rings. Commun. Algebra 50 (2022), 1613-1631. | DOI | MR | JFM

[17] Gilmer, R.: Multiplicative Ideal Theory. Pure and Applied Mathematics 12. Marcel Dekker, New York (1972). | MR | JFM

[18] Gilmer, R.: Some finiteness conditions on the set of overrings of an integral domain. Proc. Am. Math. Soc. 131 (2003), 2337-2346. | DOI | MR | JFM

[19] Grothendieck, A.: Éléments de géométrie algébrique. Publ. Math., Inst. Hautes Étud. Sci. 4 (1960), 1-228 French. | DOI | MR | JFM

[20] Jaballah, A.: Subrings of $Q$. J. Sci. Technology 2 (1997), 1-13.

[21] Jaballah, A.: A lower bound for the number of intermediary rings. Commun. Algebra 27 (1999), 1307-1311. | DOI | MR | JFM

[22] Jaballah, A.: Finiteness of the set of intermediary rings in normal pairs. Saitama Math. J. 17 (1999), 59-61. | MR | JFM

[23] Jaballah, A.: The number of overrings of an integrally closed domain. Expo. Math. 23 (2005), 353-360. | DOI | MR | JFM

[24] Jaballah, A.: Ring extensions with some finiteness conditions on the set of intermediate rings. Czech. Math. J. 60 (2010), 117-124. | DOI | MR | JFM

[25] Jaballah, A.: Numerical characterizations of some integral domains. Monatsh. Math. 164 (2011), 171-181. | DOI | MR | JFM

[26] Jaballah, A.: Maximal non-Prüfer and maximal non-integrally closed subrings of a field. J. Algebra Appl. 11 (2012), Article ID 1250041, 18 pages. | DOI | MR | JFM

[27] Jaballah, A.: Graph theoretic characterizations of maximal non-valuation subrings of a field. Beitr. Algebra Geom. 54 (2013), 111-120. | DOI | MR | JFM

[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. | MR | JFM

[29] Jaballah, A.: The dimension-overrings equation and maximal ideals of integral domains. (to appear) in Beitr. Algebra Geom. | DOI

[30] Jaballah, A., Jarboui, N.: From topologies of a set to subrings of its power set. Bull. Aust. Math. Soc. 102 (2020), 15-20. | DOI | MR | JFM

[31] Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge (1989). | DOI | MR | JFM

[32] Picavet, G., Picavet-L'Hermitte, M.: FIP and FCP products of ring morphisms. Palest. J. Math. 5 (2016), 63-80. | MR | JFM

[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