Czechoslovak Mathematical Journal, first online, pp. 1-10


On strongly affine extensions of commutative rings

Nabil Zeidi

Received May 14, 2018.   Published online November 18, 2019.

Abstract:  A ring extension $R\subseteq S$ is said to be strongly affine if each $R$-subalgebra of $S$ is a finite-type $R$-algebra. In this paper, several characterizations of strongly affine extensions are given. For instance, we establish that if $R$ is a quasi-local ring of finite dimension, then $R\subseteq S$ is integrally closed and strongly affine if and only if $R\subseteq S$ is a Prüfer extension (i.e. $(R,S)$ is a normal pair). As a consequence, the equivalence of strongly affine extensions, quasi-Prüfer extensions and INC-pairs is shown. Let $G$ be a subgroup of the automorphism group of $S$ such that $R$ is invariant under action by $G$. If $R\subseteq S$ is strongly affine, then $R^G\subseteq S^G$ is strongly affine under some conditions.
Keywords:  strongly affine; Prüfer extension; finitely many intermediate algebras property extension; finite chain propery extension; normal pair; integrally closed pair; ring of invariants
Classification MSC:  13B02, 13A15, 13A50, 13E05
DOI:  10.21136/CMJ.2019.0240-18

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References:
[1] D. D. Anderson, D. E. Dobbs, B. Mullins: The primitive element theorem for commutative algebra. Houston J. Math. 25 (1999), 603-623; corrigendum ibid. 28 217-219 (2002). MR 1829123 | Zbl 0999.13003
[2] A. Ayache, D. E. Dobbs: Finite maximal chains of commutative rings. J. Algebra Appl. 14 (2015), Article ID 1450075, 27 pages. DOI 10.1142/S0219498814500753 | MR 3257816 | Zbl 1310.13012
[3] A. Ayache, A. Jaballah: Residually algebraic pairs of rings. Math. Z. 225 (1997), 49-65. DOI 10.1007/PL00004598 | MR 1451331 | Zbl 0868.13007
[4] M. Ben Nasr, N. Zeidi: A special chain Theorem in the set of intermediate rings. J. Algebra Appl. 16 (2017), Article ID 1750185, 11 pages. DOI 10.1142/S0219498817501857 | MR 3703540 | Zbl 1390.13028
[5] E. D. Davis: Overrings of commutative rings III: Normal pairs. Trans. Amer. Math. Soc. 182 (1973), 175-185. DOI 10.1090/S0002-9947-1973-0325599-3 | MR 325599 | Zbl 0272.13004
[6] D. E. Dobbs: Lying-over pairs of commutative rings. Can. J. Math. 33 (1981), 454-475. DOI 10.4153/CJM-1981-040-5 | MR 617636 | Zbl 0466.13002
[7] D. E. Dobbs: On characterizations of integrality involving the lying-over and incomparability properties. J. Comm. Algebra 1 (2009), 227-235. DOI 10.1216/JCA-2009-1-2-227 | MR 2504933 | Zbl 1184.13023
[8] D. E. Dobbs, B. Mullins, G. Picavet, M. Picavet-L'Hermitte: On the FIP property for extensions of commutative rings. Commun. Algebra 33 (2005), 3091-3119. DOI 10.1081/AGB-200066123 | MR 2175382 | Zbl 1120.13009
[9] D. E. Dobbs, G. Picavet, M. Picavet-L'Hermitte: Characterizing the ring extensions that satisfy FIP or FCP. J. Algebra 371 (2012), 391-429. DOI 10.1016/j.jalgebra.2012.07.055 | MR 2975403 | Zbl 1271.13022
[10] D. E. Dobbs, G. Picavet, M. Picavet-L'Hermitte: Transfer results for the FIP and FCP properties of ring extensions. Commun. Algebra 43 (2015), 1279-1316. DOI 10.1080/00927872.2013.856440 | MR 3298132 | Zbl 1317.13016
[11] D. E. Dobbs, J. Shapiro: Descent of divisibility properties of integral domains to fixed rings. Houston J. Math. 32 (2006), 337-353. MR 2219318 | Zbl 1109.13004
[12] D. Ferrand, J.-P. Olivier: Homomorphismes minimaux d'anneaux. J. Algebra 16 (1970), 461-471. (In French.) DOI 10.1016/0021-8693(70)90020-7 | MR 271079 | Zbl 0218.13011
[13] R. Gilmer: Multiplicative Ideal Theory. Pure and Applied Mathematics 12, Marcel Dekker, New York (1972). MR 0427289 | Zbl 0248.13001
[14] R. Gilmer: Some finiteness conditions on the set of overrings of an integral domain. Proc. Am. Math. Soc. 131 (2003), 2337-2346. DOI 10.1090/S0002-9939-02-06816-8 | MR 1974630 | Zbl 1017.13009
[15] R. Gilmer, W. Heinzer: Finitely generated intermediate rings. J. Pure Appl. Algebra 37 (1985), 237-264. DOI 10.1016/0022-4049(85)90101-X | MR 797865 | Zbl 0564.13009
[16] R. Gilmer, J. Hoffmann: A characterization of Prüfer domains in terms of polynomials. Pac. J. Math. 60 (1975), 81-85. DOI 10.2140/pjm.1975.60.81 | MR 412175 | Zbl 0307.13011
[17] A. Jaballah: Finiteness of the set of intermediary rings in normal pairs. Saitama Math. J. 17 (1999), 59-61. MR 1740247 | Zbl 1073.13500
[18] A. Jaballah: Ring extensions with some finiteness conditions on the set of intermediate rings. Czech. Math. J. 60 (2010), 117-124. DOI 10.1007/s10587-010-0002-x | MR 2595076 | Zbl 1224.13011
[19] I. Kaplansky: Commutative Rings. University of Chicago Press, Chicago (1974). MR 0345945 | Zbl 0296.13001
[20] M. Knebusch, D. Zhang: Manis Valuations and Prüfer Extensions I: A New Chapter in Commutative Algebra. Lecture Notes in Mathematics 1791, Springer, Berlin (2002). DOI 10.1007/b84018 | MR 1937245 | Zbl 1033.13001
[21] R. Kumar, A. Gaur: On $\lambda$-extensions of commutative rings. J. Algebra Appl. 17 (2018), Article ID 1850063, 9 pages. DOI 10.1142/S0219498818500639 | MR 3786742 | Zbl 1395.13006
[22] I. J. Papick: Finite type extensions and coherence. Pac. J. Math. 78 (1978), 161-172. DOI 10.2140/pjm.1978.78.161 | MR 513292 | Zbl 0363.13010
[23] G. Picavet, M. Picavet-L'Hermitte: Some more combinatorics results on Nagata extensions. Palest. J. Math. 5 (2016), 49-62. MR 3477615 | Zbl 1346.13012
[24] G. Picavet, M. Picavet-L'Hermitte: Quasi-Prüfer extensions of rings. Rings, Polynomials and Modules (M. Fontana et al., eds.). Springer, Cham (2017), 307-336. DOI 10.1007/978-3-319-65874-2_16 | MR 3751703 | Zbl 1400.13012
[25] A. Schmidt: Properties of Rings and of Ring Extensions That are Invariant Under Group Action. PhD Thesis. George Mason University, Spring (2015). Available at http://hdl.handle.net/1920/9627. MR 3389167
[26] A. R. Wadsworth: Pairs of domains where all intermediate domains are Noetherian. Trans. Am. Math. Soc. 195 (1974), 201-211. DOI 10.1090/S0002-9947-1974-0349665-2 | MR 349665 | Zbl 0294.13010

Affiliations:   Nabil Zeidi, Faculty of Sciences, Department of Mathematics, Sfax University, B.P. 1171, 3000 Sfax, Tunisia, e-mail: zeidi.nabil@gmail.com, zeidi_nabil@yahoo.com


 
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