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Keywords:
comparable overring, integral closure; Prüfer extension; Pinched extension; pullback; support of a module
comparable overring, integral closure; Prüfer extension; Pinched extension; pullback; support of a module
Summary:
We say $R\subseteq S$ is pinched at some intermediate ring $R_0,$ where $R\subset R_0\subset S,$ if each intermediate ring between $R$ and $S$ is comparable to $R_0$ under inclusion. A new characterization of Prüfer extensions in terms of maximal excluding domains is given. We also characterize minimal extensions of a Prüfer domain and prove that no extension of a one-dimensional Prüfer domain can be pinched, and thereby extending old results of Gilbert on $\lambda $-extensions. Next, we show that a proper finite Galois extension is pinched if and only if the Galois group is cyclic of prime power order. Further, the preservation of comparability of the integral closure and that of $\lambda $-finiteness in pullbacks is also studied.
References:
[1] Ayache, A.: When is a fixed ring comparable to all overrings?. J. Algebra Appl. 21 (2022), Article ID 2250050, 16 pages. DOI 10.1142/S0219498822500505 | MR 4391815 | Zbl 1482.13014
[2] Ayache, A., Jaballah, A.: Residually algebraic pairs of rings. Math. Z. 225 (1997), 49-65. DOI 10.1007/PL00004598 | MR 1451331 | Zbl 0868.13007
[3] Dobbs, D. E., Picavet, G., Picavet-L'Hermitte, M.: Characterizing the ring extensions that satisfy FIP or FCP. J. Alegbra 371 (2012), 391-429. DOI 10.1016/j.jalgebra.2012.07.055 | MR 2975403 | Zbl 1271.13022
[4] Dobbs, D. E., Picavet, G., Picavet-L'Hermitte, M.: 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
[5] Dummit, D. S., Foote, R. M.: Abstract Algebra. John Wiley, Chichester (2004). MR 2286236 | Zbl 1037.00003
[6] Ferrand, D., Olivier, J.-P.: Homomorphismes minimaux d'anneaux. J. Alegbra 16 (1970), 461-471 French. DOI 10.1016/0021-8693(70)90020-7 | MR 0271079 | Zbl 0218.13011
[7] Fontana, M.: Topologically defined classes of commutative rings. Ann. Mat. Pura Appl. (4) 123 (1980), 331-355. DOI 10.1007/BF01796550 | MR 0581935 | Zbl 0443.13001
[8] Fontana, M., Huckaba, J. A., Papick, I. R.: Prüfer domains. Pure and Applied Mathematics, Marcel Dekker 203. Marcel Dekker, New York (1997). MR 1413297 | Zbl 0861.13006
[9] Gilbert, M.: Extensions of Commutative Rings With Linearly Ordered Intermediate Rings: Ph.D. Thesis. University of Tennessee, Knoxville (1996). MR 2695057
[10] Gilmer, R., Heinzer, W. J.: Intersections of quotient rings of an integral domain. J. Math. Kyoto Univ. 7 (1967), 133-150. DOI 10.1215/kjm/1250524273 | MR 0223349 | Zbl 0166.30601
[11] Gilmer, R., Huckaba, J. A.: $\Delta$-rings. J. Alegbra 28 (1974), 414-432. DOI 10.1016/0021-8693(74)90050-7 | MR 0427308 | Zbl 0278.13003
[12] Gilmer, R., Huckaba, J. A.: Maximal overrings of an integral domain not containing a given element. Commun. Algebra 2 (1974), 377-401. DOI 10.1080/00927877408822018 | MR 0357388 | Zbl 0291.13008
[13] Guerrieri, L., Loper, K. A.: Non-integrally closed Kronecker function rings and integral domains with a unique minimal overring. Ann. Mat. Pura Appl. (4) 203 (2024), 1483-1511. DOI 10.1007/s10231-023-01410-2 | MR 4772563 | Zbl 1547.13008
[14] Isaacs, I. M.: Finite Group Theory. Graduate Studies in Mathematics 92. AMS, Providence (2008). DOI 10.1090/gsm/092 | MR 2426855 | Zbl 1169.20001
[15] Klawa, H.: Global perinormality in a generalized $D+M$ construction. Commun. Algebra 51 (2023), 3015-3019. DOI 10.1080/00927872.2023.2175844 | MR 4571083 | Zbl 1523.13038
[16] Knebusch, M., Zhang, D.: 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
[17] Lee, J.: Commutative Rings: New Research. Nova Science Publishers, New York (2009).
[18] Nagata, M.: Local Rings. Interscience Tracts in Pure and Applied Mathematics 13. John Wiley, New York (1962). MR 0155856 | Zbl 0123.03402
[19] Nasr, M., Zeidi, N.: When is the integral closure comparable to all intermediate rings. Bull. Aust. Math. Soc. 95 (2017), 14-21. DOI 10.1017/S0004972716000721 | MR 3592540 | Zbl 1365.13011
[20] Navarro, G.: On the fundamental theorem of finite Abelian groups. Am. Math. Mon. 110 (2003), 153-154. DOI 10.1080/00029890.2003.11919951 | MR 1952445 | Zbl 1027.20028
[21] Picavet, G., Picavet-L'Hermitte, M.: Quasi-Prüfer extensions of rings. Rings, Polynomials and Modules Springer, Cham (2017), 307-336. DOI 10.1007/978-3-319-65874-2_16 | MR 3751703 | Zbl 1400.13012
[22] Picavet, G., Picavet-L'Hermitte, M.: Rings extensions of length two. J. Algebra Appl. 18 (2019), Article ID 1950174, 34 pages. DOI 10.1142/S0219498819501743 | MR 3981707 | Zbl 1423.13059
[23] Picavet, G., Picavet-L'Hermitte, M.: FCP $\Delta$-extensions of rings. Arab. J. Math. 10 (2021), 211-238. DOI 10.1007/s40065-020-00298-7 | MR 4244815 | Zbl 1466.13008
[24] Picavet, G., Picavet-L'Hermitte, M.: Splitting ring extensions. Beitr. Algebra Geom. 64 (2023), 627-668. DOI 10.1007/s13366-022-00650-2 | MR 4615490 | Zbl 1524.13035
[25] Singh, M., Singh, R.: On ring extensions pinched at an arbitrary intermediate ring. Ric. Mat. 74 (2025), 2351-2366. DOI 10.1007/s11587-024-00911-3 | MR 4961478 | Zbl 08101731
[26] Singh, M., Singh, R.: On ring extensions pinched at the integral closure. Beitr. Algebra Geom. 66 (2025), 465-477. DOI 10.1007/s13366-024-00752-z | MR 4905095 | Zbl 08043099
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