Previous |  Up |  Next


integral domain; intermediate ring; overring; integrally closed; Prüfer domain; residually algebraic pair; normal pair; primitive extension; a.c.c.; d.c.c.; minimal condition; maximal condition; affine extension; Dilworth number; width of an ordered set
A ring extension $R\subseteq S$ is said to be FO if it has only finitely many intermediate rings. $R\subseteq S$ is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension $R\subseteq S$ to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair with only finitely many intermediate rings. We also obtain as a corollary several new and old characterizations of Prüfer and integral domains satisfying the corresponding finiteness conditions.
[1] Ayache, A., Jaballah, A.: Residually algebraic pairs of rings. Math. Z. 225 (1997), 49-65. DOI 10.1007/PL00004598 | MR 1451331 | Zbl 0868.13007
[2] Badawi, A., Jaballah, A.: Some finiteness conditions on the set of overrings of a $\phi $-ring. Houston J. Math. 34 (2008), 397-408. MR 2417400 | Zbl 1143.13010
[3] Nasr, M. B., Jaballah, A.: Counting intermediate rings in normal pairs. Expo. Math. 26 (2008), 163-175. DOI 10.1016/j.exmath.2007.09.002 | MR 2413833 | Zbl 1142.13004
[4] Davis, E. D.: Overrings of commutative rings. III: Normal pairs. Trans. Amer. Math. Soc. 182 (1973), 175-185. MR 0325599 | Zbl 0272.13004
[5] Dobbs, D., Fontana, M.: Universally incomparable ring homomorphisms. Bull. Aust. Math. Soc. 29 (1984), 289-302. DOI 10.1017/S0004972700021547 | MR 0748722 | Zbl 0535.13006
[6] Fontana, M., Huckaba, J. A., Papick, I. J.: Prüfer Domains. Marcel Dekker New York (1997). MR 1413297 | Zbl 0861.13006
[7] Gilmer, R.: 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
[8] Gilmer, R., Hoffman, J.: A characterization of Prüfer domains in terms of polynomials. Pacific J. Math. 60 (1975), 81-85. DOI 10.2140/pjm.1975.60.81 | MR 0412175 | Zbl 0307.13011
[9] Jaballah, A.: A lower bound for the number of intermediary rings. Commun. Algebra 27 (1999), 1307-1311. DOI 10.1080/00927879908826495 | MR 1669083 | Zbl 0972.13008
[10] Jaballah, A.: Finiteness of the set of intermediary rings in normal pairs. Saitama Math. J. 17 (1999), 59-61. MR 1740247 | Zbl 1073.13500
[11] Jaballah, A.: The number of overrings of an integrally closed domain. Expo. Math. 23 (2005), 353-360. DOI 10.1016/j.exmath.2005.02.003 | MR 2186740 | Zbl 1100.13008
[12] Schröder, Bernd S. W.: Ordered Sets: an Introduction. Birkhäuser Boston (2003). MR 1944415
Partner of
EuDML logo