Ring extensions with some finiteness conditions on the set of intermediate rings

Ali Jaballah

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 1, page 117-124
  • ISSN: 0011-4642

Abstract

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A ring extension R ⊆ S is said to be FO if it has only finitely many intermediate rings. R ⊆ 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 ⊆ 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.

How to cite

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Jaballah, Ali. "Ring extensions with some finiteness conditions on the set of intermediate rings." Czechoslovak Mathematical Journal 60.1 (2010): 117-124. <http://eudml.org/doc/37994>.

@article{Jaballah2010,
abstract = {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.},
author = {Jaballah, Ali},
journal = {Czechoslovak Mathematical Journal},
keywords = {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; integral domain; intermediate ring; overring; integrally closed; Prüfer domain; residually algebraic pair},
language = {eng},
number = {1},
pages = {117-124},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Ring extensions with some finiteness conditions on the set of intermediate rings},
url = {http://eudml.org/doc/37994},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Jaballah, Ali
TI - Ring extensions with some finiteness conditions on the set of intermediate rings
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 1
SP - 117
EP - 124
AB - 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.
LA - eng
KW - 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; integral domain; intermediate ring; overring; integrally closed; Prüfer domain; residually algebraic pair
UR - http://eudml.org/doc/37994
ER -

References

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  9. Jaballah, A., 10.1080/00927879908826495, Commun. Algebra 27 (1999), 1307-1311. (1999) Zbl0972.13008MR1669083DOI10.1080/00927879908826495
  10. Jaballah, A., Finiteness of the set of intermediary rings in normal pairs, Saitama Math. J. 17 (1999), 59-61. (1999) Zbl1073.13500MR1740247
  11. Jaballah, A., 10.1016/j.exmath.2005.02.003, Expo. Math. 23 (2005), 353-360. (2005) Zbl1100.13008MR2186740DOI10.1016/j.exmath.2005.02.003
  12. Schröder, Bernd S. W., Ordered Sets: an Introduction, Birkhäuser Boston (2003). (2003) MR1944415

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