Geodesic
Ring extensions with some finiteness conditions on the set of intermediate rings
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 117-124.

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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.
Classification : 13B02, 13B22, 13E15, 13E99, 13F05, 13G05
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 
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     author = {Jaballah, Ali},
     title = {Ring extensions with some finiteness conditions on the set of intermediate rings},
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     pages = {117--124},
     publisher = {mathdoc},
     volume = {60},
     number = {1},
     year = {2010},
     mrnumber = {2595076},
     zbl = {1224.13011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010__60_1_a9/}
}
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Jaballah, Ali. Ring extensions with some finiteness conditions on the set of intermediate rings. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 1, pp. 117-124. http://geodesic.mathdoc.fr/item/CMJ_2010__60_1_a9/