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Residually algebraic pairs of rings. (English) Zbl 0868.13007

The main purpose of this paper is to study residually algebraic extensions and pairs. The ring extension \(R\subset S\) is said to be residually algebraic, if for each prime \(Q\) of \(S\) and \(P=Q \cap R\), the extension \(R/P \subset S/Q\) is algebraic. The pair of rings \((R,S)\) is said to be residually algebraic if for each ring \(T\) in \([R,S]\), the set of intermediary rings, the extension \(R\subset T\) is residually algebraic. – This paper contains three main results:
1. A generalisation of the \((u,u^{-1})\)-lemma. It is shown in this paper that the residually algebraic pairs satisfy the \((u,u^{-1})\)-lemma, which enable the authors to give several statements equivalent to this lemma;
2. Sharp upper bounds for the number of rings, and the length of chains in \([R,S]\), the set of intermediary rings, are given when \((R,S)\) is a residually algebraic pair satisfying some good finiteness conditions. These bounds applied to a Prüfer domain \(R\), with field of fractions \(k(R)\), having Krull dimension 1 and with exactly \(r\) maximal ideals, show that there are exactly \(2^r\) rings in \([R,k(R)]\); and maximal chains of rings in the set \([R,k(R)]\) have length \(r\);
3. A positive answer, in a particular case, to a question raised by A. R. Wadsworth [Trans. Am. Math. Soc. 195, 201-211 (1976; Zbl 0294.13010)]. The author show that if the ring \(R\) is Noetherian and integrally closed in \(S\), and if \((R_M,S_M)\) is a Noetherian pair for each maximal ideal \(M\) of \(R\), then \((R,S)\) is also a Noetherian pair.
Beside the above mentioned results, the paper also contains a study of the properties of the residually algebraic extensions and pairs. These pairs are compared to extensions that are going up, incomparable, lying over, integral, going down, satisfying the dimension formula or the dimension inequality. They are also compared to valuation and Noetherian pairs, as well as to pairs of rings \((R,S)\) where \(R\) is a maximal subring of \(S\). Several characterisations of residually algebraic pairs in both the local and global case are given. Some known results for Prüfer domains are also generalised to residually algebraic pairs, as well as new results concerning Prüfer rings are obtained. Finally some examples and counterexamples raised by some results in this paper are given. These examples are based on pullback constructions.

MSC:

13B02 Extension theory of commutative rings

Citations:

Zbl 0294.13010
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