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Relatively Prüfer pairs of commutative rings. (English) Zbl 0747.13013

A Prüfer pair of commutative rings \((R,U)\) has \(R\subseteq U\) with a common identity and every ring between \(R\) and \(U\) integrally closed in \(U\). This note provides the first comprehensive study of such pairs; recent work of many authors has included results on Prüfer pairs in special cases [see chapter II of J. A. Huckaba’s book, “Commutative rings with zero divisors” (1988; Zbl 0637.13001)]. In particular the present work demonstrates common extensions and generalizations of results on \(I\)-rings by N. Eggert [J. Reine Angew. Math. 282, 88-95 (1976; Zbl 0318.13008)] and Prüfer rings by D. D. Anderson and J. Pascual [Commun. Algebra 15, 1287-1295 (1987; Zbl 0624.13015)] in which \(U\) is the complete ring of quotients of \(R\), and \(U\) is the total ring of quotients of \(R\), respectively. — The main theorem provides a set of 15 conditions on \((R,U)\) each of which is equivalent to the above Prüfer property and the author comments on possible variants to the statements of some of the conditions. The way in which the Prüfer property is affected by taking factors or quotients is investigated, as are the obvious transitivity properties. The connections between the Prüfer and the Bézout and Marot conditions are studied and the note ends with relationships between \((R,U)\) being a Prüfer pair and \(R\) being a Manis valuation subring of \(U\).

MSC:

13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F30 Valuation rings
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