Rhodes, C. P. L. Relatively Prüfer pairs of commutative rings. (English) Zbl 0747.13013 Commun. Algebra 19, No. 12, 3423-3445 (1991). 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 \(U\). Reviewer: D.Kirby (Southampton) Cited in 1 ReviewCited in 8 Documents MSC: 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 13F30 Valuation rings Keywords:regular submodule; invertible submodule; Prüfer pair; \(I\)-rings; ring of quotients; Manis valuation subring Citations:Zbl 0637.13001; Zbl 0318.13008; Zbl 0624.13015 PDFBibTeX XMLCite \textit{C. P. L. Rhodes}, Commun. Algebra 19, No. 12, 3423--3445 (1991; Zbl 0747.13013) Full Text: DOI References: [1] Akiba T., J.Math. Kyoto Univ. 9 pp 205– (1969) · Zbl 0187.00402 · doi:10.1215/kjm/1250523941 [2] DOI: 10.1080/00927878708823469 · Zbl 0624.13015 · doi:10.1080/00927878708823469 [3] DOI: 10.1016/0021-8693(71)90058-5 · Zbl 0218.13019 · doi:10.1016/0021-8693(71)90058-5 [4] DOI: 10.1016/0022-4049(87)90063-6 · Zbl 0623.13008 · doi:10.1016/0022-4049(87)90063-6 [5] DOI: 10.2140/pjm.1972.40.7 · Zbl 0202.32703 · doi:10.2140/pjm.1972.40.7 [6] Boisen M. B., Proc. Amer Math. Soc 40 pp 87– (1973) [7] DOI: 10.1216/RMJ-1971-1-4-667 · Zbl 0226.13001 · doi:10.1216/RMJ-1971-1-4-667 [8] Brase C. H., J. Reine u. Angew. Math. 265 pp 170– (1974) [9] DOI: 10.1007/BF01111523 · Zbl 0153.37003 · doi:10.1007/BF01111523 [10] Davis E. D., Trans. Amer. Math. Soc. 182 pp 175– (1973) [11] DOI: 10.4153/CMB-1980-005-8 · Zbl 0432.13007 · doi:10.4153/CMB-1980-005-8 [12] DOI: 10.4153/CJM-1981-040-5 · Zbl 0466.13002 · doi:10.4153/CJM-1981-040-5 [13] DOI: 10.1515/crll.1976.282.88 · Zbl 0318.13008 · doi:10.1515/crll.1976.282.88 [14] Eggert N., J. Reine u. Angew. Math 250 pp 109– (1971) [15] Gilmer R.W., Multiplicative ideal theory (1972) · Zbl 0248.13001 [16] DOI: 10.1016/0021-8693(74)90050-7 · Zbl 0278.13003 · doi:10.1016/0021-8693(74)90050-7 [17] DOI: 10.1515/crll.1969.239-240.55 · Zbl 0185.09801 · doi:10.1515/crll.1969.239-240.55 [18] Griffin M., Queen’s Univ (1940) [19] DOI: 10.4153/CJM-1974-042-1 · Zbl 0259.13008 · doi:10.4153/CJM-1974-042-1 [20] DOI: 10.1090/S0002-9939-1973-0318134-2 · doi:10.1090/S0002-9939-1973-0318134-2 [21] Huckaba J. A., Commutative rings with zero divisors (1988) · Zbl 0637.13001 [22] Lambek J., Waltham, Mass. (1966) [23] Larsen M. D., Multiplicative theory of ideals (1971) · Zbl 0237.13002 [24] DOI: 10.4153/CMB-1979-041-3 · Zbl 0448.13011 · doi:10.4153/CMB-1979-041-3 [25] DOI: 10.1090/S0002-9904-1966-11514-8 · Zbl 0136.31405 · doi:10.1090/S0002-9904-1966-11514-8 [26] DOI: 10.1017/S0017089500007606 · Zbl 0668.13015 · doi:10.1017/S0017089500007606 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.