Olberding, Bruce Almost maximal Prüfer domains. (English) Zbl 0977.13010 Commun. Algebra 27, No. 9, 4433-4458 (1999). Extending a definition of I. Kaplansky, W. Brandal [Trans. Am. Math. Soc. 183, 203-222 (1973; Zbl 0273.13011)] defined a commutative ring \(R\) to be almost maximal if every proper homomorphic image of \(R\) is linearly compact. He proved that if \(D\) is an almost maximal Prüfer domain with quotient field \(K\), then every homomorphic image of \(D\) is an injective \(D\)-module. In the paper under review, the converse of Brandal’s theorem is established. It is also shown that a Prüfer domain \(D\) is almost maximal if and only if each nonzero prime ideal of \(D\) has injective dimension one. Applications involving reflexive domains are also considered. Reviewer: Tiberiu Dumitrescu (Bucureşti) Cited in 6 Documents MSC: 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 13C11 Injective and flat modules and ideals in commutative rings 18G05 Projectives and injectives (category-theoretic aspects) 13J99 Topological rings and modules Keywords:Prüfer domain; almost maximal ring; injective module; linearly compact ring; homomorphic image; injective dimension Citations:Zbl 0273.13011 PDFBibTeX XMLCite \textit{B. Olberding}, Commun. Algebra 27, No. 9, 4433--4458 (1999; Zbl 0977.13010) Full Text: DOI References: [1] DOI: 10.1006/jabr.1996.0353 · Zbl 0873.13020 · doi:10.1006/jabr.1996.0353 [2] DOI: 10.1090/S0002-9947-1973-0325609-3 · doi:10.1090/S0002-9947-1973-0325609-3 [3] Cartan H., Homological algebra (1956) [4] DOI: 10.1016/0022-4049(94)90030-2 · Zbl 0845.13005 · doi:10.1016/0022-4049(94)90030-2 [5] Fontana M., Prüfer domains (1997) [6] Gilmer R.W., Multiplicative Ideal Theory 90 (1992) · Zbl 0804.13001 [7] DOI: 10.1016/0021-8693(67)90073-7 · Zbl 0156.04304 · doi:10.1016/0021-8693(67)90073-7 [8] Goeters, H.P. Warfield duality and extensions of modules over an integral domain, Abelian Groups and Modules. Proceedings of the Padova Conference. 1995. Kluwer Academic Press. · Zbl 0873.13019 [9] DOI: 10.1080/00927879808826325 · Zbl 0920.13001 · doi:10.1080/00927879808826325 [10] DOI: 10.1090/S0002-9947-1952-0046349-0 · doi:10.1090/S0002-9947-1952-0046349-0 [11] Matlis E., Nagoya Math. J 15 pp 57– (1959) [12] DOI: 10.1016/0021-8693(68)90031-8 · Zbl 0191.32301 · doi:10.1016/0021-8693(68)90031-8 [13] Matlis E., Torsion-free modules (1972) · Zbl 0298.13001 [14] DOI: 10.1006/jabr.1997.7406 · Zbl 0928.13013 · doi:10.1006/jabr.1997.7406 [15] Olberding B., Modules of injective dimension one over Prüfer domains · Zbl 0978.13009 · doi:10.1016/S0022-4049(99)00166-8 [16] Reid J.D., Abelian groups and non-commutative rings, Contemporary Math 130 pp 361– (1992) [17] DOI: 10.1112/jlms/s2-12.1.103 · Zbl 0318.12109 · doi:10.1112/jlms/s2-12.1.103 [18] DOI: 10.1007/BF01110257 · Zbl 0169.03602 · doi:10.1007/BF01110257 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.