Flat modules over valuation rings. (English) Zbl 1123.13016

A number of ring and module properties are explored in stride; these properties include P-flatness, content flatness, local projectivity, finite projectivity and single projectivity, separable, projective and free modules. Most of these properties are discussed in the setting where the base ring \(R\) is a valuation ring. Some of the results are as follows:
Theorem 2. The ring \(R\) is left perfect if and only if each flat left module is a content module.
Theorem 7. Let \(R\) be a commutative ring with a self-FP-injective quotient ring \(Q\). Then each flat \(R\)-module is finitely projective iff \(Q\) is perfect.
Theorem 16 gives a characterization of commutative PP-Baer rings and singly projective rings.
Theorem 30. If \(R\) is a valuation ring, then the following three conditions are equivalent:
(1) \(R\) is maximal;
(2) each singly projective \(R\)-module is separable;
(3) each flat content module is locally projective.
Furthermore, characterizations are given for a valuation ring (with non-trivial zero divisors) to be strongly coherent and \(\pi\)-coherent.


13F30 Valuation rings
16D40 Free, projective, and flat modules and ideals in associative algebras
13C11 Injective and flat modules and ideals in commutative rings
Full Text: DOI arXiv


[1] Azumaya, G., Finite splitness and finite projectivity, J. Algebra, 106, 114-134 (1987) · Zbl 0607.16017
[2] Bass, H., Finistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc., 95, 466-488 (1960) · Zbl 0094.02201
[3] Camillo, V., Coherence for polynomial rings, J. Algebra, 132, 72-76 (1990) · Zbl 0701.16023
[4] Couchot, F., Injective modules and \(f p\)-injective modules over valuations rings, J. Algebra, 267, 359-376 (2003) · Zbl 1060.13003
[5] Couchot, F., Pure-injective hulls of modules over valuation rings, J. Pure Appl. Algebra, 207, 63-76 (2006) · Zbl 1101.13034
[6] Cox, H.; Pendleton, R., Rings for which certain flat modules are projective, Trans. Amer. Math. Soc., 150, 139-156 (1970) · Zbl 0215.07802
[7] Facchini, A., Relative injectivity and pure-injective modules over Prüfer rings, J. Algebra, 110, 380-406 (1987) · Zbl 0629.13008
[8] Franzen, B., On the separability of a direct product of free modules over a valuation domain, Arch. Math., 42, 131-135 (1984) · Zbl 0522.13007
[9] Fuchs, L.; Salce, L., (Modules over Valuation Domains. Modules over Valuation Domains, Lecture Notes in Pure and Appl. Math, vol. 97 (1985), Marcel Dekker: Marcel Dekker New York) · Zbl 0578.13004
[10] Fuchs, L.; Salce, L., (Modules over Non-Noetherian Domains. Modules over Non-Noetherian Domains, Mathematical Surveys and Monographs, vol. 84 (2001), American Mathematical Society: American Mathematical Society Providence) · Zbl 0973.13001
[11] Garfinkel, G. S., Universally torsionless and trace modules, Trans. Amer. Math. Soc., 215, 119-144 (1976) · Zbl 0319.16020
[12] Gill, D. T., Almost maximal valuation rings, J. London Math. Soc., 4, 140-146 (1971) · Zbl 0219.13016
[13] Gruson, L.; Raynaud, M., Critères de platitude et de projectivité, Invent. Math., 13, 1-89 (1971) · Zbl 0227.14010
[14] Jain, S., Flatness and \(F P\)-injectivity, Proc. Amer. Math. Soc., 41, 2, 437-442 (1973) · Zbl 0246.16013
[15] Jones, M. F., Flatness and \(f\)-projectivity of torsion-free modules and injective modules, (Advances in Non-commutative Ring Theory. Advances in Non-commutative Ring Theory, Lecture Notes in Mathematics, vol. 951 (1982), Springer-Verlag: Springer-Verlag New York, Berlin), 94-116 · Zbl 0498.16020
[16] Kemper, R., Product-trace-rings and a question of G.S. Garfinkel, Proc. Amer. Math. Soc., 128, 3, 709-712 (2000) · Zbl 0942.13019
[17] Klatt, G. B.; Levy, L. S., Pre-self injective rings, Trans. Amer. Math. Soc., 137, 407-419 (1969) · Zbl 0176.31603
[18] Kobayashi, S., A note on regular self-injective rings, Osaka J. Math., 21, 3, 679-682 (1984) · Zbl 0542.16011
[19] Ramamurthi, V. S.; Rangaswamy, K. M., On finitely injective modules, J. Aust. Math. Soc., XVI, 2, 239-248 (1973) · Zbl 0267.16014
[20] Shenglin, Z., On rings over which every flat left module is finitely projective, J. Algebra, 139, 311-321 (1991) · Zbl 0736.16004
[21] Warfield, R. B., Purity and algebraic compactness for modules, Pacific J. Math., 28, 3, 689-719 (1969) · Zbl 0172.04801
[22] Zimmermann-Huisgen, B., Pure submodules of direct products of free modules, Math. Ann., 224, 233-245 (1976) · Zbl 0321.16015
[23] B. Zimmermann-Huisgen, Direct products of modules and algebraic compactness, Habilitationsschrift, Tech. Univ. München, 1980; B. Zimmermann-Huisgen, Direct products of modules and algebraic compactness, Habilitationsschrift, Tech. Univ. München, 1980
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.