On finitely injective modules and locally pure-injective modules over Prüfer domains

Salce, Luigi

doi:10.1090/S0002-9939-07-08906-X

On finitely injective modules and locally pure-injective modules over Prüfer domains
HTML articles powered by AMS MathViewer

by Luigi Salce

Proc. Amer. Math. Soc. 135 (2007), 3485-3493

DOI: https://doi.org/10.1090/S0002-9939-07-08906-X

Published electronically: June 29, 2007

PDF | Request permission

Abstract:

Over Matlis valuation domains there exist finitely injective modules which are not direct sums of injective modules, as well as complete locally pure-injective modules which are not the completion of a direct sum of pure-injective modules. Over Prüfer domains which are either almost maximal, or $h$-local Matlis, finitely injective torsion modules and complete torsion-free locally pure-injective modules correspond to each other under the Matlis equivalence. Almost maximal Prüfer domains are characterized by the property that every torsion-free complete module is locally pure-injective. It is derived that semi-Dedekind domains are Dedekind.

References

S. Bazzoni and L. Salce, An independence result on cotorsion theories over valuation domains, J. Algebra 243 (2001), no. 1, 294–320. MR 1851665, DOI 10.1006/jabr.2001.8800
Willy Brandal, Almost maximal integral domains and finitely generated modules, Trans. Amer. Math. Soc. 183 (1973), 203–222. MR 325609, DOI 10.1090/S0002-9947-1973-0325609-3
Stephen U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473. MR 120260, DOI 10.1090/S0002-9947-1960-0120260-3
Paul C. Eklof and Alan H. Mekler, Almost free modules, Revised edition, North-Holland Mathematical Library, vol. 65, North-Holland Publishing Co., Amsterdam, 2002. Set-theoretic methods. MR 1914985
Alberto Facchini, Torsion-free covers and pure-injective envelopes over valuation domains, Israel J. Math. 52 (1985), no. 1-2, 129–139. MR 815608, DOI 10.1007/BF02776086
Alberto Facchini, Absolutely pure modules and locally injective modules, Commutative ring theory (Fès, 1992) Lecture Notes in Pure and Appl. Math., vol. 153, Dekker, New York, 1994, pp. 105–109. MR 1261882
László Fuchs and Luigi Salce, Modules over valuation domains, Lecture Notes in Pure and Applied Mathematics, vol. 97, Marcel Dekker, Inc., New York, 1985. MR 786121
—, Modules over non-Noetherian domains, Mathematical Surveys and Monographs, vol. 84, American Mathematical Society, Providence, RI, 2001.
Phillip Griffith, A note on a theorem of Hill, Pacific J. Math. 29 (1969), 279–284. MR 245613
Laurent Gruson and Christian U. Jensen, Deux applications de la notion de $L$-dimension, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 1, Aii, A23–A24. MR 401880
Paul Hill, On the decomposition of groups, Canadian J. Math. 21 (1969), 762–768. MR 249507, DOI 10.4153/CJM-1969-087-6
T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294, DOI 10.1007/978-1-4612-0525-8
Sang B́um Lee, $h$-divisible modules, Comm. Algebra 31 (2003), no. 1, 513–525. MR 1969238, DOI 10.1081/AGB-120016774
Eben Matlis, Divisible modules, Proc. Amer. Math. Soc. 11 (1960), 385–391. MR 116044, DOI 10.1090/S0002-9939-1960-0116044-8
V. S. Ramamurthi and K. M. Rangaswamy, On finitely injective modules, J. Austral. Math. Soc. 16 (1973), 239–248. Collection of articles dedicated to the memory of Hanna Neumann, II. MR 0332882
Luigi Salce and Peter Vámos, On some classes of divisible modules, Rend. Sem. Mat. Univ. Padova 115 (2006), 125–136. MR 2245591
R. Warfield, Relatively injective modules, unpublished manuscript, 1969.
Wolfgang Zimmermann, Rein injektive direkte Summen von Moduln, Comm. Algebra 5 (1977), no. 10, 1083–1117 (German). MR 450327, DOI 10.1080/00927877708822211
Wolfgang Zimmermann, On locally pure-injective modules, J. Pure Appl. Algebra 166 (2002), no. 3, 337–357. MR 1870625, DOI 10.1016/S0022-4049(01)00011-1
Birge Zimmermann-Huisgen, On the abundance of $\aleph _1$-separable modules, Abelian groups and noncommutative rings, Contemp. Math., vol. 130, Amer. Math. Soc., Providence, RI, 1992, pp. 167–180. MR 1176118, DOI 10.1090/conm/130/1176118