It is shown that if a submodule N of M is co-coatomically supplemented and M/N has no maximal submodule, then M is a co-coatomically supplemented module. If a module M is co-coatomically supplemented, then every finitely M-generated module is a co-coatomically supplemented module. Every left R-module is co-coatomically supplemented if and only if the ring R is left perfect. Over a discrete valuation ring, a module M is co-coatomically supplemented if and only if the basic submodule of M is coatomic. Over a nonlocal Dedekind domain, if the torsion part T(M) of a reduced module M has a weak supplement in M, then M is co-coatomically supplemented if and only if M/T (M) is divisible and T P (M) is bounded for each maximal ideal P. Over a nonlocal Dedekind domain, if a reduced module M is co-coatomically amply supplemented, then M/T (M) is divisible and T P (M) is bounded for each maximal ideal P. Conversely, if M/T (M) is divisible and T P (M) is bounded for each maximal ideal P, then M is a co-coatomically supplemented module.
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References
R. Aizade, G. Bilhan, and P. F. Smith, “Modules whose maximal submodules have supplements,” Comm. Algebra, 29, 2389–2405 (2001).
E. Büyükaşık, Weakly and CofinitelyWeak Supplemented Modules over Dedekind Ring: PhD Thesis, Dokuz Eylül University (2005).
E. Büyükaşık and C. Lomp, “Rings whose modules are weakly supplemented are perfect. Applications to certain ring extensions,” Math. Scand., 105, 25–30 (2009).
E. Büyükaşık and D. Pusat-Yılmaz, “Modules whose maximal submodules are supplements,” Hacet. J. Math. Stat., 39, No. 4, 477–487 (2010).
J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer, Lifting Modules, Birkhäuser-Verlag (2006).
I. Kaplansky, “Modules over Dedekind rings and valuation rings,” Trans. Amer. Math. Soc., 72, 327–340 (1952).
T. Y. Lam, Lectures on Modules and Rings, Springer (1999).
P. F. Smith, “Finitely generated supplemented modules are amply supplemented,” Arab. J. Sci. Eng., 25, 69–80 (2000).
R. Wisbauer, Foundations of Modules and Rings, Gordon & Breach (1991).
H. Zöschinger, “Komplementierte moduln über Dedekindringen,” J. Algebra, 29, 42–56 (1974).
H. Zöschinger, “Moduln die in jeder Erweiterung ein Komplement haben,” Math. Scand., 35, 267–287 (1974).
H. Zöschinger and F. Rosenberg, “Koatomare moduln,” Math. Z., 170, 221–232 (1980).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 7, pp. 867–876, July, 2017.
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Alizade, R., Güngӧr, S. Co-Coatomically Supplemented Modules. Ukr Math J 69, 1007–1018 (2017). https://doi.org/10.1007/s11253-017-1411-x
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DOI: https://doi.org/10.1007/s11253-017-1411-x