Abstract
In this paper, it is shown that any non-\(M\)-cosingular \(\oplus\)-supplemented module \(M\) is \((D_3)\) if and only if \(M\) has the summand intersection property. Let \(N\in\sigma[M]\) be any module such that \({\overline { Z}}_M(N)\) has a coclosure in \(N\). Then we prove that \(N\) is (completely) \(\oplus\)-supplemented if and only if \(N={\overline { Z}}_M^2(N) \oplus K\) for some submodule \(K\) of \(N\) such that \({\overline { Z}}_M^2(N)\) and \(K\) both are (completely) \(\oplus\)-supplemented.
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References
Alkan, M., Harmanci, A.: On summand sum and summand intersection property of modules. Turkish J. Math. 26, 131–147 (2002)
Clark, J., Lomp, C., Vanaja, N., Wisbauer, R.: Lifting modules. In: Supplements and Projectivity in Module Theory Series: Frontiers in Mathematics (2006)
Ganesan, L., Vanaja, N.: Strongly discrete modules. Comm. Algebra (to appear)
Garcia, J.L.: Properties of direct summands of modules. Comm. Algebra 17, 73–92 (1989)
Hausen, J.: Modules with the summand intersection property. Comm. Algebra 17, 135–148 (1989)
Harmanci, A., Keskin, D., Smith, P.F.: On ⊕-supplemented modules. Acta Math. Hungar. 83(1-2), 161–169 (1999)
Idelhadj. A., Tribak, R.: On some properties of ⊕-supplemented modules. Int. J. Math. Math. Sci. 69, 4373–4387 (2003)
Keskin, D., Xue, W.: Generalizations of lifting modules. Acta Math. Hungar. 91(3), 253–261 (2001)
Keskin, D.: On lifting modules. Comm. Algebra 28(7), 3427–3440 (2000)
Lam, T.Y.: Lectures on Modules and Rings. Springer, Berlin Heidelberg New York (1999)
Mohamed, S.H., Müller, B.J.: Continuous and discrete modules. In: London Math. Soc. LNS 147. Cambridge University Press, Cambridge (1990)
Smith, P.F., Tercan, A.: Generalizations of CS-modules. Comm. Algebra 21(6), 1809–1847 (1993)
Talebi, Y., Vanaja, N.: The Torsion theory cogenerated by \(M\)-small modules. Comm. Algebra 30(3), 1449–1460 (2002)
Wisbauer, R.: Foundations of Module and Ring Theory. Gordon and Breach, Philadelphia (1991)
Wang, Y.: A note on modules with \((D_{11}^+)\). Southeast Asian Bull. Math. 29(5), 999–1002 (2005)
Zhou, D.: On non-M-singular modules with (\(C^+_{11}\)). Acta Math. Hungar. 97(3), 265–271 (2002)
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Tütüncü, D.K. On Non-M-Cosingular Completely ⊕-Supplemented Modules. Appl Categor Struct 16, 249–254 (2008). https://doi.org/10.1007/s10485-006-9025-8
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DOI: https://doi.org/10.1007/s10485-006-9025-8