We investigate some properties of supplement submodules. Some relations between lying-above and weak supplement submodules are also studied. Let V be a supplement of a submodule U in M. Then it is possible to define a bijective map between the maximal submodules of V and the maximal submodules of M that contain U. Let M be an R-module, U ≤ M, let V be a weak supplement of U, and let K ≤ V. In this case, K is a weak supplement of U if and only if V lies above K in M. We prove that an R-module M is amply supplemented if and only if every submodule of M lies above a supplement in M. We also prove that M is semisimple if and only if every submodule of M is a supplement in M.
Similar content being viewed by others
References
R. Alizade and A. Pancar, Homoloji Cebire Giriş, Ondokuz Mayıs Üniv., Samsun (1999).
R. Alizade, G. Bilhan, and P. F. Smith, “Modules whose maximal submodules have supplements,” Comm. Algebra, 29, No. 6, 2389–2405 (2001).
F. V. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag (1992).
J. Clark, C. Lomp, N. Vanaja, and R. Wisbauer, Lifting Modules, Birkhüuser, Basel etc. (2006).
J. Hausen and J. A. Johnson, “On supplements in modules,” Comment. Math. Univ. St. Pauli, 31, 29–31 (1982).
T. W. Hungerford, Algebra, Springer, New York (1973).
F. Kasch, Modules and Rings, Academic Press, New York (1982).
D. Keskin, A. Harmancι, and P. F. Smith, “On ⊕-supplemented modules,” Acta Math. Hung., 83, No. 1-2, 161–169 (1999).
D. Keskin, P. F. Smith, and W. Xue, “Rings whose modules are ⊕-supplemented,” J. Algebra, 218, 470–487 (1999).
C. Lomp, “On semilocal modules and rings,” Comm. Algebra, 27, No. 4, 1921–1935 (1999).
C. Lomp, On Dual Goldie Dimension, Ph. D. Thesis, Düsseldorf (1996).
S. H. Mohamed and B. J. Müller, “Continuous and discrete modules,” London Math. Soc., Cambridge Univ. Press, Cambridge, 147 (1990).
C. Nebiyev and A. Pancar, “Strongly ⊕-supplemented modules,” Int. J. Comput. Cognit., 2, No. 3, 57–61 (2004).
R. Wisbauer, Foundations of Module and Ring Theory, Gordon & Breach, Philadelphia (1991).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 7, pp. 961–966, July, 2013.
Rights and permissions
About this article
Cite this article
Nebiyev, C., Pancar, A. On Supplement Submodules. Ukr Math J 65, 1071–1078 (2013). https://doi.org/10.1007/s11253-013-0842-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-013-0842-2