Abstract
An S-closed submodule of a module M is a submodule N for which M/N is nonsingular. A module M is called a generalized CS-module (or briefly, GCS-module) if any S-closed submodule N of M is a direct summand of M. Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right R-modules are projective if and only if all right R-modules are GCS-modules.
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References
G. F. Birkenmeier, B. J. Müller, S. Tariq Rizvi: Modules in which every fully invariant submodule is essential in a direct summand. Commun. Algebra 30 (2002), 1395–1415.
A. W. Chatters, S. M. Khuri: Endomorphism rings of modules over non-singular CS rings. J. Lond. Math. Soc., II. Ser. 21 (1980), 434–444.
C. Faith: Algebra. Vol. II: Ring Theory. Grundlehren der Mathematischen Wissenschaften 191, Springer, Berlin, 1976. (In German.)
K. R. Goodearl: Ring Theory. Nonsingular Rings and Modules. Pure and Applied Mathematics 33, Marcel Dekker, New York, 1976.
S. McAdam: Deep decompositions of modules. Commun. Algebra 26 (1998), 3953–3967.
V. D. Nguyen, V. H. Dinh, P. F. Smith, R. Wisbauer: Extending Modules. Pitman Research Notes in Mathematics Series 313, Longman Scientific & Technical, Harlow, 1994.
R. Wisbauer: Foundations of Module and Ring Theory. Algebra, Logic and Applications 3, Gordon and Breach Science Publishers, Philadelphia, 1991.
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Project (No. 10874122) supported by the Natural Science Foundation of China.
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Zeng, Q. On generalized CS-modules. Czech Math J 65, 891–904 (2015). https://doi.org/10.1007/s10587-015-0215-0
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DOI: https://doi.org/10.1007/s10587-015-0215-0