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Một M mô-đun được gọi là đóng cửa yếu, bổ sung nếu vì bất kỳ submodule đóng N của M, K submodule M M = K + N và K ∩ N M. Bất kỳ summand trực tiếp của một mô-đun khép kín, bổ sung yếu cũng đóng cửa yếubổ sung. | Vietnam Journal of Mathematics 34 1 2006 17-30 Viet n a m J o u r n a I of MATHEMATICS VAST 2006 Closed Weak Supplemented Modules Qingyi Zeng 1 Dept. of Math. Zhejiang University Hangzhou 310027 China 2Dept. of Math. Shaoguan University Shaoguan 512005 China Received June 13 2004 Revised June 01 2005 Abstract. A module M is called closed weak supplemented if for any closed submodule of M there is a submodule of M such that M and. Any direct summand of a closed weak supplemented module is also closed weak supplemented. Any finite direct sum of local distributive closed weak supplemented modules is also closed weak supplemented. Any nonsingular homomorphic image of a closed weak supplemented module is closed weak supplemented. R is a closed weak supplemented ring if and only if M n R is also a closed weak supplemented ring for any positive integer . 1. Introduction Throughout this paper unless otherwise stated all rings are associative rings with identity and all modules are unitary right R -modules. A submodule of M is called an essential submodule denoted by e M if for any nonzero submodule of. 0. A closed submodule of M denoted by c M is a submodule which has no proper essential extension in M .If c and c M then c M see 2 . A submodule of M is small in M denoted by M if M implies M . Let and be submodules of M . is called a supplement of in M if it is minimal with respect to M or equivalently M This work was supported by the Natural Science Foundation of Zhejiang Province of China Project No.102028 . 18 Qingyi Zeng and. see 6 . A module is called supplemented if for any submodule of there is a submodule of such that and. see 3 . A module is called weak supplemented if for each submodule of there is a submodule of such that and .A module is called -supplemented if every submodule of has a supplement in which is also a direct summand of see 8 . A module is called extending or a CS module if every submodule is essential in a direct summand of or equivalently every .
