TAILIEUCHUNG - Báo cáo "Some results on countably 1 Σ− uniform - extending modules "

A module M is called a uniform extending if every uniform submodule of M is essential in a direct summand of M . A module M is called a countably Σ− uniform extending if M (N) is uniform extending. In this paper, we discuss the question of when a countably Σ− uniform extending module is Σ− quasi - injective? We also characterize QF rings by the class of countably Σ− uniform extending modules. | VNU Journal of Science Mathematics - Physics 25 2009 9-14 Some results on countably S-uniform - extending modules Le Van An1 Ngo Sy Tung2 1 Highschool of Phan Boi Chau Vinh city Nghe An Vietnam 2Vnh University Vinh city Nghe An Vietnam Received 10 July 2008 Abstract A module M is called a uniform extending if every uniform submodule of M is essential in a direct summand of M .A module M is called a countably S- uniform extending if M N is uniform extending. In this paper we discuss the question of when a countably S-uniform extending module is S- quasi - injective We also characterize QF rings by the class of countably S- uniform extending modules. 1. Introduction Throughout this note all rings are associative with identity and all modules are unital right modules. The Jacobson radical and the injective hull of M are denoted by J M and E M . If the composition length of a module M is finite then we denote its length by l M . For a module M consider the following conditions C1 Every submodule of M is essential in a direct summand of M. C2 Every submodule isomorphic to a direct summand of M is itself a direct summand. C3 If A and B are direct summands of M with A n B 0 then A B is a direct summand of M. Call a module M a CS - module or an extending module if it satisfies the condition C1 a continuous module if it satisfies C1 and C2 and a quasi-continuous if it satisfies C1 and C3 . We now consider a weaker form of CS - modules. A module M is called a uniform extending if every uniform submodule of M is essential in a direct summand of M . We have the following implications Injective quasi - injective continuous quasi - continuous CS uniform extending. C2 C3 We refer to 1 and 2 for background on CS and quasi- continuous modules. A module M is called a countably S-uniform extending CS quasi - injective injective module if M A respectively M N is uniform extending CS quasi - injective injective for any set A. Note that N denotes the set of all natural numbers. In this

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