tailieunhanh - Báo cáo " Characterized rings by pseudo - injective modules "

It is shown that: (1) Let R be a simple right Noetherian ring, then the following conditions are equivalent: (i) R is a right SI ring; (ii) Every cyclic singular right R - module is pseudo - injective. (2) Let R be a right artinian ring such that every finite generated right R - module is a direct sum of a projective module and a pseudo - injective module. Then: (i) R/Soc(RR ) is a semisimple artinian ring; (ii) J (R) ⊂ Soc(RR ); (iii) J 2 (R) = 0. (3) Let R be a ring with condition (∗ ),. | VNU Journal of Science Mathematics - Physics 24 2008 67-71 Characterized rings by pseudo - injective modules Le Van An1 Dinh Duc Tai2 1 Highschool of Phan Boi Chau Vinh city Nghe An Vietnam 2HaTinh University Hatinh city Ha tinh Vietnam Received 25 July 2007 received in revised form 25 July 2008 Abstract It is shown that 1 Let R be a simple right Noetherian ring then the following conditions are equivalent i R is a right SI ring ii Every cyclic singular right R - module is pseudo - injective. 2 Let R be a right artinian ring such that every finite generated right R - module is a direct sum of a projective module and a pseudo - injective module. Then i R Soc Rr is a semisimple artinian ring ii J R c Soc Rr iii J2 R 0. 3 Let R be a ring with condition then every singular right R - module is isomorphic with a direct sum of pseudo - injective modules. 1. Introduction Throughout this note all rings are associative with identity and all modules are unital right modules. The socle and the Jacobson radical of M are denoted by Soc M and J M . Given two R - modules M and N N is called M - injective if for every submodule A of M any homomorphism a A N can be extended to a homomorphism f M N. A moduleN is called injective if it is M - injective for every R - module M . On the other hand N is called quasi - injective if N is N -injective. For basic properties of injective modules we refer to 1-4 . We say N is M - pseudo - injective or pseudo - injective relative to M if for every submodule X of M any monomorphism a A N can be extended to a homomorphism f M N. N is called pseudo - injective if N is N - pseudo - injective. We have the following implications Injective quasi - injective pseudo - injective. We refer to 5-8 for background on pseudo - injective modules. Let M be a module. A module Z M is called singular submodule of M if Z M x M xI 0 for some essential right ideal of R . If Z M M then M is called singular module while if Z A 0 Corresponding author. Tel 84-0383569442 .