Principally quasi-injective modules |
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Authors: | WK Nicholson JK Park MF Yousif |
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Institution: | 1. Departmen of Mathematics , University of Calgary , Calgary, T2N 1N4, Canada E-mail: wknichol@acs.ucalgary.ca;2. Departmen of Mathematics Education , Pusan National University , Pusan, 609-735, South Korea E-mail: jkpark@hyowon.cc. pusan. ac.kr;3. Departmen of Mathematics , Ohio State University , Lima, Ohio, 45804, USA E-mail: yousif.l@osu.edu |
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Abstract: | An R-module M is called principally quasi-injective if each R-hornomorphism from a principal submodule of M to M can be extended to an endomorphism of M. Many properties of principally injective rings and quasi-injective modules are extended to these modules. As one application, we show that, for a finite-dimensional quasi-injective module M in which every maximal uniform submodule is fully invariant, there is a bijection between the set of indecomposable summands of M and the maximal left ideals of the endomorphism ring of M Throughout this paper all rings R are associative with unity, and all modules are unital. We denote the Jacobson radical, the socle and the singular submodule of a module M by J(M), soc(M) and Z(M), respectively, and we write J(M) = J. The notation N ?ess M means that N is an essential submodule of M. |
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