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Vietnam Journal of Mathematics 35:1(2007) 73-80

 On Hopfian and Co-Hopfian Modules

Yang Gang and Liu Zhong-kui

Abstract.  A R-module M is said to be Hopfian (respectively Co-Hopfian) in case any surjective (respectively injective) R-homomorphism is automatically an isomorphism. In this paper we study sufficient and necessary conditions of Hopfian and Co-Hopfian modules. In particular, we show that the weakly Co-Hopfian regular module _RR is Hopfian, and the left R-module M is Co-Hopfian if and only if the left R[x]/(xⁿ⁺¹) -module M[x]/(xⁿ⁺¹) is Co-Hopfian, where n is a positive integer.

Keywords: Hopfian modules, Co-Hopfian modules, weakly Co-Hopfian modules, generalized Hopfian modules.

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