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Vietnam Journal of Mathematics 35:3(2007) 275-283  Hopfian and Co-Hopfian Modules Over Commutative Rings Kamran Divaani-Aazar and Amir Mafi Abstract.  The structures of Hopfian and co-Hopfian modules over commutative rings are studied. The notation of semi Hopfian (resp. semi co-Hopfian) modules as a generalization of that of Hopfian (resp. co-Hopfian) modules was introduced in [2]. A characterization of semi Hopfian modules by using certain sets of prime ideals is given. Also, it is shown the analogue of Hilbert's Basis Theorem is valid for semi Hopficity, to the effect that an $R$-module $M$ is semi Hopfian if and only if $M[X]$ is a semi Hopfian $R[X]$-module. Moreover, we shall prove the dual of these results for semi co-Hopfian modules.   2000 Mathematics Subject Classification: 13C05, 13E, 16D10. Keywords: Hopfian and co-Hopfian modules, good modules, representable modules, semi Hopfian and semi co-Hopfian modules.

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