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On Generalized Hopficity of Pairs of Modules

HAO La-la,ZHOU De-xu(School of Mathematics and Computer Science,Fujian Normal University,Fuzhou 350007,China)  
Using the concepts of δ-small and σ-essential,introduce δ-generalized Hopfian(i.e.,δ-gH) pairs of modules and σ-weak co-Hopfian(i.e.,σ-wcH) pairs of modules,give various characterizations of δ-gH pairs of modules and show that being a δ-gH pair is a Morita invariant.Under Morita duality,prove that δ-gH pairs and σ-wcH pairs are dual each other.
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