Survey on generalizations of Hopficity of modules
Abstract
The main aim of this paper is the Hopficity of module classes, the study of modules (rings) by properties of their endomorphisms is a classical research subject. In 1986, Hiremath Hi introduced the concepts of Hopfian modules and rings, the notion of Hopfian modules are defined as a generalization of modules of finite length as the modules whose surjective endomorphisms are isomorphisms. Later, the dual concepts co-Hopfian modules and rings were given. Hopfian and co-Hopfian modules (rings) have been investigated by several authors. For example, Hiremath Hi, Varadarajan Va, Va1, Xue Xu, Haghany Hag, Liu Li, and Yang and Liu Yl. In 2001, Haghany and Vedadi Ha, and in 2002, Ghorbani and Haghany Gh, respectively, introduced and investigated the weakly co-Hopfian and generalized Hopfian modules. These modules and several generalizations of them are extensively studied also by several authors.
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