Survey on generalizations of Hopficity of modules

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 \cite{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 \cite{Hi}, Varadarajan \cite{Va}, \cite{Va1}, Xue \cite{Xu}, Haghany \cite{Hag}, Liu \cite{Li}, and Yang and Liu \cite{Yl}. In 2001, Haghany and Vedadi \cite{Ha}, and in 2002, Ghorbani and Haghany \cite{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.