Totally Hopfian Modules

In the present paper, a right R-module M is said to be totally Hopfian if any f ∈ EndR(M) with this property that, for each positive integer n, Imf n is an essential submodule of M, is a monomorphism. The class of totally Hopfian modules lies properly between the class of strongly Hopfian modules and the class of Hopfian modules. We show that for endoregular modules, the concepts of the Hopficity, co-Hopficity, totally Hopficity and strongly Hopficity coincide.