Variants of Hopficity in modules (Q1263651)

Let R be a ring with 1 and let M be a unitary left R-module. M is said to be Hopfian if every epi-endomorphism of M is an automorphism. M is said to be ``strongly Hopfian'' if every non-zero endomorphism of M is a monomorphism. The authors show, in this paper, that a direct sum of Hopfian modules is not necessarily Hopfian. A class of modules whose strong Hopficity is preserved under taking injective hulls, is given. A characterization has been obtained for the super-Hopficity of a quasi- injective module M (i.e. \(Hom_ R(M,M)\) is a division ring).