In this paper, we introduce the notion of µ-weakly Hopfian modules which is a new
generalization of Hopfian modules. It is shown that over a ring R, every quasi-projective …

The concept of Hopfian modules has been extensively studied in the literature. In this paper,
we introduce and study the notion of δ-weakly Hopfian modules. The class of δ-weakly …

The structures of Hopfian and co-Hopfian modules over commutative rings are studied. The
notation of semi Hopfian (resp. semi co-Hopfian) modules as a generalization of that of …

Let R be a ring and M a left R‐module. M which satisfies DCC on essential submodules is
GCH, and M which satisfies ACC on small submodules is WH. If M [X] is GCH R [X]‐module …

An R-module M is called weakly Hopfian (respectively generalized co-Hopfian) if for every
small epimorphism (respectively essential monomorphism) f of M is isomorphism. If M is …

Ar-module M is said to be Hopfian (respectively Co-Hopfian) in case any surjective
(respectively injective) R-homomorphism is automatically an isomorphism. In this paper we …

The study of modules by properties of their endomorphisms has long been of interest. In this
paper, we introduce the notion of jacobson weakly Hopfian modules. It is shown that over a …

An R-module M is called strongly Hopfian (respectively strongly co-Hopfian) if for every
endomorphism f of M the chain Kerf⊆ Kerf2⊆⋯(respectively Imf⊇ Imf2⊇⋯) stabilizes. The …

A module M is called semi co-Hopfian (resp. semi Hopfian) if any injective (resp. surjective)
endomorphism of M has a direct summand image (resp. kernel). We show that if M is semi …

A module is called generalized Hopfian (gH) if any of its surjective endomorphisms has a
small kernel. Such modules are in a sense dual to weakly co-Hopfian modules that were …