About 𝒆 -gH modules

This article introduced and explored the concept of the 𝑒 -gH module and its relation to many


Introduction
Throughout this paper, all modules are unitary left -modules and  is an associative ring with identity.A nonzero submodule  ≤  is said to be essential in  denoted by  ⊴ , if  ∩  ≠ 0 for every nonzero submodule  of  [2].A submodule  of  is called small (-small) denoted by  ≪  (resp. ≪  ) if for every (essential) submodule  of  with the property  =  +  implies  =  [13].A module  ≠ 0 is called uniform if for every submodule  of  with  ≠ 0, then  is essential [2]. is called generalized hollow if any proper submodule of  is -small in  [4].The endomorphisms of modules it has been studied in many authors.V.A. Hiremath introduced the concept of Hopfian module, defined as a module  is called Hopfian if for every surjective -endomorphism of  is an isomorphism [5].In [1] Gorbani and Haghany introduced generalized for Hopfian called generalized Hopfian (gH), a module, is said to be gH if it has a small kernel for every surjective -endomorphism of .K. Varadarajan in 1992 introduced the concept of co-Hopfian module, defined as a module  is called Hopfian if for every injective -endomorphism of  is an isomorphism [12].In [8] introduced a proper generalized for Hopfian called -gH module.A module is said to be -gH if for every surjective -endomorphism of  has an -small kernel.In section 2. We proved some relation between -gH module and some other concepts.We show that every semisimple module is -gH, we give a case that make the concepts e-gH and gH modules are identical.Theorem 2.12 showed the equivalent between Hopfian and -gH.Also introduced a new definition in section 2 called it right -domin ring defined as, a nonzero ring  is called a right -domain if,   () ≪    for any nonzero element  ∈  where   () denote the right annihilator of  in .In the same section showed the relation between Hopfian module, semi-Hopfian (see [10]) and -gH.We also showed the -gH property of Theorem 2.9.Let  be an e-gH module.If :  → ⨁ ́ is an epimorphism for some module  ́, then  ́ is semisimple.
By compare between Proposition 2.5 and Theorem 2.9, we have: Corollary 2.10.Let  be an e-gH module.If :  → ⨁ ́ is an epimorphism for some module  ́, then  ́ is e-gH.
Theorem 2.11.Let  be a ring.Then the following are equivalent.

Proposition 2.13.
Every nonzero e-small quasi-Dedekind module is e-gH.Proof.Let  ≠ 0 be an e-small quasi-Dedekind module, and let  ∈ () be a surjective.Then  ≠ 0. By assumption,  ≪   and hence  is e-gH.
Recall that an -module  is anti-Hopfian if  is non-simple and all nonzero factor modules of  are isomorphic to  Proposition 2.15.Let  be an anti-Hopfian module.Then  is e-gH if and only if  is generalized Hollow.Proof.Assume that  is an e-gH module and  any proper submodule of .If  = 0, then  is e-small in .Let  ≠ 0. We have   ⁄ ≠ , so by assumption   ⁄ ≅ .By [8, Theorem 2.19],  is an e-small submodule of .Therefore  is generalized Hopfian.The converse is proved in Proposition 2.1.
Corollary 2.18.Let  be a projective module.Then the following are equivalent.

( 3 )
if  ∈ () has a right inverse, then  ≪  .(4)  is e-gH.Now, we present the following definition.Definition 2.19.A nonzero ring  is called a right -domain if,   () ≪    for any nonzero element  ∈ .Proposition 2.20.Let  be a nonzero right e-domain ring.Then every nonzero principal right ideal  of  is e-small quasi-Dedekind.

Corollary 2.7. Every simple module is e-gH. Proof: It follows directly by Proposition 2.5. Remark 2.8. The reverse of
[] as an [] module.Finally, we investigate the behavior of e-gH module under localization.