Notes on Generalizations of Hopfian and Co-Hopfian Modules

Abderrahim El Moussaouy ⁽¹⁾ and M’hammed Ziane ⁽²⁾

(1) Mathematics Department, Sciences Faculty, Mohammed First University, BP 717, 60000 Oujda, Morocco

Email address: a.elmoussaouy@ump.ac.ma

(2) Mathematics Department, Sciences Faculty, Mohammed First University, BP 717, 60000 Oujda, Morocco
Email address: ziane12001@yahoo.fr

Doi : https://doi.org/10.47013/15.1.4

Cited by : Jordan J. Math & Stat., 15 (1) (2022), 43 - 54

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 Abstract. 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 Hopfian strongly co-Hopfian or semi co-Hopfian strongly Hopfian module, then EndR(M) is strongly π-regular ring. As a consequence we obtain a version of Hopkins-Levitzki Theorem extend to semi Hopfian module and to semi co-Hopfian module. The semi Hopficity and semi co-Hopficity of modules over truncated polynomial rings are considered.

Keywords: Semi Hopfian modules, Semi co-Hopfian modules, Dedekind finite modules.