Jacobson Hopfian modules-Reference-Cited by-同舟云学术

Jacobson Hopfian modules

Published:2022 Issue:1 Volume:33 Page:116-127
ISSN:1726-3255
Container-title:Algebra and Discrete Mathematics
language:
Short-container-title:ADM

Author:

Moussaouy Abderrahim El, ,Moniri Hamzekolaee Ali Reza,Ziane M’Hammed, ,

Abstract

The study of modules by properties of their endomorphisms has long been of interest. In this paper we introduce a proper generalization of that of Hopfian modules, called Jacobson Hopfian modules. A right R-module M is said to be Jacobson Hopfian, if any surjective endomorphism of M has a Jacobson-small kernel. We characterize the rings R for which every finitely generated free R-module is Jacobson Hopfian. We prove that a ring R is semisimple if and only if every R-module is Jacobson Hopfian. Some other properties and characterizations of Jacobson Hopfian modules are also obtained with examples. Further, we prove that the Jacobson Hopfian property is preserved under Morita equivalences.

Publisher

Luhansk Taras Shevchenko National University

Subject

Discrete Mathematics and Combinatorics,Algebra and Number Theory

Reference11 articles.

1. [1]Anderson, F. W., Fuller, K. R.: Rings and Categories of Modules, Grad. Texts inMath. Springer-Verlag, New York. vol.13, (1992).

2. [2]Armendariz, E.P., Fisher, J.W., and Snider, R.L.: On injective and surectiveendomorphisms of őnitely generated modules, Comm. Alg.6(7), 659-672 (1978).

3. [3]Aydogdu, P., Ozcan, A.C.: Semi co-Hopőan and Semi Hopőan Modules, East WestJ. Math.,10(1), 57-72 (2008).

4. [4]Baumslag, G.: Hopőcity and abelian groups, in: J. Irwin, E.A. Walker (Eds.),Topics in Abelian Groups, Scott Foresmann and Company, 331-335 (1963).

5. [5]Ebrahimi Atani, Sh., Khoramdel, M., and Pishhesari, S. D.: Modules in whichevery surjective endomorphism has aδ-small kernel, Algebra Discrete Math.26(2),170-189 (2018).