Hopficity of Modules and Rings (Survey)

  • L’Moufadal Ben YakoubEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 1133)


Under the impulse of an elementary result that characterizes the finite dimensional vector spaces (a linear application is injective if, and only if it is surjective) and partial results which are already put in place on the commutative groups by R.A. Beaumont (1945), P. Hill and C. Megibben (1966) and P. Crawly (1968). Then, for finitely generated modules over commutative rings by J. Strooker (1966), and independently by W.V. Vasconcelos (1969–1970). Finally, towards the end of the sixties, for noetherian and artinian modules by P. Ribenboim.

In the beginning of eighties, A. Kaidi and M. Sangharé introduced the concept of modules satisfying the properties (I), (S) and (F). We say that an A-module M satisfies the property (I) (resp., (S)), if each injective (resp., surjective) endomorphism of M is an automorphism of M, and we say that M satisfies the property (F), if for each endomorphism f of M there exists an integer \(n \ge 1\) such that \(M = Im(f^n) \oplus Ker(f^n)\). In 1986, V. A. Hiremath introduced the concept of Hopfian modules to designate modules satisfying the property (S).

A bit later, K. Varadarajan introduced the notion of co-Hopfian modules to designate modules satisfying the property (I). Hopficity has been studied in many categories as abelian groups, rings, modules and topological spaces. In the context of the hopficity of rings and modules, K. Varadarajan studied the analogue of Hilbert’s basic theorem quite extensively, that is, the transfer of Hopficity to certain polynomial extensions. He also examined various aspects of Hopkins-Levitzki’s theorem related to Hopfian rings, co-Hopfian and its variants. This is a research topic, where different directions are discussed.

The subject is also of interest to several international research teams in the context of other notions related to hopficity of modules and its relationships with other classes of larger modules. In this context, we give a survey of the different notions related to the hopficity of modules, the main results of such notions and its relationships with other classes of larger modules.


Hopfian module Co-hopfian module Hopfian ring Co-hopfian ring 


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Authors and Affiliations

  1. 1.Faculté des SciencesUniversité Abdelmalek EssaâdiTétouanMaroc

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