On Mono-correct Modules
Anta Niane GUEYE
Laboratoire d’Alg `ebre, de Cryptologie, de G´eom´ etrie Alg`ebrique et Applications (LACGAA), D´epartement de Math´ematiques et Informatique, Facult ´e des Sciences et Techniques, Universit ´e Cheikh Anta Diop de DAKAR, SENEGAL.
Cheikh Thi´ecoumba GUEYE *
Laboratoire d’Alg `ebre, de Cryptologie, de G´eom´ etrie Alg`ebrique et Applications (LACGAA), D´epartement de Math´ematiques et Informatique, Facult ´e des Sciences et Techniques, Universit ´e Cheikh Anta Diop de DAKAR, SENEGAL.
*Author to whom correspondence should be addressed.
Abstract
Let R be a commutative ring. It is well known that any artinian module is co-hopfian and any artinian module is mono-correct, but the converse is not true. Furthermore, commutative rings on which co-hopfian modules are artinian have been characterized. The aim of this work is to study the existence of commutative rings R on which mono-correct R-modules are artinian.
We establish that if there exists a commutative ring on which mono-correct R-modules are artinian, then it is an artinian ideal principal one. And on a non-zero commutative artinian principal ideal ring R, we have shown the existence of R-modules which are mono-correct but are not artinian. Hence a non-singleton unital commutative ring R such that every mono-correct R-module is artinian does not exist.
Keywords: Mono-correct module, artinian, co-hopfian, artinian principal ideal ring