Localization of Hopfian and Cohopfian Objects in the Categories of A − Mod, AGr(A − Mod) and COMP(AGr(A − Mod))

Authors

  • Seydina Ababacar Balde Université Gaston BERGER de Saint-Louis du SENEGAL, UFR-SAT
  • Mohamed Ben Faraj Ben Maaouia Université Gaston BERGER de Saint-Louis du SENEGAL, UFR-SAT
  • Ahmed Ould Chbih Université Gaston BERGER de Saint-Louis du SENEGAL, UFR-SAT

DOI:

https://doi.org/10.29020/nybg.ejpam.v14i2.3889

Keywords:

graded ring, a saturated multiplicative part formed by the non-zero homogeneous elements of A, Ore conditions, hopfian, cohopfian, sequence complex, chain complex, quasi-injective and quasi-prjective.

Abstract

The aim of this paper is to study the localization of hopfian and cohopfian objects in the categories A-Mod of left A-modules, AGr(A-Mod) of graded left A-modules and COMP(AGr(A-Mod)) of complex  sequences associated to graded left A-modules.

We have among others  the main following results :

1. Let M be a noetherian graded left A-module, S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions, N a submodule of M, M_{*} is a noetherian quasi-injective complex sequence associated with M and N_{*} is an essential and completely invariant complex sub\--sequence of M_{*}. Then, S^{-1}(N_{*}) the complex sequence of morphisms of left S^{-1}A\--modules is cohopfian if, and only, if S^{-1}(M_{*}) is cohopfian ;


2. let M be a graded left A\--module and S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions. If M_{*} is a hopfian, noetherian and quasi-injective complex sequence associated with M, then the complex sequence of morphisms of left S^{-1}(A)-modules S^{-1}(M_{*}) has the following property :
{any epimorphism of sub-complex S^{-1}(N_{*}) of S^{-1}(M_{*}) is an isomorphism } ;

3. let M be a graded left A-module, N a graded submodule of M, S a saturated multiplicative part formed by the non-zero homogeneous elements of A verifying the left Ore conditions. M_{*} the quasi-projective complex sequence associated with M and $N_{*}$ a superfluous and completely invariant complex sub\--sequence of $M_{*}$. Then the complex morphism sequence of left $S^{-1}(A)$\--modules $S^{-1}(N_{*})$ is hopfian if, and only if, $S^{-1}(M_{*}/N_{*})$ the complex sequence associated with S^{-1}(M/N) is hopfian.

 

Author Biographies

Seydina Ababacar Balde, Université Gaston BERGER de Saint-Louis du SENEGAL, UFR-SAT

Mathématiques Appliquées

Mohamed Ben Faraj Ben Maaouia, Université Gaston BERGER de Saint-Louis du SENEGAL, UFR-SAT

Mathématiques Appliquées

Ahmed Ould Chbih, Université Gaston BERGER de Saint-Louis du SENEGAL, UFR-SAT

Mathematiques Appliquées

Downloads

Published

2021-05-18

How to Cite

Balde, S. A., Ben Maaouia, M. B. F., & Chbih, A. O. (2021). Localization of Hopfian and Cohopfian Objects in the Categories of A − Mod, AGr(A − Mod) and COMP(AGr(A − Mod)). European Journal of Pure and Applied Mathematics, 14(2), 404–422. https://doi.org/10.29020/nybg.ejpam.v14i2.3889

Issue

Section

Nonlinear Analysis