Localization of Hopfian and Cohopfian Objects in the Categories of A − Mod, AGr(A − Mod) and COMP(AGr(A − Mod))
DOI:
https://doi.org/10.29020/nybg.ejpam.v14i2.3889Keywords:
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.
Â
Downloads
Published
Issue
Section
License
Upon acceptance of an article by the journal, the author(s) accept(s) the transfer of copyright of the article to European Journal of Pure and Applied Mathematics.
European Journal of Pure and Applied Mathematics will be Copyright Holder.