Factorization of graded modules of fractions

Abstract

The main results of this paper are: 1. Lets A = n∈Z A n be a graded domain and S is the multiplicative set satisfying the left conditions of Ore formed of homogeneous elements of A, then we have: The relation S −1 Gr (−) : AG r − M od −→ S −1 Gr A − M od which that for any graded left A−module M we correspond the graded left S −1 Gr (A)-module S −1 Gr (M) and for all graded morphism of graded left A−modules f : M −→ N we correspond the graded morphism S −1 Gr (f) of graded left S −1 Gr (A)-modules is a exact covariant functor that preserves degrees; 2. Lets A = n∈N A n be a positively graded domain and S is the multiplicative set satisfying the left conditions of Ore formed of homogeneous elements of A, M = n∈N M n , N = n∈N N n and L = n∈N L n three positively graded left A−modules then :

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

“…In this article, we study the localization of hopfian and cohopfian objects in the categories A−M od of left A-modules, AGr(A−M od) of graded left A-modules and COM P (AGr(A-− M od)) of complex sequences associated to graded left A-modules. We rely on the articles, Graduation of Module of Fraction on a Graded Domain Ring not Necessarily Commutative [2], Factorization of Graded Modules of Fractions [3], Module de Fractions, Sous-modules S−saturée et Foncteur S −1 [18] and Hopfian and Cohopfian Objects in the Categories of Gr(A − M od) and COM P (Gr(A − M od)) [23], which are used as a basis for studying the notions of localization, hopficity and cohopficity. The transition to localization and the study of hopficity and cohopficity from the category of left A − M od whose objects are the left A-modules and the morphisms are the left A-module morphisms to the category AGr(A − M od) whose objects are the graded left A-module and the morphisms are the graded left A-module graded morphisms and AGr(A − M od) to the category of COM P (AGr(A − M od)) whose the objects are the complex sequences of graded left A-modules and the morphisms are the chain complexes associated to the graded morphims of graded left A-modules is not easy.…”