Category of $n$-FCP-gr-projective modules with respect to special copresented graded modules

Abstract

Let R be a ring graded by a group G and n ≥ 1 an integer. We introduce the notion of n-FCP-gr-projective R-modules and by using of special finitely copresented graded modules, we investigate that (1) there exist some equivalent characterizations of n-FCP-gr-projective modules and graded right modules of n-FCP-gr-projective dimension at most k over n-gr-cocoherent rings, (2) R is right n-gr-cocoherent if and only if for every short exact sequence 0 → A → B → C → 0 of graded right R-modules, where B and C are n-FCP-gr-projective, it follows that A is n-FCP-gr-projective if and only if (gr-F CPn, gr-F CP ⊥ n ) is a hereditary cotorsion theory, where gr-F CPn denotes the class of n-FCP-gr-projective right modules. Then we examine some of the conditions equivalent to that each right R-module is n-FCP-gr-projective.

Graded Lie-Rinehart Algebras

“…On the other hand, gradings appear elsewhere in the theory of Lie algebras, for example in the Cartan decomposition of a finite-dimensional complex semisimple Lie algebra (see for instance [1,6,10,15,16,19,23]). Also, graded modules have attracted the attention of many researchers in the last years (see [2,4,5,8,28,31]). The concept of graded Lie-Rinehart algebra will allow us to study, under a unique structure, a graded Lie algebra which is also a module over a graded associative algebra.…”