on certain classes of group derivations

Let γ be a group and x denote a class of (left) γ-modules which contains all projectives. assume that the trivial γ-module z admits an x -precover. in this paper, we introduce and study a class of the derivations from a group γ to a γ-module a that arises from x -precovers of the γ-module z. these derivations, that will be called x - derivations, form a subgroup of the group of derivations der(γ, a). we study this group explicitly when x is the class of gorenstein projective γ-modules, where x -derivations are called gorenstein derivations. corresponding results will be discussed when x is the class of flat γ-modules.