Kernel functors for which the associated idempotent kernel functor is stable

Let R be a ring with unity and let denote the category of unital right R-modules. A preradical γ of is a functor γ: → such that (i) γ(M) ⊆ M for each R-module M, (ii) for f:M → N, γ(f) is the restriction of f to γ(M).