In this paper

the notion of strongly nilpotent Γ-rings is generalized by introducing the class of almost strongly nilpotent Γ-rings

that is

of Γ-rings every proper Γ-homomorphie image of which

is strongly nilpotent Γ-ring

the simple Γ-ring are assumed not to be almost strongly nilpotent

and following results are proved:(Ⅰ) the prime radical class is the smallest one of all partitions containing nonzero almost strongly nilpotent Γ-rings of the class of almost strongly nilpotent Γ-rings.(Ⅱ)the lower radical property determined by the class of all almost strongly nilpotent Γ-rings Coineids with the upper radical property determined by the class of all Γ-rings without nonzero almost strongly nilpotent ideals.