1. The purpose of the present paper, which in some sense is a continuation of the author's paper [10], is to study the upper radical property W, determined by the class Er of all rings without admissible right ideals which are not right ideals. At first, it is proved that Wr is a nonhereditary almost subidempotent radical property and that Y" < Wr holds, where Y" is the upper radical property determined by the class E of all rings without admissible ideals which are not ideals (G. Tzintzis [10]). Next, it is proved that Wr coincides with the upper radical determined by the class of all division rings and of all zerorings, which means that Y', coincides with the class Ie of all idempotent ~-radical rings where ~ is the upper radical determined by all division rings (G. Tzintzis [9]). From this last result it is evident that Y'r is independent of the choice of right or left ideals. However, by examples it is shown that the classes E,, Et (the left analogue of E~) and E are mutually incomparable. Finally, E~-rings are examined.