Nil properties for rings which are sums of their additive subgroups

Suppose that a ring R is a direct sum of a finite number of its additive subgroups, and the union of these subgroups is closed under multiplication. We show that if all rings among these subgroups are nilpotent (left T-nilpotent, locally nilpotent or Baer radical), then the whole ring R satisfies the same property.