Finiteness conditions and sums of rings

Let R be a ring, which is a sum of its additive subgroups Rs, s ∈ S. Suppose that all rings among the Rs satisfy one of the following classical finiteness properties: right or left Artinian or Noetherian, right or left perfect, semilocal or semiprimary. We provide new conditions on the interaction of the Rs sufficient for the whole ring R to enjoy the same property.