Rickart-type Annihilator Conditions on Formal Power Series

Let R be an α-rigid ring and R0[[x; α]] be the nearring of a formal skew powerseries in which addition and substitution are used as operations. It is shown that Ris Rickart and any countable family of idempotents of R has a join in I(R) if andonly if R0[[x;α]] ∈ Rr1 if and only if R0[[x;α]]∈ Rl1 if and only if R0[[x;α]] ∈ qRr2.An example to show that, α-rigid condition on R is not superfluous, is provided.