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 power series in which addition and substitution are used as operations. It is shown that R is Rickart and any countable family of idempotents of R has a join in I(R )i f and only if R0[[x; α]] ∈R r1 if and only if R0[[x; α]] ∈R � 1 if and only if R0[[x; α]] ∈ qRr2. An example to show that, α-rigid condition on R is not superfluous, is provided.