In this paper, the authors study various annihilator conditions over nearrings of polynomials and formal power series over an abelian ring, which were originally used by Rickart and Kaplansky to abstract the algebraic properties of von Neumann algebra. These results are somewhat surprising since, in contrast to the polynomial ring case, the nearrings of polynomials and formal power series have substitution for its "multiplication" operation. As a consequence we obtain a generalization of [10].