For a ring endomorphism ${alpha}$ and an ${alpha}-derivation;{delta}$ of a ring R, we study relation between the set of annihilators in R and the set of annihilators in nearring $R[x;{alpha},{delta}];and;R_0[[x;{alpha}]]$. Also we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of skew polynomials. These results are somewhat surprising since, in contrast to the skew polynomial ring and skew power series case, the nearring of skew polynomials and skew power series have substitution for its "multiplication" operation.