  In the context of his theory of numberings, Ershov showed that Kleene's
recursion theorem holds for any precomplete numbering. We discuss various
generalizations of this result. Among other things, we show that Arslanov's
completeness criterion also holds for every precomplete numbering, and we
discuss the relation with Visser's ADN theorem, as well as the uniformity or
nonuniformity of the various fixed point theorems. Finally, we base numberings
on partial combinatory algebras and prove a generalization of Ershov's theorem
in this context.
