Prime O-ideals of distributive lattices

In this paper we derive some sufficient conditions for a prime ideal of a distributive lattice to become an O-ideal. A set of equivalent conditions are established for every O-ideal to become an annihilator ideal. An equivalency is obtained between prime O-ideals and minimal prime ideals of a distributive lattice. Finally, the concept of co-maximality is introduced and obtained that any two distinct prime O-ideals of a distributive lattice are co-maximal.