  Prime objects were defined as generalization of simple objects in the
categories of rings (modules). In this paper we introduce and investigate what
turns out to be a suitable generalization of simple corings (simple comodules),
namely fully coprime corings (fully coprime comodules). Moreover, we consider
several primeness notions in the category of comodules of a given coring and
investigate their relations with the fully coprimeness and the simplicity of
these comodules. These notions are applied then to study primeness and
coprimeness properties of a given coring, considered as an object in its
category of right (left) comodules.
