An element of a ring (or rig) or lattice is called prime if the ideal that it generates a prime ideal.
Note that by this definition zero is a prime element of the rig of natural numbers, although it is not a prime number. Prime numbers correspond more closely to maximal ideals than to prime ideals.
Classically (using excluded middle), a prime element of a frame corresponds precisely to a point of the corresponding locale. For a constructive treatment, however, one must use the completely prime filters of the frame instead.