prime element

An element of a ring (or rig) or lattice (or proset) is called **prime** if the ideal that it generates is a prime ideal (thus a principal 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 as such correspond more closely to maximal ideals or irreducible 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.

Last revised on August 29, 2018 at 01:04:43. See the history of this page for a list of all contributions to it.