nLab improper ideal

Idea

An ideal is improper if it is the whole thing … whatever thing the ideals are ideals in.

Definitions

As there are different (but related) kinds of ideal, there are different (but very similar) kinds of improper ideal.

Definition (improper ideal of a ring or other rig)

If $R$ is a ring or even a rig, then $R$ is a two-sided ideal of itself, the improper ideal.

Definition (improper ideal of a lattice or other proset)

If $L$ is a lattice or even a proset, then $L$ is an ideal of itself, the improper ideal.

Properties

The improper ideal contrasts with proper ideals (all of the other ideals).

The improper ideal does not count as a prime ideal or a maximal ideal, because it is too simple to be simple, although it may satisfy careless or naïve definitions of those concepts (which must be required to be proper).

Conversely, sometimes one defines ‘ideal’ to exclude the improper ideal, but this makes the set of all ideals less nice to work with.