Teaching the doctrine of idealism; as, the ideal theory or philosophy.

The word ideal appears in Mathematics in at least a couple of contexts.

of or relating to the philosophical doctrine of the reality of ideas

a subset of elements in a ring that forms an additive group and has the property that, whenever belongs to and belongs to , then and belong to

a subset X of a poset P that is a directed lower set

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept generalizes in an appropriate way some important properties of integers like "even number" or "multiple of 3".

In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory.