ideal

Ideals are like stars; you will not succeed in touching them with your hands. But like the seafaring man on the desert of waters, you choose them as your guides, and following them you will reach your destiny - Carl Schurz

Let ℤ be the ring of integers and let 2ℤ be its ideal of even integers. Then the quotient ring ℤ/2ℤ is a Boolean ring.

An ideal A of L is called complete if it contains all least upper bounds of its subsets that exist in L. Bishop and Schreiner [80] studied conditions under which joins of ideals in the lattices of all ideals and of all complete ideals coincide.

Formally, an ideal I of a given set X is a nonempty subset of the powerset 𝒫(X) such that: (1)∅∈I, (2)A∈I and B⊆A⟹B∈I and (3)A,B∈I⟹A∪B∈I.

If 𝖌 is a Lie algebra, 𝖍 is an ideal and the Lie algebras 𝖍 and 𝖌/𝖍 are solvable, then 𝖌 is solvable.

The set of natural numbers with multiplication as the monoid operation (instead of addition) has multiplicative ideals, such as, for example, the set {1, 3, 9, 27, 81, ...}. If any member of it is multiplied by a number which is not a power of 3 then the result will not be a power of three.