Ideal

Ideal is a term with several related meanings in language, philosophy, and mathematics. In everyday use, an ideal is something regarded as perfect or most suitable, or a standard by which others are judged. The word comes from Latin idealis, via French and English, ultimately from Greek idea or eidos, meaning form or image.

In philosophy and ethics, ideals are normative standards or goals that guide thought and conduct. They express aspirational aims—what an individual or society strives to become—often contrasted with actuality. Ideals can be debated or criticized for being unreachable, biased, or culturally specific.

In mathematics, ideal has a precise technical meaning. In a ring R, an ideal I is an additive subgroup that absorbs multiplication by any element of R: for any r in R and i in I, both ri and ir lie in I. In commutative algebra this leads to quotient rings R/I, which formalize “modding out” by the ideal. Prime and maximal ideals play key roles in connecting algebra to geometry and number theory. In lattice and order theory, an ideal of a partially ordered set is a nonempty downward-closed directed subset; a dual notion is a filter.

Common examples include the principal ideal generated by an element or a set S, denoted typically by ⟨S⟩, consisting of all linear combinations with coefficients in the ring. For instance, the set of even integers forms an ideal 2Z in the ring of integers Z.

Beyond mathematics, the term appears in contexts describing models, standards, or goals—ideals in social, political, or personal realms—reflecting the broad sense of striving toward perceived perfection. Related terms include idealism and, in some uses, ideologies that articulate guiding beliefs.