An ideal is a special kind of subset within a ring (see that entry) that "absorbs" multiplication — multiplying any of its elements by anything else in the ring keeps the result inside the ideal. Originally invented by Ernst Kummer in the 1840s to rescue a mysteriously broken pattern of unique factorization, ideals became one of the most important and versatile organising concepts in all of modern algebra.
Every whole number greater than 1 factors into primes in exactly one way (2×3×5 and no other combination gives you 30) — this is the Fundamental Theorem of Arithmetic (see the Prime Number entry), and it feels like it should just always be true in any reasonable number system. In the 1840s, while working on Fermat's Last Theorem (see that entry), Ernst Kummer discovered, to considerable surprise, that this unique factorization pattern mysteriously breaks down in certain natural extensions of the ordinary integers — the same number could genuinely be factored into "primes" in two entirely different, equally valid ways.
Think of an ideal as a kind of "black hole" subset within a larger number system: once you multiply anything into it, from anywhere else in the system, the result gets pulled back inside. The set of all multiples of 3 within the ordinary integers (…,−6,−3,0,3,6,9,…) is a simple, familiar example of an ideal — multiply any multiple of 3 by any integer whatsoever, and you always land back on another multiple of 3.
Ideals let mathematicians precisely capture and generalise the everyday notion of "divisibility" (an ideal generated by a number n is essentially just "the multiples of n") to far more exotic and abstract algebraic settings where the ordinary elements themselves may not behave as simply or as predictably as ordinary whole numbers do.
A subset I of a ring R is an ideal if: (1) I is closed under addition (a+b∈I whenever a,b∈I), and (2) for any element r in the whole ring R and any element a in I, both r·a and a·r are in I — the "absorption" property. This second condition is considerably stronger than merely being a subring, precisely capturing the multiplication-absorbing behaviour central to the concept.
The simplest kind of ideal, called a principal ideal, consists of all multiples of a single chosen element a, written (a) = {ra : r∈R}. In the integers, every single ideal turns out to be principal — generated by one specific number — a special, comparatively simple property that fails in many other, more exotic rings, which is precisely where the theory becomes genuinely interesting and useful.
A prime ideal generalises the notion of a prime number: an ideal P (not equal to the whole ring) such that whenever a product ab lies in P, at least one of a or b must also lie in P. A maximal ideal is an ideal that is not properly contained within any other ideal besides the whole ring itself — maximal ideals are always prime, though the reverse doesn't always hold.
Given an ideal I within a ring R, you can construct an entirely new ring, called the quotient ring R/I, by treating every element of I as if it were equal to zero (formally, by grouping elements of R into equivalence classes that differ from one another only by something in I). This construction — directly analogous to how quotient groups are built from normal subgroups (see the Homomorphism & Isomorphism entry) — is one of the single most useful and frequently applied techniques throughout modern abstract algebra, letting mathematicians systematically "simplify" a complicated ring by deliberately collapsing a chosen ideal down to zero.
Dedekind's fully developed theory shows that in an important and widely-studied class of rings called Dedekind domains (which includes the "rings of integers" arising in algebraic number theory, exactly the setting where Kummer's original factorization troubles first appeared), every ideal factors uniquely into a product of prime ideals — even in situations where the ordinary elements of the ring themselves stubbornly refuse to factor uniquely. This is the fully rigorous, modern realisation of Kummer's original 1840s insight: unique factorization isn't actually false in these exotic settings, it was simply operating one conceptual level higher (in the ideals) than anyone had originally been looking.
Kummer's ideal numbers were developed specifically in a (ultimately incomplete, but highly influential) attempt to prove Fermat's Last Theorem (see that entry) for a broad class of prime exponents called "regular primes." Though Kummer's approach fell short of a complete general proof (Andrew Wiles's eventual full 1995 proof used an entirely different, considerably more sophisticated set of techniques from elliptic curves and modular forms), Kummer's work on ideal numbers remains foundational to all of modern algebraic number theory in its own right, entirely independent of its original, narrower motivating purpose.
A ring is called Noetherian (after Emmy Noether — see the Noether's Theorem entry — who identified and systematically studied the crucial importance of this finiteness property in the 1920s) if every ideal within it can be generated by only finitely many elements, or equivalently, if every strictly increasing chain of ideals must eventually stop growing. This comparatively simple-sounding finiteness condition turns out to be an enormously important and useful property, and most rings arising naturally throughout ordinary algebra, number theory, and algebraic geometry (including the integers and any polynomial ring built over them or over a field) are indeed Noetherian.
In algebraic geometry, there is a deep and precise correspondence (formalised by Hilbert's Nullstellensatz, 1893) between geometric objects (algebraic varieties — the solution sets of polynomial equations, see the Conic Sections entry for elementary examples) and specific ideals within the corresponding polynomial ring. This correspondence is the essential conceptual bridge connecting the entirely algebraic theory of ideals to genuinely geometric objects and geometric intuition, and underlies virtually the whole of modern algebraic geometry as a discipline — including the framework in which the still-unsolved Hodge Conjecture (see the Clay Millennium Problems entry) is properly formulated.