Ideal

An ideal is a special subset of a ring, closed under addition and absorbent under multiplication, that serves as the kernel of a homomorphism and the object by which one forms quotient rings.

Definition

An ideal of a ring R is an additive subgroup that absorbs multiplication by elements of R; in a commutative ring a subset I is an ideal if it is closed under addition and ri lies in I for every r in R and i in I.

Scope

This topic covers left, right, and two-sided ideals; principal, maximal, and prime ideals; operations on ideals such as sums, products, and intersections; quotient rings and the correspondence theorem; and the characterization of fields and integral domains by their maximal and prime ideals.

Core questions

• How do ideals relate to the kernels of ring homomorphisms?
• What distinguishes prime and maximal ideals, and what do their quotients look like?
• How are new ideals built from old ones by sums, products, and intersections?
• How does the lattice of ideals reflect the structure of the ring?

Key theories

Ideals as kernels

A subset of a ring is the kernel of some ring homomorphism if and only if it is an ideal, and quotienting by an ideal yields the universal homomorphism killing it, mirroring normal subgroups in group theory.

Prime and maximal ideals

In a commutative ring with identity, an ideal is prime exactly when its quotient is an integral domain and maximal exactly when its quotient is a field, so maximal ideals are prime.

Lattice correspondence

The ideals of a quotient ring correspond bijectively to the ideals of the original ring containing the chosen ideal, allowing structural questions to be transferred between a ring and its quotients.

Clinical relevance

Ideals are the central organizing concept of ring theory: prime ideals are the points of algebraic geometry's spectra, ideals encode systems of polynomial equations, and quotient constructions by ideals build new rings such as finite fields and coordinate rings of varieties.

History

The word ideal comes from Kummer's ideal numbers, invented to restore unique factorization in algebraic number theory; Dedekind reformulated them as sets, the modern ideals. Emmy Noether's chain conditions on ideals later made them the backbone of abstract ring theory.

Key figures

• Richard Dedekind
• Ernst Kummer
• Emmy Noether
• David Hilbert

Seminal works

• dummit2004
• atiyah1969
• hungerford1974

Frequently asked questions

Why can you quotient a ring by an ideal but not by an arbitrary subring?

The multiplication on a quotient is well defined only when the subset absorbs multiplication by all ring elements, which is exactly the ideal condition. A subring closed only under the ring operations does not generally give a well-defined quotient ring.

How are prime and maximal ideals different?

An ideal is prime when its quotient is an integral domain and maximal when its quotient is a field. Since every field is an integral domain, maximal ideals are always prime, but not conversely; the gap reflects the dimension of the ring.