Ideals in Rings

Why ideals matter

Ideals are how rings produce new rings. Every quotient construction, every "mod something" identification, every kernel-image decomposition of a homomorphism, runs through an ideal. The reason ring theory feels so much like number theory is that Dedekind invented ideals specifically to fix the failure of unique factorization for algebraic integers — and the abstraction turned out to drive most of 20th-century algebra.

• Quotient construction. Want a ring where x² + 1 = 0? Take ℝ[x]/(x² + 1) — the quotient by the ideal generated by x² + 1. The result is ℂ. Want a ring where 12 = 0? Take ℤ/12ℤ. Ideals are the universal "set this to zero" tool.
• Algebraic geometry. Affine varieties V ⊆ kⁿ correspond to ideals I(V) ⊆ k[x₁, ..., xₙ]. Points are maximal ideals (Hilbert Nullstellensatz). Irreducible subvarieties are prime ideals. Schemes (Grothendieck) put a topological space on the prime spectrum Spec(R) of any commutative ring.
• Number theory. Failure of unique factorization in ℤ[√−5] — 6 = 2·3 = (1+√−5)(1−√−5) — is repaired by ideal factorization: (6) = (2, 1+√−5)(2, 1−√−5)(3, 1+√−5)(3, 1−√−5). Dedekind's class group measures how far a ring of integers is from a UFD.
• Coding theory. Cyclic codes are ideals in 𝔽_q[x]/(xⁿ − 1). Generator polynomials of BCH and Reed-Solomon codes are essentially generators of these ideals.
• Cryptography. Lattice-based cryptosystems (NTRU, Ring-LWE) work in ℤ[x]/(xⁿ + 1). Hardness assumptions translate into questions about short vectors in ideal lattices.
• Functional analysis. Ideals in C(X) are exactly closures of zero sets of continuous functions. Maximal ideals correspond to points of X (Gelfand-Naimark for commutative C*-algebras).

Building ideals — generators and operations

For any subset S ⊆ R of a (commutative) ring, the ideal generated by S is

(S) = { r₁s₁ + r₂s₂ + ... + r_n s_n : n ≥ 0, r_i ∈ R, s_i ∈ S }

— the set of finite R-linear combinations. If S = {a}, write (a) for the principal ideal. Standard operations on ideals — sum I + J = {i + j : i ∈ I, j ∈ J}, product IJ = (finite sums of products ij), intersection I ∩ J, and quotient (I : J) = {r ∈ R : rJ ⊆ I}. These satisfy useful identities — e.g. in ℤ, (a) + (b) = (gcd(a, b)), (a) ∩ (b) = (lcm(a, b)), (a)(b) = (ab).

Special classes of ideals

Class	Definition	Quotient R/I	Example in ℤ
Principal	I = (a)	varies	nℤ for any n
Prime	ab ∈ I ⇒ a ∈ I or b ∈ I	Integral domain	(0), (p) for p prime
Maximal	No ideal strictly between I and R	Field	(p) for p prime
Radical	aⁿ ∈ I ⇒ a ∈ I	Reduced (no nilpotents)	(0), (n) for squarefree n
Primary	ab ∈ I, a ∉ I ⇒ bⁿ ∈ I some n	Has unique min prime	(p^k) for p prime
Nilpotent	Iⁿ = 0 for some n	—	(0); none others

The chain of containments — every maximal ideal is prime, every prime is radical, but principal is independent of these. In a PID (principal ideal domain) like ℤ or k[x], every ideal is principal, and the nonzero prime ideals are exactly the maximal ideals.

Quotient rings — the universal "modding out"

For a two-sided ideal I ⊆ R, the quotient R/I has cosets a + I as elements with

• (a + I) + (b + I) = (a + b) + I
• (a + I)(b + I) = (ab) + I
• The natural map π : R → R/I, a ↦ a + I, is a surjective ring homomorphism with kernel I.

Universal property — every ring homomorphism φ : R → S that vanishes on I (φ(I) = 0) factors uniquely through π. This is what "modding out by I" means categorically.

The first isomorphism theorem says R/ker(φ) ≅ Im(φ) for any ring homomorphism φ : R → S. The correspondence theorem says ideals of R/I are in bijection with ideals of R containing I — so questions about R/I lift to questions about R.

Hilbert's basis theorem and Noetherian rings

A ring R is Noetherian if every ascending chain I₁ ⊆ I₂ ⊆ I₃ ⊆ ... of ideals eventually stabilizes — equivalently, every ideal is finitely generated. Fields are trivially Noetherian; ℤ, k[x], and rings of integers in number fields are Noetherian.

Hilbert's basis theorem (1888) — if R is Noetherian, so is R[x]. By induction, R[x₁, ..., xₙ] is Noetherian for any Noetherian R. This is what makes algebraic geometry work — every variety is cut out by finitely many polynomial equations.

Most rings encountered in geometry, number theory, and combinatorics are Noetherian. Notable non-Noetherian examples — k[x₁, x₂, x₃, ...] (countably many indeterminates) and the ring of all algebraic integers ℤ̄.

Common misconceptions

• "Every subring is an ideal." No — ideals must absorb multiplication by R, and subrings need not. ℤ ⊆ ℚ is a subring but not an ideal. Fundamentally, ideals contain "almost no" units (only 0 if I is proper), while subrings often share units with R.
• "Principal = maximal." Different concepts. (2) ⊆ ℤ is principal and maximal; (4) ⊆ ℤ is principal but not maximal (contained in (2)); (x, y) ⊆ k[x, y] is maximal but not principal.
• "Noetherian rings are commutative." Not necessarily — Noetherian-ness is defined for any ring with the ascending chain condition. M_n(k), the Weyl algebra, and group algebras of finite groups are noncommutative Noetherian rings.
• "Prime ideals are generated by prime elements." True in PIDs and UFDs (a principal ideal (p) is prime iff p is a prime element), but in general prime ideals need not be principal — (x, y) ⊆ k[x, y] is a maximal hence prime ideal that requires two generators.
• "R/I always has the same cardinality as R minus I." No — R/I has cardinality |R|/|I| when R is finite. For infinite R, the cardinality of cosets can be anything from finite (k[x]/(f) is finite-dimensional over k for f of finite degree) to |R|.
• "Every ideal is generated by one element." True only in PIDs (ℤ, k[x], ℤ[i]). In k[x, y], the ideal (x, y) is not principal — no single polynomial f generates both x and y as multiples.