Ring Homomorphisms: Kernels, Images & Isomorphisms

A ring homomorphism is a map φ: R → S preserving both operations: φ(a + b) = φ(a) + φ(b) and φ(ab) = φ(a)φ(b). (Many authors also require φ(1_R) = 1_S.) It is the structure-respecting map of ring theory — the group-homomorphism idea, now guarding two operations.

Ring homomorphisms transport structure: reduction mod n, polynomial evaluation at a point, and embedding ℤ into larger rings are all homomorphisms. Their kernels are ideals — the bridge from maps to quotients.

Key Concepts

• Homomorphism: preserves + and ·; usually sends 1 to 1.
• Kernel: ker φ = {a : φ(a) = 0} — always an ideal of R.
• Image: im φ is a subring of S. Isomorphism: bijective homomorphism.
• Injectivity test: φ is injective iff ker φ = {0}. First isomorphism theorem: R / ker φ ≅ im φ.
• Examples: ℤ → ℤ/nℤ (reduction); R[x] → R, f ↦ f(c) (evaluation); ℤ ↪ ℚ (inclusion).

Formulas

• Defining identities: φ(a + b) = φ(a) + φ(b) and φ(ab) = φ(a)φ(b). (Abstract algebra; no physical units.)
• Consequences: φ(0) = 0, φ(−a) = −φ(a), φ(a^n) = φ(a)^n for n ≥ 1.
• Kernel is an ideal: if φ(a) = 0 then φ(ra) = φ(r)φ(a) = φ(r)·0 = 0, so ra ∈ ker φ — the computation that makes quotients work.
• Limit: requiring φ(1) = 1 rules out some maps — for instance, no unital homomorphism ℤ/6ℤ → ℤ/4ℤ exists, since 6·1 = 0 must map to 6·1 = 2 ≠ 0 in ℤ/4ℤ.

Worked Example

Take φ: ℤ → ℤ/5ℤ, φ(k) = k mod 5. Addition: φ(7 + 8) = φ(15) = 0; φ(7) + φ(8) = 2 + 3 = 0 mod 5. Multiplication: φ(7·8) = φ(56) = 1 (56 = 55 + 1); φ(7)·φ(8) = 2·3 = 1 mod 5. Kernel: 5ℤ, an ideal; image: all of ℤ/5ℤ, so φ is onto. By the first isomorphism theorem, ℤ/5ℤ ≅ ℤ/5ℤ — and the same argument gives ℤ/nℤ ≅ ℤ/nℤ for every n.

Common Misconceptions

• "The kernel of a ring homomorphism is a subring." It is an ideal — strictly stronger. In fact, ideals are exactly the kernels of ring homomorphisms.
• "φ(1) = 1 is automatic." It follows from the two operation rules only in special cases (for example when S is a domain and φ is nonzero); in general it is an extra axiom you must check or assume.
• "An injective homomorphism means the rings are equal." Injective gives R ≅ im φ, an isomorphism onto the image — ℤ injects into ℚ, but ℤ ≠ ℚ.

Real-World Applications

• Modular reduction: ℤ → ℤ/nℤ underlies all of computational number theory and RSA.
• Polynomial evaluation: f ↦ f(c) connects abstract polynomials to concrete values; its kernel is the ideal (x − c).
• Complex numbers: ℝ[x] → ℂ, x ↦ i, has kernel (x² + 1) — exhibiting ℂ as a quotient.
• Algebraic number theory: homomorphisms between rings of integers encode how primes split in extensions.

A ring homomorphism preserves addition and multiplication, its kernel is always an ideal, and the first isomorphism theorem turns every such map into an isomorphism between a quotient and an image.