Kernel and Image for Rings

Abstract Algebra · Axiom Academy

LESSON Kernel and Image for Rings Understanding how ring homomorphisms preserve structure through kernels and images, leading to the First Isomorphism Theorem for rings. 1. Kernel and Image Definitions Let φ: R → S be a ring homomorphism. We define two crucial sets: The kernel consists of all elements that map to the zero element in S. The image is the set of all outputs of φ. The kernel isn't just a subring—it has the stronger property of being an ideal. This means it "absorbs" multiplication from both sides by any element of R. Proof of absorption: If k ∈ Ker(φ), then φ(k) = 0. For any r ∈ R: φ(r · k) = φ(r) · φ(k) = φ(r) · 0 = 0, so r · k ∈ Ker(φ) φ(k · r) = φ(k) · φ(r) = 0 · φ(r) = 0, so k · r ∈ Ker(φ) The image Im(φ) forms a subring of S. It's closed under addition, multiplication, contains the multiplicative identity (if φ preserves 1), and contains additive inverses. Closure under +: If φ(r₁), φ(r₂) ∈ Im(φ), then φ(r₁) + φ(r₂) = φ(r₁ + r₂) ∈ Im(φ) Closure under ·: φ(r₁) · φ(r₂) = φ(r₁ · r₂) ∈ Im(φ) Contains 0: φ(0 R ) = 0 S ∈ Im(φ) Additive inverses: -φ(r) = φ(-r) ∈ Im(φ) 4. First Isomorphism Theorem for Rings The First Isomorphism Theorem establishes a fundamental connection between the quotient ring R/Ker(φ) and the image Im(φ). Why this works: Elements in the same coset r + Ker(φ) all map to the same element φ(r) under φ. The quotient ring identifies these equivalent elements, creating a perfect copy of the image.

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