Ring Homomorphism

A ring homomorphism is a structure-preserving map between rings, the morphism of ring theory whose kernel is an ideal and whose image is a subring, governed by the isomorphism theorems.

Definition

A ring homomorphism is a function between rings that preserves addition, multiplication, and (by convention) the multiplicative identity, so that the algebraic operations are respected.

Scope

This topic covers the definition of ring homomorphisms and isomorphisms, kernels and images, the four isomorphism theorems for rings, the characteristic and prime subring, and the universal properties of quotient rings and evaluation maps.

Core questions

• What does it mean for a map to preserve ring structure?
• How do the kernel and image of a homomorphism relate to ideals and subrings?
• How do the isomorphism theorems factor a homomorphism through a quotient?
• How do evaluation and reduction maps arise as ring homomorphisms?

Key theories

First isomorphism theorem for rings

Every ring homomorphism factors as a surjection onto its image followed by an inclusion, and its image is isomorphic to the quotient of the domain by its kernel, which is an ideal.

Correspondence and isomorphism theorems

Quotienting by an ideal sets up a bijection between the ideals containing it and the ideals of the quotient, and the second, third, and fourth isomorphism theorems describe how subrings, ideals, and quotients interact under homomorphisms.

Universal property of quotients

A homomorphism whose kernel contains a given ideal factors uniquely through the quotient by that ideal, so quotient rings are universal among homomorphic images killing the ideal.

Clinical relevance

Ring homomorphisms formalize the basic operations of algebra: reduction modulo an integer or polynomial, evaluation of polynomials, and inclusion of a ring into a larger one are all homomorphisms. They make rings into a category and are the maps along which structure and computation transfer in number theory and algebraic geometry.

History

The homomorphism and isomorphism theorems were abstracted from group theory to rings as part of Emmy Noether's program of structural algebra in the 1920s, unifying constructions that had previously been treated case by case in number theory and the theory of equations.

Key figures

• Emmy Noether
• Richard Dedekind
• Emil Artin

Seminal works

• dummit2004
• hungerford1974
• lang2002

Frequently asked questions

Why must the kernel of a ring homomorphism be an ideal?

The kernel is closed under addition and, because the map sends products to products and the image of a kernel element is zero, it absorbs multiplication by any ring element. That absorption property is exactly the definition of an ideal.

What is an example of a ring homomorphism in everyday algebra?

Reduction of integers modulo n, evaluation of a polynomial at a fixed number, and complex conjugation are all ring homomorphisms. Each preserves sums and products, and the isomorphism theorems describe their images as quotient rings.