Ring Homomorphisms

Introduction

Ring homomorphisms are structure-preserving maps between rings, analogous to group homomorphisms in group theory. They are fundamental to understanding the relationships between different rings and provide the foundation for isomorphism theorems.

Definition

A ring homomorphism is a map $\varphi :R\to S$ between two rings that preserves both addition and multiplication:

• $\varphi (a+b)=\varphi (a)+\varphi (b)$
• $\varphi (ab)=\varphi (a)\varphi (b)$
• $\varphi (1_R)=1_S$ (if the rings have identities)

Properties

Kernel and Image

For a ring homomorphism $\varphi :R\to S$:

• Kernel: $\ker (\varphi )=\{r\in R\mid \varphi (r)=0_S\}$ is an ideal of $R$
• Image: $\text{Im}(\varphi )=\{\varphi (r)\mid r\in R\}$ is a subring of $S$

Injectivity and Surjectivity

• $\varphi$ is injective if and only if $\ker (\varphi )=\{0\}$
• $\varphi$ is surjective if and only if $\text{Im}(\varphi )=S$
• $\varphi$ is bijective if it is both injective and surjective

Composition

The composition of two ring homomorphisms is again a ring homomorphism.

Examples

Example 1: Evaluation Homomorphism

Consider the homomorphism $\varphi :\mathbb{Z}[x]\to \mathbb{Z}$ defined by $\varphi (f(x))=f(0)$. Then:

• $\ker (\varphi )=(x)$ (the ideal generated by $x$)
• $\text{Im}(\varphi )=\mathbb{Z}$

Example 2: Reduction Modulo n

The map $\varphi :\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ defined by $\varphi (a)=a+n\mathbb{Z}$ is a ring homomorphism with kernel $n\mathbb{Z}$.

Example 3: Complex Conjugation

The map $\varphi :\mathbb{C}\to \mathbb{C}$ defined by $\varphi (a+bi)=a-bi$ is a ring homomorphism.

Example 4: Matrix Trace

The map $\text{tr}:M_n(\mathbb{R})\to \mathbb{R}$ defined by $\text{tr}(A)=\sum _{i=1}^n{a}_{ii}$ is a ring homomorphism.

First Isomorphism Theorem for Rings

Theorem: Let $\varphi :R\to S$ be a ring homomorphism. Then:

Proof Sketch

The proof involves:

1. Defining a map $\psi :R/\ker (\varphi )\to \text{Im}(\varphi )$ by $\psi (a+\ker (\varphi ))=\varphi (a)$
2. Showing that $\psi$ is well-defined
3. Proving that $\psi$ is a ring homomorphism
4. Establishing that $\psi$ is bijective

Applications

Application 1: Quotient Ring Construction

The First Isomorphism Theorem is fundamental for understanding quotient rings and their properties.

Application 2: Ring Classification

Ring homomorphisms help classify rings by establishing relationships between different ring structures.

Application 3: Field Extensions

Ring homomorphisms are crucial in field theory for understanding field extensions and embeddings.

Application 4: Algebraic Geometry

Ring homomorphisms correspond to geometric maps, establishing the connection between algebra and geometry.

Special Types of Homomorphisms

Isomorphisms

A ring isomorphism is a bijective ring homomorphism. Two rings are isomorphic if there exists an isomorphism between them.

Endomorphisms and Automorphisms

• A ring endomorphism is a homomorphism from a ring to itself
• A ring automorphism is an isomorphism from a ring to itself

Embeddings

A ring embedding is an injective ring homomorphism, which allows us to view one ring as a subring of another.

Related Concepts

• Rings
• Ideals
• Quotient Rings
• First Isomorphism Theorem
• Isomorphisms