Quotient Rings

Introduction

Quotient rings are fundamental constructions in ring theory, analogous to quotient groups in group theory. They allow us to "factor out" ideals and create new rings with simplified structure.

Definition

Definition 9.2: Let $I$ be a two-sided ideal of a ring $R$. The quotient ring $R/I$ is the set of cosets $\{a+I\mid a\in R\}$ with operations:

• $(a+I)+(b+I)=(a+b)+I$
• $(a+I)(b+I)=(ab)+I$

These operations are well-defined precisely because $I$ is a two-sided ideal.

Properties

Well-Defined Operations

The operations in $R/I$ are well-defined because $I$ is an ideal. This means that if $a+I=a^{\prime }+I$ and $b+I=b^{\prime }+I$, then $(a+b)+I=(a^{\prime }+b^{\prime })+I$ and $(ab)+I=(a^{\prime }{b}^{\prime })+I$.

Ring Structure

The quotient ring $R/I$ inherits the ring structure from $R$:

• It is an abelian group under addition
• It has a multiplicative identity $1+I$
• Multiplication is associative and distributive

Natural Homomorphism

The map $\pi :R\to R/I$ defined by $\pi (a)=a+I$ is a ring homomorphism with kernel $I$.

Key Examples

Example 1: Integers Modulo n

The ring of integers modulo $n$, $\mathbb{Z}/n\mathbb{Z}$, is the quotient of the ring $\mathbb{Z}$ by the ideal $n\mathbb{Z}$ consisting of all multiples of $n$. This construction is the foundation of modular arithmetic.

Elements: $\{0+n\mathbb{Z},1+n\mathbb{Z},\dots ,(n-1)+n\mathbb{Z}\}$

Operations:

• Addition: $(a+n\mathbb{Z})+(b+n\mathbb{Z})=(a+b)+n\mathbb{Z}$
• Multiplication: $(a+n\mathbb{Z})(b+n\mathbb{Z})=(ab)+n\mathbb{Z}$

Example 2: Complex Numbers

The complex numbers can be constructed as the quotient ring $\mathbb{R}[x]/(x^2+1)$, where $(x^2+1)$ is the ideal generated by the polynomial $x^2+1$.

In this construction:

• Elements are cosets of the form $a+bx+(x^2+1)$
• The relation $x^2+1\equiv 0$ (mod $(x^2+1)$) gives $x^2\equiv -1$
• This allows us to identify $x$ with the imaginary unit $i$

Example 3: Polynomial Quotients

Consider the ring $\mathbb{R}[x]/(x^2-2)$. This quotient ring is isomorphic to $\mathbb{R}[\sqrt{2}]$, the ring of real numbers of the form $a+b\sqrt{2}$ where $a,b\in \mathbb{R}$.

Example 4: Field Extensions

The quotient ring $\mathbb{Q}[x]/(x^2-2)$ is isomorphic to $\mathbb{Q}[\sqrt{2}]$, which is a field extension of $\mathbb{Q}$.

First Isomorphism Theorem for Rings

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

Example: Using the First Isomorphism Theorem

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 $x$)
• $\text{Im}(\varphi )=\mathbb{Z}$
• By the First Isomorphism Theorem: $\mathbb{Z}[x]/(x)\cong \mathbb{Z}$

Applications

Application 1: Modular Arithmetic

Quotient rings provide the algebraic foundation for modular arithmetic, which is essential in number theory and cryptography.

Application 2: Algebraic Number Theory

Quotient rings of polynomial rings are used to construct algebraic number fields and study algebraic integers.

Application 3: Algebraic Geometry

Quotient rings correspond to geometric objects, establishing a deep connection between algebra and geometry.

Application 4: Field Theory

Quotient rings are fundamental to field theory and the construction of field extensions.

Properties of Quotient Rings

Zero Divisors

If $R$ is an integral domain and $I$ is a prime ideal, then $R/I$ is an integral domain.

Fields

If $I$ is a maximal ideal, then $R/I$ is a field.

Units

An element $a+I$ is a unit in $R/I$ if and only if there exists $b\in R$ such that $ab-1\in I$.

Related Concepts

• Rings
• Ideals
• Ring Homomorphisms
• First Isomorphism Theorem
• Fields