Rings, Ideals, and Quotient Rings

Introduction

A ring is an algebraic structure that generalizes the arithmetic of the integers. It is a set where one can add, subtract, and multiply, subject to a set of familiar rules.

Definition of a Ring

A ring is a set $R$ equipped with two binary operations, typically called addition $(+)$ and multiplication $(\cdot )$, satisfying the following axioms:

Axioms for Addition

$(R,+)$ is an abelian group:

• Associativity of addition: $(a+b)+c=a+(b+c)$
• Commutativity of addition: $a+b=b+a$
• Additive identity: There exists an element $0\in R$ such that $a+0=a$
• Additive inverse: For each $a\in R$, there exists $-a\in R$ such that $a+(-a)=0$

Axioms for Multiplication

$(R,\cdot )$ is a monoid:

• Associativity of multiplication: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$
• Multiplicative identity: There exists an element $1\in R$ such that $a\cdot 1=1\cdot a=a$

Distributivity

Multiplication is distributive over addition:

• Left distributivity: $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$
• Right distributivity: $(b+c)\cdot a=(b\cdot a)+(c\cdot a)$

Commutative Rings

A ring is commutative if its multiplication is commutative. Throughout this part, unless otherwise specified, rings are assumed to be commutative.

Examples of Rings

Example 1: The Integers

$(\mathbb{Z},+,\cdot )$ is a commutative ring with identity.

Example 2: Polynomial Rings

For any ring $R$, the set $R[x]$ of polynomials with coefficients in $R$ forms a ring under polynomial addition and multiplication.

Example 3: Matrix Rings

The set $M_n(R)$ of $n\times n$ matrices with entries in a ring $R$ forms a ring under matrix addition and multiplication (non-commutative for $n>1$).

Example 4: Function Rings

The set of all functions from a set $X$ to a ring $R$ forms a ring under pointwise addition and multiplication.

Ideals and Quotient Rings

In group theory, normal subgroups are the special subgroups that allow for the construction of quotient groups. In ring theory, the analogous concept is that of an ideal.

Definition of an Ideal

Definition 9.1: A subset $I$ of a ring $R$ is a two-sided ideal if it satisfies:

1. $(I,+)$ is a subgroup of $(R,+)$
2. For every $r\in R$ and every $x\in I$, both $rx$ and $xr$ are in $I$

This second condition is called the absorption property. For commutative rings, left and right ideals are the same.

Properties of Ideals

Just as normal subgroups are the kernels of group homomorphisms, two-sided ideals are precisely the kernels of ring homomorphisms. This property allows for the construction of quotient rings.

Quotient Rings

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.

Key Examples of Quotient Rings

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 $a,b\in \mathbb{R}$.

Ring Homomorphisms

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)

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 $S$

First Isomorphism Theorem for Rings

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

Special Types of Rings

Integral Domains

A ring $R$ is an integral domain if it is commutative, has an identity, and has no zero divisors (i.e., if $ab=0$, then either $a=0$ or $b=0$).

Examples:

• $\mathbb{Z}$ is an integral domain
• $\mathbb{Z}[x]$ is an integral domain
• $\mathbb{Z}/6\mathbb{Z}$ is not an integral domain (since $2\cdot 3=0$)

Fields

A field is a commutative ring with identity where every non-zero element has a multiplicative inverse.

Examples:

• $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$ are fields
• $\mathbb{Z}/p\mathbb{Z}$ is a field when $p$ is prime
• $\mathbb{Z}$ is not a field

Examples and Applications

Example 1: Ideals in $\mathbb{Z}$

The ideals in $\mathbb{Z}$ are precisely the sets $n\mathbb{Z}=\{nk\mid k\in \mathbb{Z}\}$ for some $n\in \mathbb{Z}$. The quotient ring $\mathbb{Z}/n\mathbb{Z}$ is the ring of integers modulo $n$.

Example 2: Ideals in Polynomial Rings

In $\mathbb{R}[x]$, the ideal $(x^2+1)$ consists of all polynomials that are multiples of $x^2+1$. The quotient ring $\mathbb{R}[x]/(x^2+1)$ is isomorphic to $\mathbb{C}$.

Example 3: 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.

Summary

Rings generalize the arithmetic of integers by allowing addition, subtraction, and multiplication. Ideals play the role of normal subgroups in ring theory, allowing the construction of quotient rings. These concepts provide the foundation for field theory and ultimately Galois theory, which connects group theory to the study of polynomial equations.