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

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

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.

Example 5: Modular Arithmetic

The set $\mathbb{Z}/n\mathbb{Z}$ of integers modulo $n$ forms a ring under modular addition and multiplication.

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

Division Rings

A division ring (or skew field) is a ring where every non-zero element has a multiplicative inverse, but multiplication need not be commutative.

Examples:

• The quaternions $\mathbb{H}$ form a division ring
• All fields are division rings

Properties

Zero Divisors

A zero divisor in a ring $R$ is a non-zero element $a$ such that there exists a non-zero element $b$ with $ab=0$ or $ba=0$.

Units

A unit in a ring $R$ is an element $a$ that has a multiplicative inverse, i.e., there exists $b\in R$ such that $ab=ba=1$.

Characteristic

The characteristic of a ring $R$ is the smallest positive integer $n$ such that $n\cdot 1=0$, or $0$ if no such integer exists.

Applications

Application 1: Number Theory

Rings provide the algebraic foundation for number theory, particularly through the study of integers and modular arithmetic.

Application 2: Algebraic Geometry

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

Application 3: Linear Algebra

Matrix rings are fundamental to linear algebra and the study of linear transformations.

Related Concepts

• Ideals
• Ring Homomorphisms
• Integral Domains
• Fields
• Quotient Rings