# Notes on Ring Theory and Field Theory

Last Modified 2022-03-15

# Introduction

In these notes, we introduce a new class of algebraic structures, called rngs and rings, whose study is collectively called * ring theory*. Rngs and rings are more complicated than groups because their definition involves not one, but two binary operations.

**Definition: Rng**

A * rng* (pronounced as “rung”) is an algebraic structure

- a set
$R$ , called the;**underlying set** - a distinguished element
$0 \in R$ , called the;**zero element** - a unary operation
$-: R \to R$ , written as$x \mapsto -x$ , called;**negation** - a binary operation
$+: R \times R \to R$ , written as$(x, y) \mapsto x + y$ , called;**addition** - a binary operation
$\cdot: R \times R \to R$ , written as$(x, y) \mapsto x \cdot y$ , called;**multiplication**

satisfying the following requirements:

:**Additive structure**$\alg{R; 0, -, +}$ is an abelian group.:**Associativity**$(x \cdot y) \cdot z = x \cdot (y \cdot z)$ for all$x, y, z \in R$ .:**Left distributivity**$x \cdot (y + z) = (x \cdot y) + (x \cdot z)$ for all$x, y, z \in R$ .:**Right distributivity**$(x + y) \cdot z = (x \cdot z) + (y \cdot z)$ for all$x, y, z \in R$ .

The key ingredient in the definition of a rng is distributivity, which establishes a link between two different binary operations. We begin our study of rngs by proving a simple (but important) result to demonstrate the utility of the distributive property.

**Multiplying by Zero Yields Zero**

**Theorem:** To do.

*Proof:* To do. □

**Definition: Zero Rng**

The * zero rng* is the rng

**Definition: Trivial Rng, Nontrivial Rng**

- A rng is
if its underlying set contains only one element.**trivial** - A rng is
if its underlying set contains more than one element.**nontrivial**

**Definition: Commutative Rng**

A * commutative rng* is a rng

:**Commutativity**$x \cdot y = y \cdot x$ for all$x, y \in R$ .

The word “commutative” in the term “commutative rng” emphasizes that the *multiplication* operation is commutative. By definition, the addition operation is always commutative in every rng.

**Definition: Ring**

A * ring* is an algebraic structure

- a set
$R$ , called the;**underlying set** - a distinguished element
$0 \in R$ , called the;**zero element** - a distinguished element
$1 \in R$ , called the;**identity element** - a unary operation
$-: R \to R$ , written as$x \mapsto -x$ , called;**negation** - a binary operation
$+: R \times R \to R$ , written as$(x, y) \mapsto x + y$ , called;**addition** - a binary operation
$\cdot: R \times R \to R$ , written as$(x, y) \mapsto x \cdot y$ , called;**multiplication**

satisfying the following requirements:

:**Rng structure**$\alg{R; 0, -, +, \cdot}$ is a rng.:**Identity**$1 \cdot x = x \cdot 1 = x$ for all$x \in R$ .

# Zero Divisors and Inverses

Every rng *into* the operations of addition and multiplication, but they leave unspecified when *output*. This is the topic that we will study in this section.

The equation

**Definition: Zero Divisor, Left/Right/Two-Sided Zero Divisor**

Let

- An element
$a \in R$ is aif there exists**left zero divisor**$x \in R$ such that$ax = 0_R$ . - An element
$b \in R$ is aif there exists**right zero divisor**$x \in R$ such that$xb= 0_R$ . - An element of
$R$ is aif it is a left zero divisor or a right zero divisor.**zero divisor** - An element of
$R$ is aif is it both a left zero divisor and a right zero divisor.**two-sided zero divisor**

The name “zero divisor” is a rather unfortunate historical convention. We will later learn a definition of the word “divides” under which every element of a rng divides

Zero divisors are so pernicious that mathematicians have concocted a series of names for rings in which they don’t exist. Note that these names apply to *rings*, not *rngs*.

**Definition: Domain**

A * domain* is a nontrivial ring in which the product of any two nonzero elements is itself nonzero.

In other words, a domain is a ring that does not contain any (left or right) zero divisors except

**Definition: Integral Domain**

An * integral domain* is a nontrivial commutative ring in which the product of any two nonzero elements is itself nonzero.

When we pass from rngs to rings, we gain the new distinguished element

**Definition: Invertible, Left-Invertible, Right-Invertible**

Let

$x$ isif there exists**left-invertible**$\ell \in R$ such that$\ell x = 1_R$ . We call such an element$\ell$ aof**left inverse**$x$ .$x$ isif there exists**right-invertible**$r \in R$ such that$xr = 1_R$ . We call such an element$r$ aof**right inverse**$x$ .$x$ is(or**invertible**) if there exists**two-sided invertible**$y \in R$ such that$xy = yx = 1_R$ . We call such an element$y$ an(or**inverse**) of**two-sided inverse**$x$ , and we write$y = x^{-1}$ to denote this relationship.

Note that the terms “zero divisor” and “invertible” have opposite usage conventions. When we call something a “zero divisor” without further clarification, we mean that it is a left or right zero divisor; but when we call something “invertible” without further clarification, we mean “two-sided invertible.”

In addition, the terms “zero divisor” and “invertible” also have opposite emotional connotations. When we learn that an element of a rng is a zero divisor, we must put ourselves on high alert and remember that the usual laws of arithmetic might not apply (i.e.,

**Definition: Unit, Group of Units, $R^\times$**

Let * unit* if there exists an element

*of*

**group of units**In other words, *group* of units because it is group under the ring’s multiplication operation. In particular, we always have

Just like domains and integral domains, mathematicians have also concocted names for rings in which every element is invertible. (Actually, it’s too much to ask that *every* element be invertible, since *nonzero* element be invertible.)

**Definition: Division Ring, Skew Field**

A * division ring* (or

*) is a nontrivial ring in which every nonzero element is invertible.*

**skew field**In other words, a ring

**Definition: Field**

A * field* is a nontrivial commutative ring in which every nonzero element is invertible.

# Under Construction

**Definition: Associate Elements in a Ring**

Let * associates* if there exist

Note that, in a commutative ring, this condition simplifies to

**Associatedness is an Equivalence Relation**

**Theorem:** In any ring

*Proof:* We need to verify that associatedness is reflexive, symmetric, and transitive. Let

*Reflexivity*:$a = 1_R a 1_R$ .*Symmetry*: If$a = ubv$ , then$b = u^{-1} a v^{-1}$ .*Transitivity*: If$a = ubv$ and$b = xcy$ , then$a = (ux)c(yv)$ . □

This shows that every ring can be partitioned into associate classes. Note that

**Definition: Principal Ideal, Generator**

Let * principal ideal* generated by

An ideal * principal* if there exists an element

*of*

**generator**Note that a principal ideal may have more than one generator. For example, the ideal of even numbers in

**Two Generators of the Same Ideal in an Integral Domain are Associates**

**Theorem:** Let

*Proof:* If

In a commutative ring with zero divisors, it can even be the case that a single principal ideal has multiple generators which are not associate to each other. **[TO DO: Add an example.]**

**Definition: Principal Ideal Domain (PID)**

A * principal ideal domain*, also known as a

*, is an integral domain in which every ideal is principal.*

**PID****Definition: Divides, Divisibility Relation, $a \mid b$**

Let **divides*** divisibility relation* on

The divisibility relation

**Divisibility is Transitive**

**Theorem:** The divisibility relation on any rng is transitive.

*Proof:* Let

**Definition: Prime Element**

Let * prime* if

The prime elements of the commutative ring

**Definition: Irreducible Element**

Let * irreducible* if

Note that, while the notion of an irreducible element is defined in any ring, the notion of a prime element is only defined in a *commutative* ring.

**Definition: Characteristic, $\fchar R$**

Let * characteristic* of

If no such

For example,