Notes on Ring Theory and Field Theory

David K. Zhang
Last Modified 2022-03-25

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 $\alg{R; 0, -, +, \cdot}$ consisting of:

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 $\alg{\{0\}; 0, -, +, \cdot}$ consisting of a single element $0$ . The negation, addition, and multiplication operations in the zero rng are defined by $-0 = 0 + 0 = 0 \cdot 0 = 0$ .

Definition: Trivial Rng, Nontrivial Rng

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

Definition: Commutative Rng

A commutative rng is a rng $\alg{R; 0, -, +, \cdot}$ that satisfies the following additional requirement:

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 $\alg{R; 0, 1, -, +, \cdot}$ consisting of:

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 $R$ has a distinguished element $0_R$ . The rng axioms specify what happens when $0_R$ is fed into the operations of addition and multiplication, but they leave unspecified when $0_R$ is produced as an output. This is the topic that we will study in this section.

The equation $x + y = 0_R$ is familiar territory—this simply means that $x$ and $y$ are additive inverses, i.e., $x = -y$ . On the other hand, the equation $xy = 0_R$ is far more interesting. In the familiar rngs of grade-school arithmetic, including $\Z$ , $\Q$ , $\R$ , and $\C$ , the equation $xy = 0$ implies that $x = 0$ or $y = 0$ . However, when we study rngs in general, it turns out to be possible for two nonzero elements to multiply into zero. This situation is completely foreign and allows bizarre things to happen in general rngs that have no analogue in $\Z$ , $\Q$ , $\R$ , or $\C$ . In fact, this phenomenon is so dangerous that mathematicians have assigned it a cautionary name.

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

Let $R$ be a rng.

An element $a \in R$ is a left zero divisor if there exists $x \in R$ such that $ax = 0_R$ .
An element $b \in R$ is a right zero divisor if there exists $x \in R$ such that $xb= 0_R$ .
An element of $R$ is a zero divisor if it is a left zero divisor or a right zero divisor.
An element of $R$ is a two-sided zero divisor if is it both a left zero divisor and a right 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 $0_R$ . (You may have already learned this definition if you have previously studied number theory.) You need to remember that, in the context of rng theory, the phrase “zero divisor” is not synonymous with “divides $0_R$ .”

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 $0_R$ itself. The most well-studied domains happen to be commutative rings, so there is a special term for a commutative domain.

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 $1_R$ . This prompts us to ask when $1_R$ is produced as an output of addition or multiplication. Again, the equation $x + y = 1_R$ is rather uninteresting, since it means nothing more than $y = 1_R - x$ . However, the equation $xy = 1_R$ merits further investigation. In $\Q$ , $\R$ , or $\C$ , this would simply mean that $x$ and $y$ are inverses, i.e., $x = y^{-1}$ ; but in a general ring, it is not guaranteed that every nonzero element has an inverse.

Definition: Invertible, Left-Invertible, Right-Invertible

Let $R$ be a ring, and let $x \in R$ .

$x$ is left-invertible if there exists $\ell \in R$ such that $\ell x = 1_R$ . We call such an element $\ell$ a left inverse of $x$ .
$x$ is right-invertible if there exists $r \in R$ such that $xr = 1_R$ . We call such an element $r$ a right inverse of $x$ .
$x$ is invertible (or two-sided invertible) if there exists $y \in R$ such that $xy = yx = 1_R$ . We call such an element $y$ an inverse (or two-sided inverse) of $x$ , 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., $xy = 0$ does not imply $x = 0$ or $y = 0$ ). In contrast, when we learn that an element of a ring is invertible, we can breathe a sigh of relief and divide with impunity. In fact, invertible elements are so useful that we give them another special name.

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

Let $R$ be a ring. An element $u \in R$ is a unit if there exists an element $v \in R$ such that $uv = vu = 1_R$ . The set of all units in $R$ is denoted by $R^\times$ , and is called the group of units of $R$ .

In other words, $R^\times$ is the set of elements of $R$ that have a (two-sided) inverse. We call $R^\times$ the group of units because it is group under the ring’s multiplication operation. In particular, we always have $1_R \in R^\times$ , and if $u, v \in R^\times$ , then $u^{-1} \in R^\times$ (since its inverse is $u$ ) and $uv \in R^{\times}$ (since its inverse is $v^{-1} u^{-1}$ ). For example, $\Z^\times = \{\pm 1\}$ and $\Q^\times = \Q \setminus \{0\}$ .

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 $0_R$ is never invertible in any nontrivial ring $R$ , but we can ask that every nonzero element be invertible.)

Definition: Division Ring, Skew Field

A division ring (or skew field) is a nontrivial ring in which every nonzero element is invertible.

In other words, a ring $R$ is a division ring if $R^\times = R \setminus \{0_R\}$ . Like domains, the most well-studied division rings happen to be commutative, so there is a special term for a commutative division 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 $R$ be a ring. We say that two elements $a, b \in R$ are associates if there exist $u, v \in R^\times$ such that $a = ubv$ .

Note that, in a commutative ring, this condition simplifies to $a = ub$ . In standard algebra textbooks, this definition is usually only stated for commutative rings.

Associatedness is an Equivalence Relation

Theorem: In any ring $R$ , the binary relation of “associatedness” is an equivalence relation on $R$ .

Proof: We need to verify that associatedness is reflexive, symmetric, and transitive. Let $a, b, c \in R$ and $u, v, x, y \in R^\times$ .

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 $R^\times$ is the associate class of $1_R$ , and $\{0_R\}$ is its own associate class.

Definition: Principal Ideal, Generator

Let $R$ be a commutative ring, and let $r \in R$ . The principal ideal generated by $r$ , denoted by $\gen{r}$ , is the subset of $R$ containing all multiples of $r$ .

\gen{r} \coloneqq \{ rx : x \in R \}

An ideal $I \normalin R$ is principal if there exists an element $r \in R$ such that $I = \gen{r}$ . In this case, we call $r$ a generator of $I$ .

Note that a principal ideal may have more than one generator. For example, the ideal of even numbers in $\Z$ has two generators, $\gen{2} = \gen{-2}$ . Of course, these two generators are quite closely related, since $2$ and $-2$ differ only in sign. The following theorem shows that this is a general phenomenon in the absence of zero divisors.

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

Theorem: Let $R$ be an integral domain. If two elements $a, b \in R$ generate the same ideal $\gen{a} = \gen{b}$ , then $a$ and $b$ are associates.

Proof: If $\gen{a} = \gen{b} = \{0\}$ , then $a = b = 0$ , and we are done, since $0$ is certainly associate to itself. Otherwise, $a$ and $b$ are both nonzero. The hypothesis $\gen{a} = \gen{b}$ implies that $a \in \gen{b}$ and $b \in \gen{a}$ . Thus, there exist $r, s \in R$ such that $a = rb$ and $b = sa$ . It follows that $a = rsa$ , or equivalently, $(1 - rs)a = 0$ . Since $R$ is an integral domain and $a \ne 0$ , we must have $1 - rs = 0$ , which shows that $r$ and $s$ are both units in $R$ . We therefore conclude from the defining condition $a = rb$ that $a$ and $b$ are associates. ∎

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

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

Let $R$ be a commutative rng, and let $a, b \in R$ . We say that $a$ divides $b$ , denoted by $a \mid b$ , if there exists an element $q \in R$ such that $b = qa$ . This binary relation ${\mid} \subseteq R \times R$ is called the divisibility relation on $R$ .

The divisibility relation $\mid$ is always reflexive in a ring, but can fail to be reflexive in a rng. Note that $0$ does not divide any element of a rng except $0$ , but every element of a rng divides $0$ .

Divisibility is Transitive

Theorem: The divisibility relation on any rng is transitive.

Proof: Let $R$ be a rng, and let $a, b, c \in R$ . If $a \mid b$ and $b \mid c$ , then there exist $x, y \in R$ sch that $b = xa$ and $c = yb$ . It follows that $c = yxa$ , which proves that $a \mid c$ . ∎

Definition: Prime Element

Let $R$ be a commutative ring. An element $p \in R$ is prime if $p \notin R^\times \cup \{0\}$ and for all $a, b \in R$ , if $p \mid ab$ , then $p \mid a$ or $p \mid b$ .

The prime elements of the commutative ring $\Z$ are precisely the usual prime numbers $2, 3, 5, 7, \dots$ and their negatives.

Definition: Irreducible Element

Let $R$ be a ring. An element $r \in R$ is irreducible if $r \notin R^\times$ and for all $a, b \in R$ , if $r = ab$ , then $a \in R^\times$ or $b \in R^\times$ .

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 $R$ be a ring. The characteristic of $R$ , denoted by $\fchar R$ , is the smallest positive integer $n$ such that

\underbrace{1_R + 1_R + \cdots + 1_R}_{n \text{ copies of } 1_R} = 0_R.

If no such $n$ exists, then we define $\fchar R \coloneqq 0$ .

For example, $\fchar \Z = 0$ and $\fchar(\Z / 4\Z) = 4$ .