Notes on Ring Theory and Field Theory

David K. Zhang
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 R;0,,+,\alg{R; 0, -, +, \cdot} consisting of:

  • a set RR, called the underlying set;
  • a distinguished element 0R0 \in R, called the zero element;
  • a unary operation :RR-: R \to R, written as xxx \mapsto -x, called negation;
  • a binary operation +:R×RR+: R \times R \to R, written as (x,y)x+y(x, y) \mapsto x + y, called addition;
  • a binary operation :R×RR\cdot: R \times R \to R, written as (x,y)xy(x, y) \mapsto x \cdot y, called multiplication;

satisfying the following requirements:

  • Additive structure: R;0,,+\alg{R; 0, -, +} is an abelian group.
  • Associativity: (xy)z=x(yz)(x \cdot y) \cdot z = x \cdot (y \cdot z) for all x,y,zRx, y, z \in R.
  • Left distributivity: x(y+z)=(xy)+(xz)x \cdot (y + z) = (x \cdot y) + (x \cdot z) for all x,y,zRx, y, z \in R.
  • Right distributivity: (x+y)z=(xz)+(yz)(x + y) \cdot z = (x \cdot z) + (y \cdot z) for all x,y,zRx, 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 {0};0,,+,\alg{\{0\}; 0, -, +, \cdot} consisting of a single element 00. The negation, addition, and multiplication operations in the zero rng are defined by 0=0+0=00=0-0 = 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 R;0,,+,\alg{R; 0, -, +, \cdot} that satisfies the following additional requirement:

  • Commutativity: xy=yxx \cdot y = y \cdot x for all x,yRx, 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 R;0,1,,+,\alg{R; 0, 1, -, +, \cdot} consisting of:

  • a set RR, called the underlying set;
  • a distinguished element 0R0 \in R, called the zero element;
  • a distinguished element 1R1 \in R, called the identity element;
  • a unary operation :RR-: R \to R, written as xxx \mapsto -x, called negation;
  • a binary operation +:R×RR+: R \times R \to R, written as (x,y)x+y(x, y) \mapsto x + y, called addition;
  • a binary operation :R×RR\cdot: R \times R \to R, written as (x,y)xy(x, y) \mapsto x \cdot y, called multiplication;

satisfying the following requirements:

  • Rng structure: R;0,,+,\alg{R; 0, -, +, \cdot} is a rng.
  • Identity: 1x=x1=x1 \cdot x = x \cdot 1 = x for all xRx \in R.

Zero Divisors and Inverses

Every rng RR has a distinguished element 0R0_R. The rng axioms specify what happens when 0R0_R is fed into the operations of addition and multiplication, but they leave unspecified when 0R0_R is produced as an output. This is the topic that we will study in this section.

The equation x+y=0Rx + y = 0_R is familiar territory—this simply means that xx and yy are additive inverses, i.e., x=yx = -y. On the other hand, the equation xy=0Rxy = 0_R is far more interesting. In the familiar rngs of grade-school arithmetic, including Z\Z, Q\Q, R\R, and C\C, the equation xy=0xy = 0 implies that x=0x = 0 or y=0y = 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\Z, Q\Q, R\R, or C\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 RR be a rng.

  • An element aRa \in R is a left zero divisor if there exists xRx \in R such that ax=0Rax = 0_R.
  • An element bRb \in R is a right zero divisor if there exists xRx \in R such that xb=0Rxb= 0_R.
  • An element of RR is a zero divisor if it is a left zero divisor or a right zero divisor.
  • An element of RR 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 0R0_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 0R0_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 0R0_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 1R1_R. This prompts us to ask when 1R1_R is produced as an output of addition or multiplication. Again, the equation x+y=1Rx + y = 1_R is rather uninteresting, since it means nothing more than y=1Rxy = 1_R - x. However, the equation xy=1Rxy = 1_R merits further investigation. In Q\Q, R\R, or C\C, this would simply mean that xx and yy are inverses, i.e., x=y1x = 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 RR be a ring, and let xRx \in R.

  • xx is left-invertible if there exists R\ell \in R such that x=1R\ell x = 1_R. We call such an element \ell a left inverse of xx.
  • xx is right-invertible if there exists rRr \in R such that xr=1Rxr = 1_R. We call such an element rr a right inverse of xx.
  • xx is invertible (or two-sided invertible) if there exists yRy \in R such that xy=yx=1Rxy = yx = 1_R. We call such an element yy an inverse (or two-sided inverse) of xx, and we write y=x1y = 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=0xy = 0 does not imply x=0x = 0 or y=0y = 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×R^\times

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

In other words, R×R^\times is the set of elements of RR that have a (two-sided) inverse. We call R×R^\times the group of units because it is group under the ring’s multiplication operation. In particular, we always have 1RR×1_R \in R^\times, and if u,vR×u, v \in R^\times, then u1R×u^{-1} \in R^\times (since its inverse is uu) and uvR×uv \in R^{\times} (since its inverse is v1u1v^{-1} u^{-1}). For example, Z×={±1}\Z^\times = \{\pm 1\} and Q×=Q{0}\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 0R0_R is never invertible in any nontrivial ring RR, 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 RR is a division ring if R×=R{0R}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 RR be a ring. We say that two elements a,bRa, b \in R are associates if there exist u,vR×u, v \in R^\times such that a=ubva = ubv.

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

Associatedness is an Equivalence Relation

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


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

  • Reflexivity: a=1Ra1Ra = 1_R a 1_R.
  • Symmetry: If a=ubva = ubv, then b=u1av1b = u^{-1} a v^{-1}.
  • Transitivity: If a=ubva = ubv and b=xcyb = xcy, then a=(ux)c(yv)a = (ux)c(yv).

This shows that every ring can be partitioned into associate classes. Note that R×R^\times is the associate class of 1R1_R, and {0R}\{0_R\} is its own associate class.

Definition: Principal Ideal, Generator

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

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

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

Note that a principal ideal may have more than one generator. For example, the ideal of even numbers in Z\Z has two generators, 2=2\gen{2} = \gen{-2}. Of course, these two generators are quite closely related, since 22 and 2-2 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 RR be an integral domain. If two elements a,bRa, b \in R generate the same ideal a=b\gen{a} = \gen{b}, then aa and bb are associates.


Proof: If a=b={0}\gen{a} = \gen{b} = \{0\}, then a=b=0a = b = 0, and we are done, since 00 is certainly associate to itself. Otherwise, aa and bb are both nonzero. The hypothesis a=b\gen{a} = \gen{b} implies that aba \in \gen{b} and bab \in \gen{a}. Thus, there exist r,sRr, s \in R such that a=rba = rb and b=sab = sa. It follows that a=rsaa = rsa, or equivalently, (1rs)a=0(1 - rs)a = 0. Since RR is an integral domain and a0a \ne 0, we must have 1rs=01 - rs = 0, which shows that rr and ss are both units in RR. We therefore conclude from the defining condition a=rba = rb that aa and bb 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, aba \mid b

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

The divisibility relation \mid is always reflexive in a ring, but can fail to be reflexive in a rng. Note that 00 does not divide any element of a rng except 00, but every element of a rng divides 00.

Divisibility is Transitive

Theorem: The divisibility relation on any rng is transitive.


Let RR be a rng, and let a,b,cRa, b, c \in R. If aba \mid b and bcb \mid c, then there exist x,yRx, y \in R sch that b=xab = xa and c=ybc = yb. It follows that c=yxac = yxa, which proves that aca \mid c.

Definition: Prime Element

Let RR be a commutative ring. An element pRp \in R is prime if pR×{0}p \notin R^\times \cup \{0\} and for all a,bRa, b \in R, if pabp \mid ab, then pap \mid a or pbp \mid b.

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

Definition: Irreducible Element

Let RR be a ring. An element rRr \in R is irreducible if rR×r \notin R^\times and for all a,bRa, b \in R, if r=abr = ab, then aR×a \in R^\times or bR×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, charR\fchar R

Let RR be a ring. The characteristic of RR, denoted by charR\fchar R, is the smallest positive integer nn such that

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

If no such nn exists, then we define charR0\fchar R \coloneqq 0.

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