Notes on Ring Theory and Field Theory
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 consisting of:
- a set , called the underlying set;
- a distinguished element , called the zero element;
- a unary operation , written as , called negation;
- a binary operation , written as , called addition;
- a binary operation , written as , called multiplication;
satisfying the following requirements:
- Additive structure: is an abelian group.
- Associativity: for all .
- Left distributivity: for all .
- Right distributivity: for all .
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 consisting of a single element . The negation, addition, and multiplication operations in the zero rng are defined by .
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 that satisfies the following additional requirement:
- Commutativity: for all .
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 consisting of:
- a set , called the underlying set;
- a distinguished element , called the zero element;
- a distinguished element , called the identity element;
- a unary operation , written as , called negation;
- a binary operation , written as , called addition;
- a binary operation , written as , called multiplication;
satisfying the following requirements:
- Rng structure: is a rng.
- Identity: for all .
Zero Divisors and Inverses
Every rng has a distinguished element . The rng axioms specify what happens when is fed into the operations of addition and multiplication, but they leave unspecified when is produced as an output. This is the topic that we will study in this section.
The equation is familiar territory—this simply means that and are additive inverses, i.e., . On the other hand, the equation is far more interesting. In the familiar rngs of grade-school arithmetic, including , , , and , the equation implies that or . 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 , , , or . 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 be a rng.
- An element is a left zero divisor if there exists such that .
- An element is a right zero divisor if there exists such that .
- An element of is a zero divisor if it is a left zero divisor or a right zero divisor.
- An element of 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 . (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 .”
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 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 . This prompts us to ask when is produced as an output of addition or multiplication. Again, the equation is rather uninteresting, since it means nothing more than . However, the equation merits further investigation. In , , or , this would simply mean that and are inverses, i.e., ; but in a general ring, it is not guaranteed that every nonzero element has an inverse.
Definition: Invertible, Left-Invertible, Right-Invertible
Let be a ring, and let .
- is left-invertible if there exists such that . We call such an element a left inverse of .
- is right-invertible if there exists such that . We call such an element a right inverse of .
- is invertible (or two-sided invertible) if there exists such that . We call such an element an inverse (or two-sided inverse) of , and we write 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., does not imply or ). 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,
Let be a ring. An element is a unit if there exists an element such that . The set of all units in is denoted by , and is called the group of units of .
In other words, is the set of elements of that have a (two-sided) inverse. We call the group of units because it is group under the ring’s multiplication operation. In particular, we always have , and if , then (since its inverse is ) and (since its inverse is ). For example, and .
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 is never invertible in any nontrivial ring , 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 is a division ring if . 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 be a ring. We say that two elements are associates if there exist such that .
Note that, in a commutative ring, this condition simplifies to . In standard algebra textbooks, this definition is usually only stated for commutative rings.
Associatedness is an Equivalence Relation
Theorem: In any ring , the binary relation of “associatedness” is an equivalence relation on .
Proof: We need to verify that associatedness is reflexive, symmetric, and transitive. Let and .
- Reflexivity: .
- Symmetry: If , then .
- Transitivity: If and , then . □
This shows that every ring can be partitioned into associate classes. Note that is the associate class of , and is its own associate class.
Definition: Principal Ideal, Generator
Let be a commutative ring, and let . The principal ideal generated by , denoted by , is the subset of containing all multiples of .
An ideal is principal if there exists an element such that . In this case, we call a generator of .
Note that a principal ideal may have more than one generator. For example, the ideal of even numbers in has two generators, . Of course, these two generators are quite closely related, since and 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 be an integral domain. If two elements generate the same ideal , then and are associates.
Proof: If , then , and we are done, since is certainly associate to itself. Otherwise, and are both nonzero. The hypothesis implies that and . Thus, there exist such that and . It follows that , or equivalently, . Since is an integral domain and , we must have , which shows that and are both units in . We therefore conclude from the defining condition that and 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,
Let be a commutative rng, and let . We say that divides , denoted by , if there exists an element such that . This binary relation is called the divisibility relation on .
The divisibility relation is always reflexive in a ring, but can fail to be reflexive in a rng. Note that does not divide any element of a rng except , but every element of a rng divides .
Divisibility is Transitive
Theorem: The divisibility relation on any rng is transitive.
Proof: Let be a rng, and let . If and , then there exist sch that and . It follows that , which proves that . □
Definition: Prime Element
Let be a commutative ring. An element is prime if and for all , if , then or .
The prime elements of the commutative ring are precisely the usual prime numbers and their negatives.
Definition: Irreducible Element
Let be a ring. An element is irreducible if and for all , if , then or .
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,
Let be a ring. The characteristic of , denoted by , is the smallest positive integer such that
If no such exists, then we define .
For example, and .