This chapter marks our transition from groups to linear algebra. Vector spaces are among the most tractable algebraic structures: finite-dimensional vector spaces over a fixed field are classified up to isomorphism by a single integer, their dimension, whereas nothing so simple is available for groups. Before we can speak of vector spaces we need their scalars, so we begin with rings and fields, their homomorphisms, polynomial rings and the characteristic of a field. We then develop vector spaces, linear independence, bases and dimension.
A group has one operation. The familiar number systems , , and have two, addition and multiplication, linked by the distributive law. Rings and fields axiomatize this situation.
Definition 4.1.1 (Ring). A ring is a set with two binary operations, addition and multiplication , such that:
- is an abelian group, whose identity element we write ;
- multiplication is associative, , and has an identity element : for all ;
- multiplication distributes over addition: and for all .
The ring is commutative if for all .
Example 4.1.2. The integers , the rationals , the reals and the complex numbers are commutative rings under the usual operations. Most rings in this book are commutative. The main exceptions are the rings of matrices and of linear operators on a vector space, which we meet in Chapter 5.
Proposition 4.1.3 (First consequences of the axioms). Let be a ring and . Then:
- ;
- ; in particular ;
- .
Proof. (a) By distributivity, . Adding to both sides gives . The same argument gives .
(b) We have , so is the additive inverse of . Likewise . Taking gives .
(c) By (b) twice, . ∎
Proposition 4.1.4 (The zero ring). In a ring we have if and only if .
Proof. If , then every satisfies . Conversely, if then . ∎
The set with is a ring, the zero ring. By the proposition it is the only ring in which .
Definition 4.1.5 (Units). An element of a ring is a unit if there is with . Such a is unique: if as well, then . We call it the inverse of and write it . The set of units of is denoted .
Note. The units of form a group under multiplication: is a unit, the inverse of a unit is a unit, and if are units then so is , with . For example , and for , is the group of residues coprime to (Proposition 2.4.9). In a nonzero ring, is never a unit, since .
Definition 4.1.6 (Field). A field is a commutative ring in which and every nonzero element is a unit. Equivalently, , so that is an abelian group.
The condition excludes the zero ring, which by convention is not a field. The rings , and are fields; is not, since has no inverse in .
Definition 4.1.7 (Subrings and subfields). A subring of a ring is a subset that contains and and is closed under addition, negation and multiplication; it is a ring with the operations of . A subfield of a field is a subring that is closed under taking inverses of nonzero elements; it is then a field. In this situation we call a field extension of .
For instance is a subring of , and are subfields.
Definition 4.1.8 (Zero divisor). A nonzero element of a commutative ring is a zero divisor if for some nonzero .
Proposition 4.1.9 (Fields have no zero divisors). Let be a field and . If , then or . Consequently, cancellation holds: if and , then .
Proof. Suppose and . Then . For cancellation, gives , so . ∎
Note. The converse fails: has no zero divisors and is not a field. A commutative ring with and no zero divisors is called an integral domain; these return in Chapter 17. In a general ring, products of nonzero elements can vanish, and then cancellation fails; Example 4.1 shows this happening in .
4.1.1 The Rings and the Fields
Recall from Chapter 2 that is a group under addition mod . We now also multiply mod : the product of and in is the remainder of the integer on division by .
Proposition 4.1.10 ( is a ring). For every , the set with addition and multiplication mod is a commutative ring with identity . (For it is the zero ring.)
Proof. Write if divides . If and , then and , the latter because . Hence when we evaluate an expression built from sums and products, reducing mod after every operation gives the same remainder as reducing only once at the end. For instance, for , both and computed in equal the remainder of the integer , so they are equal. Each of the other ring axioms is an identity that holds in , and it holds in by the same argument. ∎
Example 4.1.11. In we have , so and are zero divisors and is not a field. Cancellation fails as well: in , although .
Proposition 4.1.12 ( is a field). Let . The ring is a field if and only if is prime.
Proof. If is not prime, write with . Then and are nonzero in and , so has zero divisors and is not a field.
Now let be prime. Since , we have in . Let be nonzero. The order of in the additive group divides by Lagrange’s theorem (Theorem 3.1.4), and it is not because ; so it equals . Hence the multiples in the additive group are distinct, and they exhaust . In particular for some . The -fold sum equals , the product of and in the ring , by distributivity. So is a multiplicative inverse of . ∎
When we regard as a field we write it . (The proposition also follows from the description of as the residues coprime to in Proposition 2.4.9: for these are all the nonzero residues.)