multiplication and addition forms a ring. 2

Example 9.4. The set R of real numbers under the usual rules of multipli-

cation and addition forms a ring. 2

Example 9.5. The set C of complex numbers under the usual rules of mul-

tiplication and addition forms a ring. Any ± ∈ C can be written (uniquely)

√

as ± = a + bi, with a, b ∈ R, and i = ’1. If ± = a + b i is another complex

number, with a , b ∈ R, then

± + ± = (a + a ) + (b + b )i and ±± = (aa ’ bb ) + (ab + a b)i.

The fact that C is a ring can be veri¬ed by direct calculation; however, we

shall see later that this follows easily from more general considerations.

Recall the complex conjugation operation, which sends ± to ± := a ’

¯

bi. One can verify by direct calculation that complex conjugation is both

additive and multiplicative; that is, ± + ± = ± + ± and ± · ± = ± · ± .

¯¯ ¯¯

9.1 De¬nitions, basic properties, and examples 213

The norm of ± is N (±) := ±± = a2 + b2 . So we see that N (±) is

¯

a non-negative real number, and is zero i¬ ± = 0. Moreover, from the

multiplicativity of complex conjugation, it is easy to see that the norm is

multiplicative as well: N (±± ) = ±± ±± = ±± ±± = N (±)N (± ). 2

¯¯

Example 9.6. Consider the set F of all arithmetic functions, that is, func-

tions mapping positive integers to real numbers. We can de¬ne addition

and multiplication operations on F in a natural, point-wise fashion: for

f, g ∈ F, let f + g be the function that sends n to f (n) + g(n), and let

f · g be the function that sends n to f (n)g(n). These operations of addition

and multiplication make F into a ring: the additive identity is the function

that is everywhere 0, and the multiplicative identity is the function that is

everywhere 1.

Another way to make F into a ring is to use the addition operation as

above, together with the Dirichlet product, which we de¬ned in §2.6, for

the multiplication operation. In this case, the multiplicative identity is the

function I that we de¬ned in §2.6, which takes the value 1 at 1 and the value

0 everywhere else. The reader should verify that the distributive law holds.

2

Note that in a ring R, if 1R = 0R , then for all a ∈ R, we have a = 1R · a =

0R · a = 0R , and hence the ring R is trivial, in the sense that it consists of

the single element 0R , with 0R + 0R = 0R and 0R · 0R = 0R . If 1R = 0R , we

say that R is non-trivial. We shall rarely be concerned with trivial rings for

their own sake; however, they do sometimes arise in certain constructions.

If R1 , . . . , Rk are rings, then the set of all k-tuples (a1 , . . . , ak ) with ai ∈ Ri

for i = 1, . . . , k, with addition and multiplication de¬ned component-wise,

forms a ring. The ring is denoted by R1 — · · · — Rk , and is called the direct

product of R1 , . . . , Rk .

The characteristic of a ring R is de¬ned as the exponent of the un-

derlying additive group (see §8.5). Note that for m ∈ Z and a ∈ R, we

have

ma = m(1R · a) = (m · 1R )a,

so that if m · 1R = 0R , then ma = 0R for all a ∈ R. Thus, if the additive

order of 1R is in¬nite, the characteristic of R is zero, and otherwise, the

characteristic of R is equal to the additive order of 1R .

Example 9.7. The ring Z has characteristic zero, Zn has characteristic n,

and Zn1 — Zn2 has characteristic lcm(n1 , n2 ). 2

For elements a, b in a ring R, we say that b divides a, or alternatively,

214 Rings

that a is divisible by b, if there exists c ∈ R such that a = bc. If b divides

a, then b is called a divisor of a, and we write b | a. Note Theorem 1.1

holds for an arbitrary ring.

When there is no possibility for confusion, one may write “0” instead of

“0R ” and “1” instead of “1R .” Also, one may also write, for example, 2R to

denote 2 · 1R , 3R to denote 3 · 1R , and so on; moreover, where the context

is clear, one may use an implicit “type cast,” so that m ∈ Z really means

m · 1R .

For a ∈ R and positive integer n, the expression an denotes the product

a · a · · · · · a, where there are n terms in the product. One may extend this

de¬nition to n = 0, de¬ning a0 to be the multiplicative identity 1R .

Exercise 9.1. Verify the usual “rules of exponent arithmetic” for a ring R.

That is, show that for a ∈ R, and non-negative integers n1 , n2 , we have

(an1 )n2 = an1 n2 and an1 an2 = an1 +n2 .

Exercise 9.2. Show that the familiar binomial theorem holds in an ar-

bitrary ring R; that is, for a, b ∈ R and positive integer n, we have

n

n n’i i

(a + b)n = a b.

i

i=0

Exercise 9.3. Show that

n m n m

ai bj = ai bj ,

i=1 j=1 i=1 j=1

where the ai and bj are elements of a ring R.

9.1.1 Units and ¬elds

Let R be a ring. We call u ∈ R a unit if it divides 1R , that is, if uu = 1R

for some u ∈ R. In this case, it is easy to see that u is uniquely determined,

and it is called the multiplicative inverse of u, and we denote it by u’1 .

Also, for a ∈ R, we may write a/u to denote au’1 . It is clear that a unit u

divides every a ∈ R.

We denote the set of units by R— . It is easy to verify that the set R—

is closed under multiplication, from which it follows that R— is an abelian

group, called the multiplicative group of units of R. If u ∈ R— , then of

course un ∈ R— for all non-negative integers n, and the multiplicative inverse

9.1 De¬nitions, basic properties, and examples 215

of un is (u’1 )n , which we may also write as u’n (which is consistent with

our notation for abelian groups).

If R is non-trivial and every non-zero element of R has a multiplicative

inverse, then R is called a ¬eld.

Example 9.8. The only units in the ring Z are ±1. Hence, Z is not a ¬eld.

2

Example 9.9. For positive integer n, the units in Zn are the residue classes

[a]n with gcd(a, n) = 1. In particular, if n is prime, all non-zero residue

classes are units, and if n is composite, some non-zero residue classes are

not units. Hence, Zn is a ¬eld if and only if n is prime. Of course, the

notation Z— introduced in this section for the group of units of the ring Zn

n

is consistent with the notation introduced in §2.3. 2

Example 9.10. Every non-zero element of Q is a unit. Hence, Q is a ¬eld.

2

Example 9.11. Every non-zero element of R is a unit. Hence, R is a ¬eld.

2

Example 9.12. For non-zero ± = a + bi ∈ C, with a, b ∈ R, we have c :=

N (±) = a2 + b2 > 0. It follows that the complex number ±c’1 = (ac’1 ) +

¯

’1 )i is the multiplicative inverse of ±, since ± · ±c’1 = (±±)c’1 = 1.

(’bc ¯ ¯

Hence, every non-zero element of C is a unit, and so C is a ¬eld. 2

Example 9.13. For rings R1 , . . . , Rk , it is easy to see that the multiplicative

— —

group of units of the direct product R1 — · · · — Rk is equal to R1 — · · · — Rk .

Indeed, by de¬nition, (a1 , . . . , ak ) has a multiplicative inverse if and only if

each individual ai does. 2

Example 9.14. Consider the rings of arithmetic functions de¬ned in Exam-

ple 9.6. If multiplication is de¬ned point-wise, then an arithmetic function f

is a unit if and only if f (n) = 0 for all n. If multiplication is de¬ned in terms

of the Dirichlet product, then by the result of Exercise 2.27, an arithmetic

function f is a unit if and only if f (1) = 0. 2

9.1.2 Zero divisors and integral domains