01 01 00 11 10

10 10 11 00 01

11 11 10 01 00

Note that 00 acts as the additive identity for E, and that as an additive

group, E is isomorphic to the additive group Z2 — Z2 .

As for multiplication in E, one has to compute the product of two poly-

nomials, and then reduce modulo f . For example, to compute 10 · 11, using

the identity X2 ≡ X + 1 (mod f ), one sees that

X · (X + 1) ≡ X2 + X ≡ (X + 1) + X ≡ 1 (mod f );

thus, 10 · 11 = 01. The reader may verify the following multiplication table

for E:

· 00 01 10 11

00 00 00 00 00

01 00 01 10 11

10 00 10 11 01

11 00 11 01 10

Observe that 01 acts as the multiplicative identity for E. Notice that every

non-zero element of E has a multiplicative inverse, and so E is in fact a ¬eld.

By Theorem 9.16, we know that E — must be cyclic (this fact also follows

from Theorem 8.32, and the fact that |E — | = 3.) Indeed, the reader may

verify that both 10 and 11 have multiplicative order 3.

234 Rings

This is the ¬rst example we have seen of a ¬nite ¬eld whose cardinality is

not prime. 2

Exercise 9.25. Let I be an ideal of a ring R, and let x and y be elements

of R with x ≡ y (mod I). Let f ∈ R[X]. Show that f (x) ≡ f (y) (mod I).

Exercise 9.26. Let p be a prime, and consider the ring Q(p) (see Exam-

ple 9.23). Show that any non-zero ideal of Q(p) is of the form (pi ), for some

uniquely determined integer i ≥ 0.

Exercise 9.27. Let R be a ring. Show that if I is a non-empty subset

of R[X] that is closed under addition, multiplication by elements of R, and

multiplication by X, then I is an ideal of R[X].

For the following three exercises, we need some de¬nitions. An ideal I of

a ring R is called prime if I R and if for all a, b ∈ R, ab ∈ I implies a ∈ I

or b ∈ I. An ideal I of a ring R is called maximal if I R and there are

no ideals J of R such that I J R.

Exercise 9.28. Let R be a ring. Show that:

(a) an ideal I of R is prime if and only if R/I is an integral domain;

(b) an ideal I of R is maximal if and only if R/I is a ¬eld;

(c) all maximal ideals of R are also prime ideals.

Exercise 9.29. This exercise explores some examples of prime and maximal

ideals.

(a) Show that in the ring Z, the ideal {0} is prime but not maximal, and

that the maximal ideals are precisely those of the form pZ, where p

is prime.

(b) More generally, show that in an integral domain D, the ideal {0} is

prime, and this ideal is maximal if and only if D is a ¬eld.

(c) Show that in the ring F [X, Y], where F is a ¬eld, the ideal (X, Y) is

maximal, while the ideals (X) and (Y) are prime, but not maximal.

Exercise 9.30. It is a fact that all non-trivial rings R contain at least one

maximal ideal. Showing this in general requires some fancy set-theoretic

notions. This exercise develops a proof in the case where R is countable

(i.e., ¬nite or countably in¬nite).

(a) Show that if R is non-trivial but ¬nite, then it contains a maximal

ideal.

9.3 Ideals and quotient rings 235

(b) Assume that R is countably in¬nite, and let a1 , a2 , a3 , . . . be an

enumeration of the elements of R. De¬ne a sequence of ideals

I0 , I1 , I2 , . . . , as follows. Set I0 := {0R }, and for i ≥ 0, de¬ne

Ii + ai R if Ii + ai R R;

Ii+1 :=

Ii otherwise.

Finally, set

∞

I := Ii .

i=0

Show that I is a maximal ideal of R. Hint: ¬rst show that I is an

R by assuming that 1R ∈ I and deriving

ideal; then show that I

a contradiction; ¬nally, show that I is maximal by assuming that

for some i = 1, 2, . . . , we have I I + ai R R, and deriving a

contradiction.

For the following three exercises, we need the following de¬nition: for

subsets X, Y of a ring R, let X · Y denote the set of all ¬nite sums of the

form

x1 y1 + · · · + x y (with xk ∈ X, yk ∈ Y for k = 1, . . . , , for some ≥ 0).

Note that X · Y contains 0R (the “empty” sum, with = 0).

Exercise 9.31. Let R be a ring, and S a subset of R. Show that S · R is

an ideal of R, and is the smallest ideal of R containing S.

Exercise 9.32. Let I and J be two ideals of a ring R. Show that:

(a) I · J is an ideal;

(b) if I and J are principal ideals, with I = aR and J = bR, then

I · J = abR, and so is also a principal ideal;

(c) I · J ⊆ I © J;

(d) if I + J = R, then I · J = I © J.

Exercise 9.33. Let S be a subring of a ring R. Let I be an ideal of R, and

J an ideal of S. Show that:

(a) I © S is an ideal of S, and that (I © S) · R is an ideal of R contained

in I;

(b) (J · R) © S is an ideal of S containing J.

236 Rings

9.4 Ring homomorphisms and isomorphisms

De¬nition 9.20. A function ρ from a ring R to a ring R is called a ring

homomorphism if it is a group homomorphism with respect to the under-

lying additive groups of R and R , and if in addition,

(i) ρ(ab) = ρ(a)ρ(b) for all a, b ∈ R, and

(ii) ρ(1R ) = 1R .

Expanding the de¬nition, we see that the requirements that ρ must satisfy

in order to be a ring homomorphism are that for all a, b ∈ R, we have

ρ(a + b) = ρ(a) + ρ(b) and ρ(ab) = ρ(a)ρ(b), and that ρ(1R ) = 1R . Note

that some texts do not require that ρ(1R ) = 1R .

Since a ring homomorphism ρ from R to R is also an additive group

homomorphism, we may also adopt the notation and terminology for image

and kernel, and note that all the results of Theorem 8.20 apply as well here.

In particular, ρ(0R ) = 0R , ρ(a) = ρ(b) if and only if a ≡ b (mod ker(ρ)),

and ρ is injective if and only if ker(ρ) = {0R }. However, we may strengthen

Theorem 8.20 as follows:

Theorem 9.21. Let ρ : R ’ R be a ring homomorphism.

(i) For any subring S of R, ρ(S) is a subring of R .

(ii) For any ideal I of R, ρ(I) is an ideal of img(ρ).

(iii) ker(ρ) is an ideal of R.

(iv) For any ideal I of R , ρ’1 (I ) is an ideal of R.

Proof. Exercise. 2

Theorems 8.21 and 8.22 have natural ring analogs ”one only has to show

that the corresponding group homomorphisms are also ring homomorphisms:

Theorem 9.22. If ρ : R ’ R and ρ : R ’ R are ring homomorphisms,

then so is their composition ρ —¦ ρ : R ’ R .

Proof. Exercise. 2

Theorem 9.23. Let ρi : R ’ Ri , for i = 1, . . . , n, be ring homomorphisms.

Then the map ρ : R ’ R1 — · · · — Rn that sends a ∈ R to (ρ1 (a), . . . , ρn (a))

is a ring homomorphism.

Proof. Exercise. 2

If a ring homomorphism ρ : R ’ R is a bijection, then it is called a ring

isomorphism of R with R . If such a ring isomorphism ρ exists, we say

9.4 Ring homomorphisms and isomorphisms 237