=

ρ is called a ring automorphism on R.

Analogous to Theorem 8.24, we have:

Theorem 9.24. If ρ is a ring isomorphism of R with R , then the inverse

function ρ’1 is a ring isomorphism of R with R.

Proof. Exercise. 2

Because of this theorem, if R is isomorphic to R , we may simply say that

“R and R are isomorphic.”

We stress that a ring isomorphism ρ of R with R is essentially just a

“renaming” of elements; in particular, ρ maps units to units and zero divisors

to zero divisors (verify); moreover, the restriction of the map ρ to R— yields

a group isomorphism of R— with (R )— (verify).

An injective ring homomorphism ρ : R ’ E is called an embedding

of R in E. In this case, img(ρ) is a subring of E and R ∼ img(ρ). If

=

the embedding is a natural one that is clear from context, we may simply

identify elements of R with their images in E under the embedding, and as

a slight abuse of terminology, we shall say that R as a subring of E.

We have already seen an example of this, namely, when we formally de-

¬ned the ring of polynomials R[X] over R, we de¬ned the map ρ : R ’ R[X]

that sends c ∈ R to the polynomial whose constant term is c, and all other

coe¬cients zero. This map ρ is clearly an embedding, and it was via this

embedding that we identi¬ed elements of R with elements of R[X], and so

viewed R as a subring of R[X].

This practice of identifying elements of a ring with their images in another

ring under a natural embedding is very common. We shall see more examples

of this later (in particular, Example 9.43 below).

Theorems 8.25, 8.26, and 8.27 also have natural ring analogs ” again, one

only has to show that the corresponding group homomorphisms are also ring

homomorphisms:

Theorem 9.25. If I is an ideal of a ring R, then the natural map ρ : R ’

R/I given by ρ(a) = a + I is a surjective ring homomorphism whose kernel

is I.

Proof. Exercise. 2

Theorem 9.26. Let ρ be a ring homomorphism from R into R . Then the

map ρ : R/ ker(ρ) ’ img(ρ) that sends the coset a + ker(ρ) for a ∈ R to

¯

ρ(a) is unambiguously de¬ned and is a ring isomorphism of R/ ker(ρ) with

img(ρ).

238 Rings

Proof. Exercise. 2

Theorem 9.27. Let ρ be a ring homomorphism from R into R . Then

for any ideal I contained in ker(ρ), the map ρ : R/I ’ img(ρ) that sends

¯

the coset a + I for a ∈ R to ρ(a) is unambiguously de¬ned and is a ring

homomorphism from R/I onto img(ρ) with kernel ker(ρ)/I.

Proof. Exercise. 2

Example 9.36. For n ≥ 1, the natural map ρ from Z to Zn sends a ∈ Z

to the residue class [a]n . In Example 8.41, we noted that this is a surjective

group homomorphism on the underlying additive groups, with kernel nZ;

however, this map is also a ring homomorphism. 2

Example 9.37. As we saw in Example 8.42, if n1 , . . . , nk are pairwise

relatively prime, positive integers, then the map from Z to Zn1 —· · ·—Znk that

sends x ∈ Z to ([x]n1 , . . . , [x]nk ) is a surjective group homomorphism on the

underlying additive groups, with kernel nZ, where n = k ni . However,

i=1

this map is also a ring homomorphism (this follows from Example 9.36 and

Theorem 9.23). Therefore, by Theorem 9.26, the map that sends [x]n ∈

Zn to ([x]n1 , . . . , [x]nk ) is a ring isomorphism of the ring Zn with the ring

Zn1 — · · · — Znk . It follows that the restriction of this map to Z— yields a

n

— and Z— — · · · — Z— (see

group isomorphism of the multiplicative groups Zn n1 nk

Example 9.13). 2

Example 9.38. As we saw in Example 8.43, if n1 , n2 are positive integers

with n1 > 1 and n1 | n2 , then the map ρ : Zn2 ’ Zn1 that sends [a]n2 to

¯

[a]n1 is a surjective group homomorphism on the underlying additive groups

with kernel n1 Zn2 . This map is also a ring homomorphism. The map ρ ¯

can also be viewed as the map obtained by applying Theorem 9.27 with the

natural map ρ from Z to Zn1 and the ideal n2 Z of Z, which is contained in

ker(ρ) = n1 Z. 2

Example 9.39. Let R be a subring of a ring E, and ¬x ± ∈ E. The

polynomial evaluation map ρ : R[X] ’ E that sends a ∈ R[X] to a(±) ∈ E

is a ring homomorphism from R[X] into E (see Theorem 9.10). The image

of ρ consists of all polynomial expressions in ± with coe¬cients in R, and is

denoted R[±]. Note that R[±] is a subring of E containing R ∪ {±}, and is

the smallest such subring of E. 2

Example 9.40. We can generalize the previous example to multi-variate

polynomials. If R is a subring of a ring E and ±1 , . . . , ±n ∈ E, then the

map ρ : R[X1 , . . . , Xn ] ’ E that sends a ∈ R[X1 , . . . , Xn ] to a(±1 , . . . , ±n ) is

9.4 Ring homomorphisms and isomorphisms 239

a ring homomorphism. Its image consists of all polynomial expressions in

±1 , . . . , ±n with coe¬cients in R, and is denoted R[±1 , . . . , ±n ]. Moreover,

this image is a subring of E containing R ∪ {±1 , . . . , ±n }, and is the smallest

such subring of E. 2

Example 9.41. For any ring R, consider the map ρ : Z ’ R that sends

m ∈ Z to m · 1R in R. This is clearly a ring homomorphism (verify). If

ker(ρ) = {0}, then img(ρ) ∼ Z, and so the ring Z is embedded in R, and

=

R has characteristic zero. If ker(ρ) = nZ for n > 0, then img(ρ) ∼ Zn , and

=

so the ring Zn is embedded in R, and R has characteristic n. Note that we

have n = 1 if and only if R is trivial.

Note that img(ρ) is the smallest subring of R; indeed, since any subring

of R must contain 1R and be closed under addition and subtraction, it must

contain img(ρ). 2

Example 9.42. Let R be a ring of prime characteristic p. For any a, b ∈ R,

we have (see Exercise 9.2)

p

p p’k k

p

(a + b) = a b.

k

k=0

However, by Exercise 1.12, all of the binomial coe¬cients are multiples of

p, except for k = 0 and k = p, and hence in the ring R, all of these terms

vanish, leaving us with

(a + b)p = ap + bp .

This result is often jokingly referred to as the “freshman™s dream,” for some-

what obvious reasons.

Of course, as always, we have

(ab)p = ap bp and 1p = 1R ,

R

and so it follows that the map ρ : R ’ R that sends a ∈ R to ap is a

ring homomorphism. It also immediately follows that for any integer e ≥ 1,

e

the e-fold composition ρe : R ’ R that sends a ∈ R to ap is also a ring

homomorphism. 2

Example 9.43. As in Example 9.34, let f be a monic polynomial over a

ring R with deg(f ) = , but now assume that > 0. Consider the natural

map ρ from R[X] to the quotient ring E := R[X]/(f ) that sends a ∈ R[X] to

[a]f . If we restrict ρ to the subring R of R[X], we obtain an embedding of R

into E. Since this is a very natural embedding, one usually simply identi¬es

240 Rings

elements of R with their images in E under ρ, and regards R as a subring

of E. Taking this point of view, we see that if a = i ai Xi , then

ai Xi ]f = ai ([X]f )i = a(·),

[a]f = [

i i

where · := [X]f ∈ E. Therefore, the map ρ may be viewed as the polynomial

evaluation map, as in Example 9.39, that sends a ∈ R[X] to a(·) ∈ E. Note