numbers of the form a/b, where gcd(a, m) = gcd(b, m) = 1. 2

Example 9.24. If R and S are non-trivial rings, then R := R — {0S }

is not a subring of R — S: although it satis¬es the ¬rst two requirements

of the de¬nition of a subring, it does not satisfy the third. However, R

does contain an element that acts as a multiplicative identity of R , namely

(1R , 0S ), and hence could be viewed as a subring of R — S under a more

liberal de¬nition. 2

Theorem 9.9. Any subring of an integral domain is also an integral do-

main.

Proof. If D is a subring of the integral domain D, then any zero divisor in

D would itself be a zero divisor in D. 2

Note that it is not the case that a subring of a ¬eld is always a ¬eld: the

subring Z of Q is a counter-example. If F is a subring of a ¬eld F , and F

is itself a ¬eld, then we say that F is a sub¬eld of F , and that F is an

extension ¬eld of F .

220 Rings

Example 9.25. Q is a sub¬eld of R, which in turn is a sub¬eld of C. 2

Exercise 9.8. Show that the set Q[i] of complex numbers of the form a+bi,

with a, b ∈ Q, is a sub¬eld of C.

Exercise 9.9. Show that if S and S are subrings of R, then so is S © S .

Exercise 9.10. Let F be the set of all functions f : R ’ R, and let C be

the subset of F of continuous functions.

(a) Show that with addition and multiplication of functions de¬ned in the

natural, point-wise fashion, F is a ring, but not an integral domain.

(b) Let a, b ∈ F. Show that if a | b and b | a, then there is a c ∈ F — such

that a = bc.

(c) Show that C is a subring of F, and show that all functions in C — are

either everywhere positive or everywhere negative.

(d) De¬ne a, b ∈ C by a(t) = b(t) = t for t < 0, a(t) = b(t) = 0 for

0 ¤ t ¤ 1, and a(t) = ’b(t) = t ’ 1 for t > 1. Show that in the ring

C, we have a | b and b | a, yet there is no c ∈ C — such that a = bc.

Thus, part (ii) of Theorem 9.4 does not hold in a general ring.

9.2 Polynomial rings

If R is a ring, then we can form the ring of polynomials R[X], consisting

of all polynomials a0 + a1 X + · · · + ak Xk in the indeterminate, or “formal”

variable, X, with coe¬cients in R, and with addition and multiplication

being de¬ned in the usual way.

Example 9.26. Let us de¬ne a few polynomials over the ring Z:

a := 3 + X2 , b := 1 + 2X ’ X3 , c := 5, d := 1 + X, e := X, f := 4X3 .

We have

a+b = 4+2X+X2 ’X3 , a·b = 3+6X+X2 ’X3 ’X5 , cd+ef = 5+5X+4X4 . 2

As illustrated in the previous example, elements of R are also polynomials.

Such polynomials are called constant polynomials; all other polynomials

are called non-constant polynomials. The set R of constant polynomials

clearly forms a subring of R[X]. In particular, 0R is the additive identity in

R[X] and 1R is the multiplicative identity in R[X].

9.2 Polynomial rings 221

For completeness, we present a more formal de¬nition of the ring R[X].

The reader should bear in mind that this formalism is rather tedious, and

may be more distracting than it is enlightening. It is technically conve-

nient to view a polynomial as having an in¬nite sequence of coe¬cients

a0 , a1 , a2 , . . . , where each coe¬cient belongs to R, but where only a ¬nite

number of the coe¬cients are non-zero. We may write such a polynomial as

an in¬nite sum ∞ ai Xi ; however, this notation is best thought of “syntac-

i=0

tic sugar”: there is really nothing more to the polynomial than this sequence

of coe¬cients. With this notation, if

∞ ∞

i

bi Xi ,

a= ai X and b =

i=0 i=0

then

∞

(ai + bi )Xi ,

a + b := (9.1)

i=0

and

∞ i

ak bi’k Xi .

a · b := (9.2)

i=0 k=0

We should ¬rst verify that these addition and multiplication operations

actually produce coe¬cient sequences with only a ¬nite number of non-zero

terms. Suppose that for non-negative integers k and , we have ai = 0R for

all i > k and bi = 0R for all i > . Then it is clear that the coe¬cient of Xi

in a + b is zero for all i > max{k, }, and it is also not too hard to see that

the coe¬cient of Xi in a · b is zero for all i > k + .

We leave it to the reader to verify that R[X], with addition and multipli-

cation de¬ned as above, actually satis¬es the de¬nition of a ring ” this is

entirely straightforward, but tedious.

For c ∈ R, we may identify c with the polynomial ∞ ci Xi , where c0 = c

i=0

∞

and ci = 0R for i > 0. Strictly speaking, c and i=0 ci Xi are not the same

mathematical object, but there will certainly be no possible confusion in

treating them as such. Thus, from a narrow, legalistic point of view, R is

not a subring of R[X], but we shall not let such let such annoying details

prevent us from continuing to speak of it as such. As one last matter of

∞ i

notation, we may naturally write X to denote the polynomial i=0 ai X ,

where a1 = 1R and ai = 0R for all i = 1.

With all of these conventions and de¬nitions, we can return to the prac-

tice of writing polynomials as we did in Example 9.26, without any loss of

precision. Note that by de¬nition, if R is the trivial ring, then so is R[X].

222 Rings

9.2.1 Polynomials versus polynomial functions

Of course, a polynomial a = k ai Xi de¬nes a polynomial function on R

i=0

k i , and we denote the value of this function

that sends ± ∈ R to i=0 ai ±

as a(±). However, it is important to regard polynomials over R as formal

expressions, and not to identify them with their corresponding functions.

In particular, two polynomials are equal if and only if their coe¬cients are

equal. This distinction is important, since there are rings R over which two

di¬erent polynomials de¬ne the same function. One can of course de¬ne the

ring of polynomial functions on R, but in general, that ring has a di¬erent

structure from the ring of polynomials over R.

Example 9.27. In the ring Zp , for prime p, by Fermat™s little theorem

(Theorem 2.16), we have ±p ’ ± = [0]p for all ± ∈ Zp . But consider the

polynomial a := Xp ’ X ∈ Zp [X]. We have a(±) = [0]p for all ± ∈ Zp , and

hence the function de¬ned by a is the zero function, yet a is de¬nitely not

the zero polynomial. 2

More generally, if R is a subring of a ring E, a polynomial a = k ai Xi ∈

i=0

R[X] de¬nes a polynomial function from E to E that sends ± ∈ E to

k i

i=0 ai ± ∈ E, and the value of this function is denoted a(±).

If E = R[X], then evaluating a polynomial a ∈ R[X] at a point ± ∈ E

amounts to polynomial composition. For example, if a = X2 + X then

= (X + 1)2 + (X + 1) = X2 + 3X + 2.

a X+1

A simple, but important, fact is the following:

Theorem 9.10. Let R be a subring of a ring E. For a, b ∈ R[X] and ± ∈ E,

if p := ab ∈ R[X] and s := a + b ∈ R[X], then we have

p(±) = a(±)b(±) and s(±) = a(±) + b(±).

Also, if c ∈ R[X] is a constant polynomial, then c(±) = c for all ± ∈ E.

Proof. Exercise. 2

Note that the syntax for polynomial evaluation creates some poten-

tial ambiguities: if a is a polynomial, one could interpret a(b + c) as

either a times b + c, or a evaluated at b + c; usually, the meaning