1. Elementar y Propertie s of Polynomial Ring s

Consider a polynomial rin g F = k[X x, . . . , Xn] ove r a ring k. The n th e propertie s

of k ar e carried ove r to o r reflecte d o n F. I n thi s section , we firs t observ e som e typica l ex -

amples o f suc h properties o f k.

First w e observe zero-divisors . Sinc e A : i s a subring o f F, i t i s obvious that zero-divisor

in k ar e zero-divisors i n F. Furthermore , t o th e convers e direction , w e have

THEOREM

1.1 . / / / ( G F) is a zero-divisor in F, then there is a nonzero element b of

k such that bf=0 (i. e., all coefficients of fare annihilated by b). If fg = 0 ( 0 ^ g £ F),

then we may choose b so that it belongs to the ideal generated by the coefficients of g in k.

PROOF. W e order monomial s b y th e lexicographica l orde r o f th e serie s of exponents ,

i: e., X\l • • • X6^ zf x • • • Xndn i f an d onl y i f ther e i s one / suc h tha t (i ) e

f

d

j

an d (ii

et = dt i f i /. I f / = 0 , then ther e i s nothing t o prove, and w e assume tha t / ^ 0 . W e

write / an d g in th e form :

s t

f = ^

a imi

8

=

X ^i

ni

( 0 ^

a

i bj€.A\ m

v

fty monomials;

m j m

2

* * • m s, n

x

n

2

•• • n

t

We use a n induction argumen t o n t an d s. I f t = 1 or s = 1, then w e may tak e b

x

a s b,

and w e assume tha t t 1. Sinc e fg = 0 , we have asbt = 0. I f a$ = £ 0, then sinc e fag = 0

and since ag ha s fewer term s tha n gy we see th e existenc e o f b b y ou r inductio n o n t. As -

sume now that ag = 0. The n with/ j = H isaimi, w e have fxg = 0. Therefor e b y ou r

double induction , w e see that ther e i s an element b ( ^ 0 ) o f th e idea l i n k generate d b y th e

coefficients o f g such tha t f1b = 0. Sinc e ag = 0, we have asb = 0. Thu s fb = 0. Q . E. D

This result implie s th e followin g corollar y whic h may b e prove d b y just adaptin g a part

of th e proo f above .

COROLLARY

1.2. If k is an integral domain, then F — k\X

v

. . . , Xn] is also an in-

tegral domain.

This is equivalent t o

COROLLARY

1.3. If P is a prime ideal in k, then PF (the ideal in F generated by P)

is a prime ideal in F.

The write r believe s that th e followin g resul t i s very familia r an d writte n i n man y text -

books, and therefor e w e state i t withou t proof .

THEOREM

1.4. If k is a unique factorization domain, then the polynomial ring k[X]

2