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
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