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Proof.
See [Mill58].
Definition.
Let I be an ideal in a commutative ring R. The
radical
of I, denoted
by
I
, is defined by
{
0.
n
I
=Œ
a
R
a
Œ
I for some n
>
An ideal I is called a
radical ideal
if
I
=
I
.
B.6.4. Proposition.
Let I be an ideal in a commutative ring R.
(1) is an ideal that divides I.
(2) is a radical ideal.
(3) If I is prime, then I is a radical ideal.
I
I
Proof.
Easy.
The next definition generalizes the notion of a prime power element.
Definition.
An ideal I in a commutative ring R is called a
primary
ideal if whenever
ab Œ I for some b that does not lie in I, then
aI
Œ
.
We can rephrase the definition to say that every zero divisor of a primary ideal
I is in its radical, that is, ab ∫ 0 (mod I) and b
0 (mod I) implies that a ∫ 0
(mod
I
).
B.6.5. Proposition.
The radical of a primary ideal in a commutative ring is a prime
ideal.
Proof.
Easy.
For example, note that the ideal <q> is primary in
Z
if and only if q = p
n
where p
is a prime.
Now the equation
kk
k
m
npp
=
◊◊◊
12
1
2
p
implies that
k
k
k
m
<
np
> = <
1
>«<
p
2
>«◊◊◊«<
p
>
.
1
2
The natural question is whether such a factorization of ideals holds in general. The
answer is yes, provided that the ring satisfies certain chain conditions.
Definition.
A commutative ring R is said to satisfy the
ascending chain condition
if
for every sequence of ideals I
i
satisfying
I
Õ
I
Õ ◊◊◊
1
2