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there is an n, so that i > n implies that I
i
= I
i+1
.
Definition.
A commutative ring R is said to be a
Noetherian ring
provided that it
satisfies the ascending chain condition.
B.6.6. Theorem.
A commutative ring R is Noetherian if and only if every ideal is
finitely generated.
Proof.
See [Jaco66] or [ZarS60], Volume I.
Definition.
An ideal I in a commutative ring is said to be
reducible
if I can be
expressed as the intersection of two ideals I
1
and I
2
, I = I
1
« I
2,
where I
j
π I. Other-
wise, I is said to be
irreducible
.
A prime ideal is irreducible, but a primary ideal need not be. An irreducible ideal
is not necessarily prime. For example, <p
k
> is irreducible in
Z
but not prime if k > 1.
On the other hand,
B.6.7. Lemma.
Every irreducible ideal in a Noetherian ring is primary.
Proof.
See [Jaco66] or [ZarS60], Volume I.
Definition.
An intersection
II I
=
«
«◊◊◊«
n
1
2
of ideals in a ring is said to be an
irredundant intersection
if I is a proper subset of
I
«
I
«◊◊◊«
I
«
I
«◊◊◊«
I
1
2
i
-
1
i
+
1
n
for i = 1, 2,..., n.
B.6.8. Theorem.
Every ideal in a Noetherian ring is a finite irredundant
intersection of primary ideals and the prime ideals associated to this factorization are
unique.
Proof.
See [Jaco66] or [ZarS60], Volume I.
B.7
Polynomial Rings
One of the important examples of rings are polynomial rings.
Definition.
Let A be a subring and S a subset of a ring R. The
polynomial ring over
A generated by S
, denoted A[S], is defined by
[]
=«
{
}
A S
B B is a subring of R and A S
,
Õ
B
.
