6.7. NOETHERIAN RINGS
311
Proposition 6.7.2.
For a ring
R
(not necessarily commutative, not neces
sarily with identity), the following are equivalent:
(a)
Every ideal of
R
is finitely generated.
(b)
R
satisfies the ascending chain condition for ideals.
Proof.
Suppose that every ideal of
R
is finitely generated. Let
I
1
I
2
I
3
be an infinite, weakly increasing sequence of ideals of
R
. Then
I
D [
n
I
n
is an ideal of
R
, so there exists a finite set
S
such that
I
is the
ideal generated by
S
. Each element of
S
is contained in some
I
n
; since
the
I
n
are increasing, there exists an
N
such that
S
I
N
. Since
I
is the
smallest ideal containing
S
, we have
I
I
N
[
n
I
n
D
I
. It follows that
I
n
D
I
N
for all
n
N
. This shows that
R
satisfies the ACC for ideals.
Suppose that
R
has an ideal
I
that is not finitely generated. Then for
any finite subset
S
of
I
, the ideal generated by
S
is properly contained in
I
. Hence there exists an element
f
2
I
n
.S/
, and the ideal generated
by
S
is properly contained in that generated by
S
[ f
f
g
. An inductive
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 Fall '08
 EVERAGE
 Algebra, Ring, ideals, Commutative ring, Emmy Noether, Noetherian ring, noetherian rings

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