| In mathematics finiteness properties are very important.
The Noetherian condition (named in honour of the great
algebraist Emmy Noether) is such a property. It plays
an important rôle in algebra and algebraic geometry.
See also the (essentially) dual notion of the Artinian condition.
First of all you will want to read ideal to find out
what a right ideal, a left ideal and a two-sided ideal
are (and to absorb some notation).
But remember for commutative rings these conditions
all coincide as do the properties of being right Noetherian,
left Noetherian, and Noetherian that I define below.
The first time you read this you should probably just think about
commutative rings and forget about left and right.
Definition A right ideal I in a ring R
is called finitely generated if there exist
finitely many elements a1,...,an of
R such that
I=a1R +...+ anR
We define the notion of finitely generated for a left ideal similarly.
Definition A ring R is called right Noetherian
if every right ideal is finitely generated. R is called left Noetherian
if every left ideal is finitely generated. R is called Noetherian
if it is both left and right Noetherian.
We could have given two alternative definitions for the Noetherian property
as the following result shows.
Theorem
Let R be a ring. Then TFAE
-
R is right Noetherian.
-
whenever I1 <= I2 <= ...
is an ascending chain (here I use "<=" to mean "is a subset of")
of right ideals it eventually becomes
stationary. i.e. for N suitably large In=IN
for all n>N. This is called the ascending chain condition.
-
Any nonempty set of right ideals of R contains
a maximal element. i.e. if S is such a nonempty set there exists
a right ideal I in S such that there is no right ideal
J in S that contains I and is not equal to I.
Note that there can be more than one such maximal element.
This is called the maximum condition.
Proof: 1 ==> 2. Let I be the union of all the
right ideals In in the chain. I is itself
a right ideal, for if a,b are elements of I then, by definition,
a is in some In and b is in some
Im. So they both lie in Imax(m,n).
Clearly then a-b and ar (for r in R)
are also in this latter right ideal and hence in I.
By 1 then we know that I has finitely many generators, say
a1,...,ar. Choose N so
that each of these generators is in IN. Then the chain
becomes stationary at this point.
2 ==> 1. Let I be a right ideal that is not finitely
generated. Suppose that we have chosen a1,...,an
in I. Then, by assumption, I is not equal to
a1R +...+ anR. So we can choose
an element
an+1 that is in I but that is not in
a1R +...+ anR.
In this way we can create an ascending chain of right ideals
that is never stationary
a1R < a1R+a2R < ...
2 ==> 3. Let S be a nonempty set of right ideals that
does not have a maximal element. Then given a right ideal
I in S there must exist another J in S such
that I is contained in but does not equal J. Repeating this
we obtain an ascending chain of right ideals that is never stationary.
3 ==> 2. The maximum condition applied to the set of right ideals
in the chain shows that the chain is eventually stationary.
A cookie to anyone who can tell me where we sneakily used the
axiom of choice in the above proof. (Actually only the axiom
of dependent choice is needed.)
Examples of Noetherian rings
-
Any field is Noetherian since any ideal must either be {0}
or R. Similarly any division ring is Noetherian.
So this applies to Q, R,
C, H, for example.
-
Any principal ideal domain is Noetherian (since this is a commutative
ring in which every ideal needs just one generator).
This applies to Z and k[x],
for a field k.
-
More generally the celebrated The Hilbert Basis Theorem tells us that the
polynomial ring in several variables over any commutative Noetherian
ring is Noetherian.
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