Noether's Rings

Date: 08/20/99 at 05:44:01
From: Sinh Ly
Subject: Noether's rings

I'm not a K-12 grade student, but a physics undergraduate at UCLA. I 
was doing some research on Emmy Noether since Noether's theorem is 
extremely powerful in physics and I came upon a subject known as 
Noether's rings. Can you explain what this is and a little of the math 
behind it?

Thanks.

Date: 08/20/99 at 15:09:44
From: Doctor Rob
Subject: Re: Noether's rings

Thanks for writing to Ask Dr. Math!

First you have to know what a "ring" is in mathematics. A ring is a 
set R, together with two operations denoted + and *, that satisfies 
the following properties:

   1) For every a and b in R, a + b is in R.

   2) There exists an element 0 in R such that for every a in R,
      a + 0 = a = 0 + a.

   3) For every a in R there is an element (denoted -a) in R such that
      a + (-a) = 0 = (-a) + a.

   4) For every a and b in R, a + b = b + a.

   5) For every a and b in R, a*b is in R.

   6) For every a, b and c in R, a*(b+c) = (a*b) + (a*c), and also
      (a+b)*c = (a*c) + (b*c).

Some rings also satisfy the following additional properties:

   7) There exists an element 1 in R such that for every a in R,
      a*1 = 1*a = a.  (We say the ring is "with unity.")

   8) For every a and b in R, a*b = b*a.  (We say the ring "is
      commutative.")

The first example of a ring satisfying these properties is the set of 
integers together with the usual addition and multiplication as 
operations.

Next you have to know what an "ideal" is. Given a ring R, an ideal I 
of the ring R is a subset of R that satisfies the following 
properties:

   A) I is a ring using the same operations as R, and its 0 is the
      same as the 0 of R.

   B) For any r in R and any i in I, r*i is in I.

Every ring has at least the following two ideals: all of R, and the 
set {0}.

In the integers, every ideal is just all the multiples of some single 
integer. As an example, all the multiples of 30 form an ideal
I = {30*n : n an integer}.

Noetherian rings are commutative rings with unity that satisfy the 
Ascending Chain Condition on ideals. That says that if you have a 
chain of ideals of a ring R, I1 contained in I2 contained in I3 ..., 
there is some point in the chain such that all ideals beyond that are 
equal. In other words, any such chain can contain only a finite number 
of different ideals.

Not every ring is a Noetherian ring. The integers are such a ring.

For example, the ideal above, of all multiples of 30, can start a 
chain no longer than 4. Try it and see!

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/