Noether's RingsDate: 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/ |
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