Emmy Noether is known as the mother of Algebra. She made many great contributions to the world of mathematics as well as physics. She faced many problems during her lifetime. She was interested in getting an education but back then most women did not even get a high school education. She struggled to be able to attend lectures at Erlangen and Gottingen Universities. She fought and won the battle in receiving an education once she earned her doctorate. Emmy then began a new struggle with being able to lecture at universities. She was able to lecture under her fatherís name as well as her friend David Hilbertís name. After making many contributions she died after an operation to remove a tumor. Emmy is most known for her contribution on the idea of ideal rings, which is now known as a Noetherian Ring.
We discussed in class the idea of Noetherian Rings. First we must start out with a set. Then you must follow the steps below in order to see if your set is a ring. Also notice that steps a,c,d,e shows that your set is also a group under addition.
a) A,B in R implies that A+B in R
b) A+B=B+A for A,B in R
c) (A+B)+C=A+(B+C) for A,B,C in R
d) There exists an element 0 in R such that A+0=A for every A in R
e) Given A in R, there exists B in R such that A+B=0
f) A,B in R implies that A*B in R
g) A(BC)=(AB)C for all A,B,C in R
(B+C)A=BA+CA for A,B,C in R
Now that we have found a ring we must decided if it is Noetherian. There is one condition that makes a ring a Noetherian ring.
1) It must satisfy the ascending chain condition on ideals such that there is no infinite increasing chain of ideals. This means that the ideals have a stopping place. They do not go on forever.
An Ideal, I, of a ring R is a subring that absorbs elements from R. To be a an Ideal the subring must satisfy the following two conditions.
1) For all A,B in I, A-B is in I.
2) For all A in I and r in R, rA is in I and Ar is in I.
This may seem like a lot properties to learn about and test but even in more elementary math there are properties that we had to learn in order to complete more advanced math. These properties that we learned today are going to help us with even more complicated math.
1) If you know that 4Z is a ring show an example as to why it satisfies the second part of the Noetherian ring definition.
2) Give a reason or example why 4Z+1 is not a ring.
3) Show why 4Z+1 does not satisfy #2.
Dummit, David and Foote, Richard. Abstract Algebra.