Finite Abelian groups have been completely classified for a long time. In the 1980s, all finite simple groups were classified in an effort that took over 100 authors and over 500 journal articles. Strangely enough, finite rings have yet to be classified. This is strange because rings are merely Abelian groups with an additional binary operation defined on them that's associative and distributive over addition.
Rings of certain order have already been classified, however. There are 2 nonisomorphic rings of prime order for all primes. If the order of the rings is squarefree, then the number of rings of that order is 2k, where k is the number of distinct prime divisors of the order. Finally, for rings of order p2 there are exactly 11 nonisomorphic types.
Here I will post my research in trying to classify finite rings of all orders.
Update: October 26, 2008. Here I have a table of the first 100 natural numbers in which the orders are color-coded according to whether or not all rings of that order have been classified.
Update: November 10, 2008. Here I have outlined all the background information required in order to understand the project.
Update: November 17, 2008. Here is a proof I have come with that there are exactly 2 nonisomorphic rings of prime order p for each prime number p. This was formed before I found other proofs of a similar result.
Here I have suggestions on how to proceed on this project.
Update: November 24, 2008. I updated the information regarding the table of ring orders. Here I have explained why this problem is especially difficult and why it may never be solved.
BIG UPDATE: December 14, 2008. Here we can finally discuss this question at length and get the flow of feedback needed in order to solve this problem.
Update: December 15, 2008. Here and Here are the files necessary to run I program I have written. This program takes in a ring order and outputs everything I know about rings of that order. When compiling the code, run the file "FintDriver."
Also, the article on the existence of 11 nonisomorphic rings of order p2 is entitled "Classification of Finite Rings of Order p2" by Benjamin Fine. It can be found on JSTOR.
Updated Monday, December 15, 2008