Rings and modules of finite order

Question

Fields of finite order are well classified, and classification of groups of finite order has taken some depth in research. Why classification of finite rings and modules is not well studied in research?

Answer 1

The following basic results can be deduced about finite rings:

• A finite simple ring is isomorphic to a matrix ring over a finite field. (Artin-Wedderburn theorem.)
• A finite ring is Artinian. Hence all the known properties of Artinian rings also hold for finite rings.
• See http://www.csc.villanova.edu/~pcesarz/
• A finite division ring is a field. (Wedderburn's little theorem.) In fact, many important basic theorems in finite ring theory concern determining conditions under which a finite ring is commutative.
• Enumerating finite rings of a given order is an important (unsolved) problem.
• In fact, a finite semisimple ring is isomorphic to a direct product of matrix rings over finite fields. (A ring is semisimple if its Jacobson radical, $J(A)$, is trivial.) Therefore, if $A$ is a ring, then $A/J(A)$ is isomorphic to a direct product of matrix rings over finite fields. Note also that $J(A)$ is a nilpotent ideal.

Of course, there are many more important results about finite rings.