For instance, the classification of finite simple groups was one of the major breakthroughs of 20th century mathematics, its proof spanning thousands of journal pages.
An important, but fairly old aspect of the theory is the classification of finite fields:[1] Despite the classification, finite fields are still an active area of research, including recent results on the Kakeya conjecture and open problems regarding the size of smallest primitive roots (in number theory).
A finite field F may be used to build a vector space of n-dimensions over F. The matrix ring A of n × n matrices with elements from F is used in Galois geometry, with the projective linear group serving as the multiplicative group of A. Wedderburn's little theorem asserts that any finite division ring is necessarily commutative: Nathan Jacobson later discovered yet another condition which guarantees commutativity of a ring: if for every element r of R there exists an integer n > 1 such that r n = r, then R is commutative.
[3] Yet another theorem by Wedderburn has, as its consequence, a result demonstrating that the theory of finite simple rings is relatively straightforward in nature.
(Warning: the enumerations in this section include rings that do not necessarily have a multiplicative identity, sometimes called rngs.)
In 1964 David Singmaster proposed the following problem in the American Mathematical Monthly: "(1) What is the order of the smallest non-trivial ring with identity which is not a field?
There is an interesting display of the discriminatory tools (nilpotents, zero-divisors, idempotents, and left- and right-identities) in Gregory Dresden's lecture notes.
The study of rings of order the cube of a prime was further developed in (Raghavendran 1969) and (Gilmer & Mott 1973).