In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent history.
The number of rings with m elements, for m a natural number, is listed under in the On-Line Encyclopedia of Integer Sequences.
The theory of finite fields is perhaps the most important aspect of finite ring theory due to its intimate connections with algebraic geometry, Galois theory and number theory. An important, but fairly old aspect of the theory is the classification of finite fields (Jacobson 1985, p. 287):
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.
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. More general conditions which guarantee commutativity of a ring are also known.
(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? Find two such rings with this minimal order. Are there more? (2) How many rings of order four are there?" One can find the solution by D.M. Bloom in a two-page proof that there are eleven rings of order 4, four of which have a multiplicative identity. Indeed, four-element rings introduce the complexity of the subject. There are three rings over the cyclic group C4 and eight rings over the Klein four-group. 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 occurrence of non-commutativity in finite rings was described in (Eldrige 1968) in two theorems: If the order m of a finite ring with 1 has a cube-free factorization, then it is commutative. And if a non-commutative finite ring with 1 has the order of a prime cubed, then the ring is isomorphic to the upper triangular 2 × 2 matrix ring over the Galois field of the prime. The study of rings of order the cube of a prime was further developed in (Raghavendran 1969) and (Gilmer & Mott 1973). Next Flor and Wessenbauer (1975) made improvements on the cube-of-a-prime case. Definitive work on the isomorphism classes came with (Antipkin & Elizarov 1982) proving that for p > 2, the number of classes is 3p + 50.
These are a few of the facts that are known about the number of finite rings (not necessarily with unity) of a given order (suppose p and q represent distinct prime numbers):