Finite Rings - Enumeration

Enumeration

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 (71:919–20) 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 (see reference).

The occasion of non-commutativity in finite rings was described in 1968 in the same journal (75:512–14) by K. Eldrige 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 by R. Raghavendran in 1969 (Compositio Mathematica 21:195–229). In 1973 the Proceedings of the Japan Academy 49:795–9 published Robert Gilmer and Joe Mott’s paper "Associative rings of order p3". Next Flor and Wessenbauer (1975) made improvements on the cube-of-a-prime case. Definitive work on the isomorphism classes came with V.G. Antipkin and V.P Elizarov (1982) writing in the Siberian Mathematical Journal (23:457–64). They prove that for p > 2, the number of classes is 3p + 50.

There are earlier references in the topic of finite rings, such as Robert Ballieu (1947) "Anneaux finis" in Ann. Soc. Sci Bruxelles (61:222–227). Earlier work by Scorza (1935) is noted by Irving Kaplansky in his review (MR0022841) of Ballieu.

These are a few of the facts that are known about the number of finite rings of a given order (suppose p and q represent distinct prime numbers):

• There are two finite rings of order p.
• There are four finite rings of order pq.
• There are eleven finite rings of order p2.
• There are twenty-two finite rings of order p2q.
• There are fifty-two finite rings of order eight.
• There are 3p + 50 finite rings of order p3, p > 2.

The number of rings with n elements is listed under  A027623 in the On-Line Encyclopedia of Integer Sequences.