Finite ring

In mathematics, more specifically abstract algebra, a finite ring is a ring (not necessarily with a multiplicative identity) 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.

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, three 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 trianglar 2 × 2 matrix ring over the Galois field of the prime. The science of rings of order the cube of a prime was further developed by R. Raghavendra 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 facts are known about the number of finite rings of a given order (suppose p and q represent prime numbers):

• 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.

Wedderburn's theorems

There are other deep aspects to the theory of finite rings, apart from mere enumeration. For instance, a remarkable theorem by Joseph Wedderburn, known as Wedderburn's little theorem, asserts that any finite division ring is necessarily commutative (and therefore a finite field). 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 rⁿ = r, then R is commutative.^[1]

If, r² = r for every r, the ring is called a Boolean ring. More general conditions which guarantee commutativity of a ring are also known.^[2]

Yet another theorem by Wedderburn has, as its consequence, a result demonstrating that the theory of finite simple rings is relatively straightforward in nature. More specifically, any finite simple ring is isomorphic to the ring of n by n matrices over a finite field of order q. This follows from two theorems of Joseph Wedderburn established in 1905 and 1907 (one of which is Wedderburn's little theorem). On the other hand, the classification of finite simple groups was one of the major breakthroughs of twentieth century mathematics, its proof spanning thousands of journal pages. Therefore, in some respects, the theory of finite rings is simpler than that of finite groups.

Finite field

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):

• The order or number of elements of a finite field equals pⁿ, where p is a prime number called the characteristic of the field, and n is a positive integer.
• For every prime number p and positive integer n, there exists a finite field with pⁿ elements.
• Any two finite fields with the same order are isomorphic.

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).

Signal coding

Since 1960 when Irving S. Reed and Gustave Solomon published “Polynomial codes over certain finite fields” the finite ring concept has been at play in algebraic coding theory. If K is a finite field, the polynomial ring K[X] is infinite since it contains Xⁿ for all n > 0. However, if a polynomial f(x) is used to form the quotient ring,Q = K[X]/(f(X)), then Q has a finite number of elements. Reed and Solomon credit their referee with noting the correspondence of their framework and "the ring of polynomials in x with coefficients in Z₂, modulo the prime ideal generated by the irreducible f(x)."^[3] An introductory approach to this application of finite rings is available in John Baylis (1998) Error Correcting Codes.^[4]. In particular, see chapter 7 "Polynomials for codes", section 7.3 "Rings and ideals", and chapter 8 "Cyclic codes".

Notes

1. ^ Jacobson 1945
2. ^ Pinter-Lucke 2007
3. ^ Journal of the Society for Industrial and Applied Mathematics 8:300–4.
4. ^ Chapman & Hall

References

• Gregory Dresden (2005) Small Rings, a research report of the work of 13 students and Prof. Sieler at a Washington & Lee University class in Abstract algebra (Math 322).
• Gregory Dresden (2005) Rings with four elements.
• Bernard A. McDonald (1974) Finite Rings with Identity, Marcel Dekker ISBN 0824761618.
• G Bini & F Flamini (2002) Finite commutative rings and their applications, ISBN 9781402070396.