In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, a finite ring is a
ring that has a finite number of elements.
Every
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
is an example of a finite ring, and the additive part of every finite ring is an example of an
abelian
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a grou ...
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or ma ...
, but the concept of finite rings in their own right has a more recent history.
Although rings have more structure than groups, the theory of finite rings is simpler than that of finite groups. For instance, the
classification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
was one of the major breakthroughs of 20th century mathematics, its proof spanning thousands of journal pages. On the other hand, it has been known since 1907 that any finite
simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a simpl ...
is isomorphic to the ring
of ''n''-by-''n'' matrices over a finite field of order ''q'' (as a consequence of Wedderburn's theorems, described below).
The number of rings with ''m'' elements, for ''m'' a natural number, is listed under in the
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to t ...
.
Finite field
The theory of
finite fields
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
is perhaps the most important aspect of finite ring theory due to its intimate connections with
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
,
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
and
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
. An important, but fairly old aspect of the theory is the classification of finite fields :
* The order or number of elements of a finite field equals ''p''
''n'', where ''p'' is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
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 ''p''
''n'' elements.
* Any two finite fields with the same order are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
.
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of n-dimensions over ''F''. The
matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...
''A'' of ''n'' × ''n'' matrices with elements from ''F'' is used in
Galois geometry
Galois geometry (so named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or ''Galois field''). More narrowly, ''a''  ...
, with the
projective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
serving as the
multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of ''A''.
Wedderburn's theorems
Wedderburn's little theorem In mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields.
The Artin–Zorn theorem generalizes the theorem to al ...
asserts that any finite
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
is necessarily commutative:
: If every nonzero element ''r'' of a finite ring ''R'' has a multiplicative inverse, then ''R'' is commutative (and therefore a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
).
Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician.
Biography
Born Nachman Arbiser in Warsaw, Jacobson emigrated to America with his family in 1918. He graduated from the University of Alabama in 1930 and was awar ...
later discovered yet another condition which guarantees commutativity of a ring: if for every element ''r'' of ''R'' there exists an integer such that , then ''R'' is commutative. More general conditions which guarantee commutativity of a ring are also known.
Yet another theorem by Wedderburn has, as its consequence, a result demonstrating that the theory of finite
simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.
The center of a simpl ...
s 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
Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a fi ...
established in 1905 and 1907 (one of which is Wedderburn's little theorem).
Enumeration
(Warning: the enumerations in this section include rings that do not necessarily have a multiplicative identity, sometimes called
rngs.) In 1964
David Singmaster
David Breyer Singmaster (born 1938) is an emeritus professor of mathematics at London South Bank University, England. A self-described metagrobologist, he has a huge personal collection of mechanical puzzles and books of brain teasers. He is mo ...
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
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
C
4 and eight rings over the
Klein four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one. ...
. There is an interesting display of the discriminatory tools (
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
s,
zero-divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
s,
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
s, and left- and right-identities) in Gregory Dresden's lecture notes.
The occurrence of ''
non-commutativity'' in finite rings was described in in two theorems: If the order m of a finite ring with 1 has a
cube-free factorization, then it is
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
. And if a
non-commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
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 and . Next Flor and Wessenbauer (1975) made improvements on the cube-of-a-prime case. Definitive work on the isomorphism classes came with proving that for ''p'' > 2, the number of classes is 3''p'' + 50.
There are earlier references in the topic of finite rings, such as Robert Ballieu and Scorza.
[Scorza (1935), see review of Ballieu by ]Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andr ...
in Mathematical Reviews
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):
*There are two finite rings of order ''p''.
*There are four finite rings of order ''pq''.
*There are eleven finite rings of order ''p''
2.
*There are twenty-two finite rings of order ''p''
2''q''.
*There are fifty-two finite rings of order eight.
*There are 3''p'' + 50 finite rings of order ''p''
3, ''p'' > 2.
The number of rings with ''n'' elements are (with )
:1, 1, 2, 2, 11, 2, 4, 2, 52, 11, 4, 2, 22, 2, 4, 4, 390, 2, 22, 2, 22, 4, 4, 2, 104, 11, 4, 59, 22, 2, 8, 2, >18590, 4, 4, 4, 121, 2, 4, 4, 104, 2, 8, 2, 22, 22, 4, 2, 780, 11, 22, ...
See also
*
Galois ring, finite commutative rings which generalize
and finite fields
*
Notes
References
*
*
* a research report of the work of 13 students and Prof. Sieler at a
Washington & Lee University
, mottoeng = "Not Unmindful of the Future"
, established =
, type = Private liberal arts university
, academic_affiliations =
, endowment = $2.092 billion (2021)
, president = William C. Dudley
, provost = Lena Hill
, city = Lexing ...
class in
Abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
(Math 322).
*
*
*
*
*
{{refend
External links
Classification of finite commutative rings
Algebraic combinatorics
Ring theory