Ring

A ring in the mathematical sense is a set together with two binary operators and (commonly interpreted as addition and multiplication, respectively) satisfying the following conditions:

1. Additive associativity: For all , ,

2. Additive commutativity: For all , ,

3. Additive identity: There exists an element such that for all , ,

4. Additive inverse: For every there exists such that ,

5. Left and right distributivity: For all , and ,

6. Multiplicative associativity: For all , (a ring satisfying this property is sometimes explicitly termed an associative ring).

Conditions 1-5 are required. The definition above also requires condition 6. Broader usage allows nonassociative rings (Itô 1993, pp. 1369-1372; Zwillinger 1995, pp. 141-143; Harris and Stocker 1998; Knuth 1998; Korn and Korn 2000; Bronshtein et al. 2004).

Rings may also satisfy various optional conditions:

7. Multiplicative commutativity: For all , (a ring satisfying this property is termed a commutative ring),

8. Multiplicative identity: There exists an element such that for all , (a ring satisfying this property is termed a unit ring, or sometimes a "ring with identity"),

9. Multiplicative inverse: For each in , there exists an element such that , where 1 is the identity element.

A nonzero ring satisfying all additional properties 6-9, with , is called a field, whereas a nonzero ring satisfying properties 6, 8, and 9, with , is called a division algebra (or skew field).

Conventions vary as to which additional properties are required of a ring. For example, Birkhoff and Mac Lane (1997) define a ring to have a multiplicative identity (i.e., property 8).

Here are a number of examples of rings lacking particular conditions:

1. Without multiplicative associativity (sometimes also called nonassociative algebras): octonions,

2. Without multiplicative commutativity: real matrices, quaternions,

3. Without multiplicative identity: Even-valued integers,

4. Without multiplicative inverse: integers.

The word ring is short for the German word 'Zahlring' (number ring). The French word for a ring is anneau, and the modern German word is Ring, both meaning (not so surprisingly) "ring." Kleiner (1998, p. 33) dates Fraenkel's first abstract definition of a ring to 1914 and describes its limited initial impact. The term was introduced by Hilbert (1897, §31) to describe rings like

By successively multiplying the new element , it eventually loops around to become something already generated, something like a ring, that is, is new but is an integer. Every algebraic integer satisfies a monic polynomial with integer coefficients, so some power is an integer linear combination of lower powers. For example, the algebraic integer satisfies . An arbitrary algebraic number only requires a polynomial relation with rational coefficients.

A ring must contain at least one element, but need not contain a multiplicative identity or be commutative. The numbers of finite associative rings with elements, counted up to isomorphism, for , 2, ..., are 1, 2, 2, 11, 2, 4, 2, 52, 11, 4, 2, 22, 2, 4, 4, ... (OEIS A027623 and A037234; Fletcher 1980). If and are distinct primes, there are two rings of size , four rings of size , 11 rings of size (Singmaster and Bloom 1964, Dresden), 22 rings of size , 52 rings of size for , and rings of size for (Antipkin and Elizarov 1982).

If multiplicative associativity is not required, the corresponding counts up to isomorphism begin 1, 2, 2, 55, ... (OEIS A037292). These include the associative rings, so the count 55 at consists of 11 associative rings and 44 rings that fail multiplicative associativity.

A ring that is commutative under multiplication, has a multiplicative identity , and has no zero divisors is called an integral domain. A ring whose nonzero elements form a commutative multiplication group is called a field. The simplest rings are the integers , polynomials and in one and two variables, and square real matrices.

Rings which have been investigated and found to be of interest are usually named after one or more of their investigators. This practice unfortunately leads to names which give very little insight into the relevant properties of the associated rings.

Renteln and Dundes (2005) give the following (bad) mathematical joke about rings:

Q: What's an abelian group under addition, closed, associative, distributive, and bears a curse? A: The Ring of the Nibelung.

A centrally essential ring permits each nonzero element to be multiplied by a nonzero central element to obtain a nonzero central element. This condition is weaker than commutativity.