A ring is a structure of abstract algebra, similar to a group or a field. A ring is a set of elements closed under two operations, usually called multiplication and addition and denoted and , for which
In other words, the following properties hold for all in :
- (associativity of addition);
- (commutativity of addition);
- For some , (existance of additive identity);
- There exists some for which (existance of additive inverses);
- (associativity of multiplication);
- For some , (existance of multiplicative identity)
- (double distributivity of multiplication over addition).
- (double distributivity of multiplication over subtraction).
Note especially that multiplicative inverses need not exist and that multiplication need not be commutative.
The elements of under addition is called the additive group of ; it is sometimes denoted . (However, this can sometimes lead to confusion when is also an ordered set.) The set of invertible elements of constitute a group under multiplication, denoted . The elements of under the multiplicative law (i.e., the opposite multiplicative law) and the same additive law constitute the opposite ring of , which can be denoted .
Let and be elements of a ring . If there exists an element of such that , then is said to be a right divisor of , and is said to be a left multiple of . Left divisors and right multiples are defined similarly. When is commutative, we say simply that is a divisor of , or divides , or is a multiple of .
Note that the relation " is a right divisor of " is transitive, for if and , then . Furthermore, every element of is a right divisor of itself. Therefore has the (sometimes trivial) structure of a partially ordered set.
Under these definitions, every element of is a left and right divisor of 0. However, by abuse of language, we usually only call an element a left (or right) divisor of zero (or left, right zero divisors) if there is a non-zero element for which (or ). The left zero divisors are precisely those elements of for which left multiplication is not cancellable. For if are distinct elements of for which , then .
Examples of Rings
The sets of integers (), rational numbers (), real numbers (), and complex numbers () are all examples of commutative rings, as is the set of Gaussian integers (). Note that of these, the integers and Gaussian integers do not have inverses; the rest do, and therefore also constitute examples of fields. All these rings are infinite, as well.
Among the finite commutative rings are sets of integers mod (), for any integer .
If is an abelian group, then the set of endomorphisms on form a ring, under the rules
Let be a ring. The set of polynomials in is also a ring.
Let be a field. The set of matrices of constitute a ring. In fact, they are the endomorphism ring of the additive group .
Let be the set of weak multiplicative functions mapping the positive integers into themselves. Then the elements of form a pseudo-ring, with multiplication defined as Dirichlet convolution, i.e., for However, there is no multiplicative identity, so this is not a proper ring.