In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations (usually called addition and multiplication), where each operation combines two elements to form a third element. To qualify as a ring, the set together with its two operations must satisfy certain conditions—namely, the set must be an abelian group under addition and a monoid under multiplication such that multiplication distributes over addition. While these operations are familiar from many mathematical structures, such as number systems or the integers—for example, they are also very general in the sense that they take a broad variety of mathematical objects into account. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of rings makes them a central organizing principle of contemporary mathematics. The branch of mathematics that studies rings is known as ring theory.
Rings share a fundamental kinship with number theory and linear algebra. The former is responsible for various analogues of number-theoretic theorems in ring theory—for instance, the fundamental theorem of arithmetic translates to a certain special class of rings known as unique factorization domains. The latter is responsible for the rich theory of algebras, including, but not restricted to, matrix rings. This theory of matrix rings, for example, is a striking consequence of the ways in which noncommutative ring theory may be used to understand fundamental physical laws underlying special relativity and symmetry phenomena in molecular chemistry.
The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880's. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920's by Emmy Noether and Wolfgang Krull. Modern ring theory—a very active mathematical discipline—studies rings in their own right. To explore rings, mathematicians have devised various notions to break rings into smaller, better-understandable pieces, such as ideals, quotient rings and simple rings. In addition to these abstract properties, ring theorists also make various distinctions between the theory of commutative rings and noncommutative rings—the former belonging to algebraic number theory and algebraic geometry. A particularly rich theory has been developed for a certain special class of commutative rings, known as fields, which lies within the realm of field theory. Likewise, the corresponding theory for noncommutative rings, that of noncommutative division rings, constitutes an active research interest for noncommutative ring theorists. Since the discovery of a mysterious connection between noncommutative ring theory and geometry during the 1980's by Alain Connes, noncommutative geometry has become a particularly active discipline in ring theory.
Definition and illustration
First example: the integers
The most familiar example of a ring is the set of all integers, Z , consisting of the numbers
together with the usual operations of addition and multiplication. These operations satisfy the following properties:
- The integers form an abelian group under addition; that is:
- Closure axiom for addition: Given two integers a and b , their sum, a + b is also an integer.
- Existence of additive identity: For any integer a , a + 0 = 0 + a = a . Zero is called the identity element of the integers because adding 0 to any integer (in any order) returns the same integer.
- Commutativity of addition: For any two integers a and b , a + b = b + a . So the order in which two integers are added is irrelevant.
- Associativity of addition: For any integers, a , b and c , ( a + b ) + c = a + ( b + c ). So, adding b to a , and then adding c to this result, is the same as adding c to b , and then adding this result to a .
- Existence of additive inverse: For any integer a , there exists an integer denoted by − a such that a + (− a ) = (− a ) + a = 0. The element, − a , is called the additive inverse of a because adding a to − a (in any order) returns the identity.
- The integers form a multiplicative monoid (a monoid under multiplication); that is:
- Closure axiom for multiplication: Given two integers a and b , their product, a · b is also an integer.
- Associativity of multiplication: Given any integers, a , b and c , ( a · b ) · c = a · ( b · c ). So multiplying b with a , and then multiplying c to this result, is the same as multiplying c with b , and then multiplying a to this result.
- Existence of multiplicative identity: For any integer a , a · 1 = 1 · a = a . So multiplying any integer with 1 (in any order) gives back that integer. One is therefore called the multiplicative identity.
- Multiplication is distributive over addition : These two structures on the integers (addition and multiplication) are compatible in the sense that
- a · ( b + c ) = ( a · b ) + ( a · c ), and
- ( a + b ) · c = ( a · c ) + ( b · c )
Formal definition
There are some differences in exactly what axioms are used to define a ring. Here one set of axioms is given, and comments on variations follow.
A ring is a set R equipped with two binary operations + : R × R → R and · : R × R → R (where × denotes the Cartesian product), called addition and multiplication . To qualify as a ring, the set and two operations, ( R , +, · ), must satisfy the following requirements known as the ring axioms .
- ( R , +, · ) is required to be an abelian group under addition:
- ( R , +, · ) is required to be a monoid under multiplication:
- The distributive laws:
This definition assumes that a binary operation on R is a function defined on R × R with values in R . Therefore, for any a and b in R , the addition a + b and the product a · b are elements of R .The most familiar example of a ring is the set of all integers, Z = {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... }, together with the usual operations of addition and multiplication.
Another familiar example is the set of real numbers R , equipped with the usual addition and multiplication.
Another example of ring is the set of all square matrices of a fixed size, with real elements, using the matrix addition and multiplication of linear algebra. In this case, the ring elements 0 and 1 are the zero matrix (with all entries equal to 0) and the identity matrix, respectively.
Notes on the definition
As with some axiomatic theories, there are often differences of usage in what axioms a ring should satisfy. Sometimes the disagreement between two definitions is minor. For instance, some authors insist that 1 ≠ 0 in a ring (in words, this means that the multiplicative identity of the ring must be different from its additive identity). In particular they do not consider the trivial ring to be a ring (see below).
A more significant disagreement is that some authors omit the existence of a multiplicative identity in a ring . For instance, this would allow the even integers to form a ring with the natural operations of addition and multiplication (all ring axioms are satisfied except for the existence of a multiplicative identity). Rings that satisfy the ring axioms as given above but do not contain a multiplicative identity are sometimes called pseudo-rings. The term rng (jocular; ring without the multiplicative i dentity) is also used for such rings. Rings which do have multiplicative identities (and also satisfy the above axioms) are sometimes referred to unital rings , unitary rings , rings with unity , rings with identity or rings with 1 . Note that one can always embed a non-unitary ring inside a unitary ring (see this for one particular construction of this embedding).
There are still other more significant differences between two particular definitions of a ring. For instance, some authors omit associativity of multiplication in the set of ring axioms;
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