In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a semigroup under multiplication such that multiplication distributes over addition. In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article.
The branch of mathematics that studies rings is known as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less wellknown mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.
Ring theory may be used to understand fundamental physical laws, such as those 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 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s 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, betterunderstandable 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 toalgebraic 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 1980s by Alain Connes, noncommutative geometry has become a particularly active discipline in ring theory.
Example: the ring Z_{4}
Consider the set Z_{4} consisting of the numbers 0, 1, 2, 3 where addition and multiplication are defined as follows (note that for any integer x, x mod 4 is defined to be the remainder when xis divided by 4):
 For any x, y in Z_{4}, x + y is defined to be their sum in Z (the set of all integers) mod 4.
 For any x, y in Z_{4}, x ⋅ y is defined to be their product in Z (the set of all integers) mod 4.
Similarly, Z_{4} is closed under multiplication as the rightmost table shows (any result above is either 0, 1, 2 or 3). Associativity of multiplication in Z_{4} follows from associativity of multiplication in the set of all integers. The multiplicative identity is 1 as can be verified by looking at the rightmost table. Therefore,Z_{4} is a monoid under multiplication.
Distributivity of the two operations over each other follow from distributivity of addition over multiplication (and viceversa) in Z (the set of all integers).
Therefore, this set does indeed form a ring under the given operations of addition and multiplication.
Properties of this ring
 In general, given any two integers, x and y, if x ⋅ y = 0, then either x is 0 or y is 0. It is interesting to note that this does not hold for the ring (Z_{4}, +, ⋅):

 2 ⋅ 2 = 0
 although neither factor is 0. In general, a nonzero element a of a ring, (R, +, ⋅) is said to be a zero divisor in (R, +, ⋅), if there exists a nonzero element b of R such that a ⋅ b = 0. So in this ring, the only zero divisor is 2 (note that 0 ⋅ a = 0 for any a in a ring (R, +, ⋅) so 0 is not considered to be a zero divisor).
 A commutative ring which has no zero divisors is called an integral domain . So Z, the ring of all integers , is an integral domain (and therefore a ring), although Z_{4} (the above example) does not form an integral domain (but is still a ring). So in general, every integral domain is a ring but not every ring is an integral domain.
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