Mathematical ring

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Definition

In mathematics a ring is a commutative group (R, +), together with a second binary operation * such that for all a, b and c in R,

a * (b * c) = (a * b) * c
a * (b + c) = (a * b) + (a * c)
(a + b) * c = (a * c) + (b * c)

and such that there exists a multiplicative identity, or unity, that is, an element 1 such that for all a in R,

a * 1 = 1 * a = a.

Some authors omit the requirement for a multiplicative identity, and call those rings which do have multiplicative identities unitary rings. In this encyclopedia, the existence of a multiplicative identity is taken to be part of the definition.

Examples

  • The motivating example is the ring of integers with the two operations of addition and multiplication.
  • The rational, real and complex numbers form rings, in fact they are even fields.
  • If n is a positive integer, then the set Zn of integers modulo n forms a ring with n elements (see modular arithmetic).
  • The set of all continuous real-valued functions defined on the interval [a, b] forms a ring (even an associative algebra). The operations are addition and multiplication of functions.
  • The set of all polynomials over some common coefficient ring forms a ring.
  • The set of all square n-by-n matrices, where n is fixed, forms a ring with matrix addition and matrix multiplication as operations.
  • If G is an abelian group, then the endomorphisms form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms.


First consequences

From the axioms, one can immediately deduce that

0 * a = a * 0 = 0
(-1) * a = -a
(-a) * b = a * (-b) = -(a * b)

for all elements a and b in R. Here, 0 is the neutral element with respect to addition +, and -x stands for the additive inverse of the element x in R.

An element a in a ring is called invertible if there is an element b such that

a * b = b * a = 1.

If that is the case, b is uniquely determined by a and we write a-1 = b. The set of all invertible elements in a ring is closed under multiplication * and therefore forms a group, the group of units of the ring. If both a and b are invertible, then we have

(a * b)-1 = b-1 * a-1

Theory of rings

Rings that sit inside other rings are called subrings. Maps between rings which respect the ring operations are called ring homomorphisms. Rings, together with ring homomorphisms, form a category. Closely related is the notion of ideals, certain subsets of rings which arise as kernels of homomorphisms and can serve to define factor rings. Basic facts about ideals, homomorphisms and factor rings are recorded in the isomorphism theorems and in the chinese remainder theorem.

A ring is called commutative if its multiplication is commutative. The theory of commutative rings resembles the theory of numbers in several respects, and various definitions for commutative rings are designed to recover properties known from the integers. Commutative rings are also important in algebraic geometry. In commutative ring theory, numbers are often replaced by ideals, and the definition of prime ideal tries to capture the essense of prime numbers. integral domains, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility. Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers. Euclidean domains are integral domains in which the Euclidean algorithm can be carried out. Important examples of commutative rings can be constructed as rings of polynomials and their factor rings.

Non-commutative rings resemble rings of matrices in many respects. Following the model of algebraic geometry, attempts have been made recently at defining non-commutative geometry based on non-commutative rings. Non-commutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules. A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces. Examples of non-commutative rings are given by rings of endomorphisms of abelian groups or modules, and by monoid rings.

Generalizations

Any ring can be seen as an additive category with a single object. It is therefore natural to consider arbitrary additive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. Functors between additive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.


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