What Are Rings?


Originally formed as a generalization of Dedekind domains, rings have proved valuable for analysis and geometry. In addition, they generalize a wide range of mathematical objects including polynomials, matrices, and fields. Rings have been studied in commutative algebra and algebraic number theory. Similarly, rings have also been studied in functional analysis, operator algebras, and representation theory.

A ring is a mathematical structure with multiplicative identity. For example, if the ring is made up of integers, the ring’s multiplicative identity is the identity element 1. In addition, the ring is commutative. Ring multiplication is the process of summing all elements of the ring. In addition, ring multiplication must be associative and must include zero. Multiplications of rings with a multiplicative identity are called ring homomorphisms. Ring homomorphisms are also known as isomorphisms.

A ring is a commutative group that generalizes modular arithmetic and polynomials. The ring is also a group under addition. However, ring multiplication is not necessarily commutative. Rings are also generalized to include a commutative multiplication group, called a field. The elements of the commutative multiplication group are nonzero, such as rational, real, and complex numbers.

Rings are formed by a set of elements, called an object, and a set of morphisms. The morphisms are also called ideals. An ideal is a set of morphisms closed under addition. These ideals are also called ring homomorphisms. For example, in the R-module of an abelian group, the object is the ring R and the morphisms are the endomorphisms. The ring is also a topological ring. Specifically, the topology makes the addition map of the ring continuous. In addition, the ring is a monoid in Ab. In the simplest rings, the addition map restricts to the operations S x S – S. In this example, the addition map is (S, +).

In commutative algebra, rings are studied in relation to abelian groups. In particular, the ring of integers is a subring of the field of real numbers. This ring is also called the elliptic curve endomorphism ring. The characteristic zero of the ring is the smallest positive integer. The ring has the multiplicative identity n. In addition, ring multiplication can be continuous.

In addition to the commutative ring, there are also nonzero commutative rings. These rings have multiplicative inverses. In the case of the ring of integers, the ring’s inverse is the element a-1. The ring of integers is a topological ring that inherits Zariski topology and product topology. The ring of integers also inherits the elliptic curve endomorphism’s topology.

In functional analysis, group rings include operator algebras. In representation theory, ring of real square matrices with n>=2 is a group ring. Rings of differential operators are also included. In topology, the ring of integers of a number field is called the coordinate ring of affine algebraic variety. The ring of integers of a number space is called the ring of p-adic integers.

Rings are generally circular. However, they can be made of any hard material. They are worn as ornaments or used as symbols of authority or high status. Some rings have a religious meaning. For example, the episcopal ring is worn by higher-ranking priests. There are also rings of dope smugglers, illegal groups, and single turns in spirals. The term “ring” is usually used to refer to finger jewelry, but it may also refer to rings of any shape or size. Rings are also worn as a symbol of marriage.