What Is a Ring?
A ring is a mathematical structure containing elements that must have the following properties: addition must be commutative, multiplication must be associative, and there must be a unique element that functions as the zero. The elements of a ring may be numbers, such as integers, real numbers, and complex numbers, but they can also be non-numerical objects, such as polynomials, square matrices, and functions. Rings are used in algebra and number theory, and many problems in those fields can be formulated as relating rings to their elements.
In a more general context, the term ring can refer to a number of things:
A figurative ring is a circle of people or things. This can be an expression of affection, friendship, or kinship. A figurative ring can also be an association of values or attitudes. For example, a person who is openly gay might be described as having a “ring of honesty”. The word ring can also mean a musical elaboration on a theme or a set of melodic ideas. The idea of a ring is also a metaphor for something that encompasses and connects, such as the encompassing power of a mother’s love.
In mathematics, the ring is a fundamental concept in algebra. All of the familiar number systems—integers, rational numbers, real numbers, and complex numbers—are rings, as are the polynomial ring of a polynomial, the symmetric polynomial ring of a matrix, and the ordinary cohomology ring of a topological space. The study of rings has had important applications in algebraic number theory, algebraic geometry, and mathematical analysis.
Another use of the word ring is in reference to an engagement or marriage ring. It is a symbol of commitment and loyalty, often made of precious metal. A ring can also be worn to signify membership in an organization, or as a badge of office or rank. An academic ring, for example, is a gold band worn by university alumni and students, as well as by priests who earn their doctorates.
There are various definitions of ring in mathematical literature, with some authors using the term to refer only to a number-theoretic structure and other using it more broadly. Books published up to about 1960 followed Noether's convention of requiring 1 for a ring, but starting in that year some more advanced books began to use the word without this requirement. This change of convention has led to confusion among mathematicians about which of several possible definitions is the correct one. In order to avoid this confusion, mathematicians should always check that the definition they are using has a natural oo-categorical version. The more natural the oo-categorical definition of a ring, the easier it is to understand and extend.