Ring (mathematics)

Ring (mathematics)

In the branch of mathematics known as abstract algebra, a ring is an algebraic concept which abstracts and generalizes the algebraic structure of integers, specifically the two operations of addition and multiplication. The concept appears in most fields of mathematics, including geometry and mathematical analysis, just as with groups. It allows mathematicians to apply theorems in elementary algebra such as the fundamental theorem of algebra and Euclidean divisions to rings consisting of non-integer objects like functions in certain cases; or more importantly, it explains how such theorems cannot hold in generality. The abstract definition of rings is a relatively recent one; this is consistent with the tendency of modern mathematics to emphasize the study of abstract structures.

Briefly, a ring is an abelian group with a second binary operation that is associative and is distributive over the abelian group operation. The abelian group operation is called "addition" and the second binary operation is called "multiplication" in analogy with the integers. One familiar example of a ring is the set of integers. The integers are a commutative ring, since a times b is equal to b times a. The set of polynomials also forms a commutative ring. An example of a non-commutative ring is the ring of square matrices of the same size. Finally, a field (such as the set of complex numbers) is a commutative ring in which one can do "division" by any nonzero element.

The ring theory, which was firmly established during the 1920s by Emmy Noether and Wolfgang Krull, acquires a distinctly different flavor depending whether it allows rings to be commutative or not. The commutative ring theory, commonly known as commutative algebra, primarily concerns itself with problems and naturally-occurring ideas in algebraic number theory and algebraic geometry. Important commutative rings include the coordinate ring of an algebraic variety and the ring of integers in number theory. Today, the study of modules and the use of homological methods are central to commutative algebra. The noncommutative ring theory, besides the rich structure theory, includes the study of rings such as operator algebras and the ring of differential operators in analysis. Starting in the 1980s with the development of noncommutative geometry and with the discovery of quantum groups (which are a certain kind of a ring), mathematicians have attempted to construct some classes of noncommutative rings in a geometric fashion, as if they were rings of functions on (non-existent) 'noncommutative spaces'.

To some extent, the scope of the ring theory goes beyond rings; ring theorists might study an algebraic structure that satisfies different or weaker axioms for rings. Lie algebras, for instance, are non-associative rings that are especially important in theoretical physics. However, a ring may still be used to study those algebras; for example, the universal enveloping algebra (certain kind of a polynomial ring) of a Lie algebra plays a crucial role in the structural study of Lie algebras.

To those who already know rings: throughout the article, a commutative ring is assumed to be unital, while a noncommutative ring may not.