|
In mathematics, and more specifically in algebra, a ring is an algebraic structure with operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects like polynomials, series, matrices and functions. Rings were first formalized as a common generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. They are also used in other branches of mathematics such as geometry and mathematical analysis. The formal definition of rings dates from the 1920s. Briefly, a ring is an abelian group with a second binary operation that is associative, is distributive over the abelian group operation and has an identity element (or in other words, it is a monoid with the second operation that is also distributive). The abelian group operation is called ''addition'' and the second binary operation is called ''multiplication'' by extension from the integers. A familiar example of a ring is the integers. The integers form a commutative ring, since the order in which a pair of elements are multiplied does not change the result. The set of polynomials also forms a commutative ring with the usual operations of addition and multiplication of functions. An example of a ring that is not commutative is the ring of ''n'' × ''n'' real square matrices with ''n'' ≥ 2. Finally, a field is a commutative ring in which one can divide by any nonzero element: an example is the field of real numbers. Whether a ring is commutative or not has profound implication on its behaviour as an abstract object, and the study of such rings is a topic in ring theory. The development of the commutative ring theory, commonly known as commutative algebra, has been greatly influenced by problems and ideas occurring naturally in algebraic number theory and algebraic geometry; important commutative rings include fields, polynomial rings, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. On the other hand, the noncommutative theory takes examples from representation theory (group rings), functional analysis (operator algebras) and the theory of differential operators (rings of differential operators), and the topology (cohomology ring of a topological space). == Definition and illustration == The most familiar example of a ring is the set of all integers, Z, consisting of the numbers :. . . , −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, . . . The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ring (mathematics)」の詳細全文を読む スポンサード リンク
|