Matches in DBpedia 2014 for { ?s ?p In mathematics, and more specifically in algebra, a ring is an algebraic structure with operations generalizing the arithmetic operations of addition and multiplication. By means of 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 is relatively recent, dating from the 1920s.Briefly, a ring is an abelian group with a second binary operation that is distributive over the abelian group operation and is associative. 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 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 in the study of rings as abstract objects, the field called the ring theory. The development of the commutative 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.). }
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- Ring_(mathematics) abstract "In mathematics, and more specifically in algebra, a ring is an algebraic structure with operations generalizing the arithmetic operations of addition and multiplication. By means of 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 is relatively recent, dating from the 1920s.Briefly, a ring is an abelian group with a second binary operation that is distributive over the abelian group operation and is associative. 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 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 in the study of rings as abstract objects, the field called the ring theory. The development of the commutative 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.)".