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Human: This article is about an algebraic structure called a ring. For geometric rings, see Annulus (mathematics). For the set theory concept, see Ring of sets. Chapter IX of David Hilbert's Die Theorie der algebraischen Zahlkorper book is about number rings. The chapter title translates to "Number Rings of the Field." The term ring comes from the contraction of Zahlring, which means "number ring." In mathematics, a fundamental algebraic structure used in abstract algebra is a ring. It consists of a set with two binary operations that generalize addition and multiplication. These operations allow us to extend arithmetic theorems to non-numerical objects like polynomials, series, matrices, and functions. A ring is an abelian group with a second binary operation. This operation is associative, distributive over the abelian group operation, and has an identity element. Some authors require an identity element, but others do not. The extension of integers implies that the abelian group operation is called addition, and the second binary operation is called multiplication. Whether or not a ring is commutative determines its behavior as an abstract object. This makes commutative ring theory, also known as commutative algebra, a key topic in ring theory. Its development was influenced by problems occurring naturally in number theory and geometry. Examples of commutative rings include integers with addition and multiplication operations, polynomials with their addition and multiplication operations, coordinate rings of affine varieties, and the set of integers of a number field. Examples of noncommutative rings include n-by-n matrices with real numbers when n is 2 or more, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, and cohomology rings of topological spaces. Rings were first conceptualized in the 1870s. By the 1920s, their formalization as a generalization of Dedekind domains was completed. They also turned out to be useful in other branches of mathematics like geometry and mathematical analysis. Key contributors included Dedekind, Hilbert, Fraenkel, and Noether.