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Human: This article is about an algebraic structure called a ring. For the geometric concept, see Annulus (mathematics). For set theory, see Ring of sets. Chapter IX of David Hilbert's Die Theorie der Algebraischen Zahlkorper defines a "Zahlring", which literally translates to number ring of a field. The term ring is a contraction of "Zahlring". A ring in mathematics is an essential algebraic structure used extensively in abstract algebra. It consists of a set with two binary operations that generalize the arithmetic operations of addition and multiplication, enabling extension of mathematical theorems from numerical objects like polynomials, series, matrices, and functions. The term "ring" originated from Chapter IX of David Hilbert's Die Theorie der Algebraischen Zahlkorper. In mathematics, it is one of the primary algebraic structures utilized in abstract algebra. A ring is an abelian group with a second binary operation that is associative, distributive over the abelian group operation, and has an identity element, although this property isn't required by some authors, as stated in § Notes on definition. By extension from the integers, the abelian group operation is known as addition and the second binary operation as multiplication. The order of these operations has significant implications for a ring's behavior as an abstract object. As a result, commutative ring theory, commonly referred to as commutative algebra, plays a vital role in ring theory. Its development has been greatly influenced by problems and ideas arising naturally from algebraic number theory and geometry. Examples of commutative rings include the set of integers equipped with addition and multiplication operations, sets of polynomials, coordinate rings of an affine algebraic variety, and rings of integers of a number field. Examples of noncommutative rings comprise the ring of n x n real square matrices for n >= 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings of topological spaces. The conceptualization of rings began in the 1870s and was completed by the 1920s. Key contributors to this development were Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains in number theory and polynomial rings in algebraic geometry.