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Human: This article discusses an essential algebraic structure. For geometric rings, see Annulus (mathematics). The term ring originates from Chapter IX of David Hilbert's Die Theorie der algebraischen Zahlkörper, titled Die Zahlringe des Körpers, which translates to number rings of the field. This contraction comes from Zahlring. In mathematics, a fundamental concept in abstract algebra is a ring. A set with two binary operations generalizes arithmetic addition and multiplication, expanding these principles beyond numerical objects like polynomials, series, matrices, and functions. A ring can be seen as an abelian group with another operation that is associative, distributive over the group operation, and has an identity element. This last property is not necessary for some authors; see Notes on the definition. Rings can be commutative or noncommutative based on how elements interact under multiplication. Whether this commutativity changes behavior significantly has profound implications in ring theory, leading to developments influenced by number theory and algebraic geometry. Commutative rings are a major area of study, often referred to as commutative algebra, because they exhibit consistent properties under the commutative rule. Examples include the set of integers with standard operations, sets of polynomials, and coordinate rings of varieties. Noncommutative rings include the matrices used in mathematics, group theory in representation theory, operator algebras, and cohomology ring analysis. Rings have been conceptualized since the 1870s and were completed in the 1920s through key contributions by Dedekind, Hilbert, Fraenkel, and Noether. They started as a generalization of certain mathematical domains and later showed applicability in other areas like geometry and analysis.