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A recent article centers on an algebraic structure. For the geometric ring concept, see Annulus (mathematics). The set theory concept known as Ring of sets is also worthy of note.\n\nDavid Hilbert's groundbreaking work "Die Theorie der algebraischen Zahlkörper" contains Chapter IX. This chapter is titled "Die Zahlringe des Körpers," or more literally "the number rings of the field." The word "ring" originates from the contraction of "Zahlring."\nIn mathematics, a fundamental algebraic structure called a ring exists within abstract algebra. A ring comprises a set with two binary operations that generalize addition and multiplication in arithmetic. By doing so, mathematical principles are applied to non-numerical entities such as polynomials, series, matrices, and functions.\n\nA ring is an abelian group that incorporates a second binary operation. This second operation is associative, is distributive over the abelian group operation, and possesses an identity element (a property not all authors require; see notes for further information). Through extension from the integers, the abelian group operation is named addition, while the second binary operation is termed multiplication.\n\nWhether a ring behaves commutatively or otherwise has significant implications for its nature as an abstract object. Consequently, research into commutative rings forms a crucial area of study within ring theory, also known as commutative algebra. Its development has been significantly influenced by problems and concepts encountered naturally in algebraic number theory and geometry.\nExamples of commutative rings include the integers along with their addition and multiplication operations, polynomials accompanied by their own operations, the coordinate ring of an affine variety, and the integers present within a number field. Examples of noncommutative rings include square matrices greater than or equal to 2, group rings in representation theory, operator algebras in analysis, rings of differential operators in differential operator theory, and the cohomology ring present in topological spaces.\n\nThe notion of rings first arose in the 1870s, gaining further form in the following decades. Dedekind, Hilbert, Fraenkel, and Noether have significantly contributed to this area of study. Rings initially came about as a generalization of Dedekind domains that appear in number theory, along with polynomial and invariant rings. As research continued, it proved they could also serve useful purposes within geometry, mathematical analysis, and other branches of mathematics.