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His article is about an algebraic structure. 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 deals with number rings. The chapter title translates to The Number Rings of the Field in English. In mathematics, a ring is an essential part of abstract algebra and one of its fundamental structures. It consists of a set with two binary operations that extend the basic arithmetic concepts of addition and multiplication. This generalization enables mathematical theorems from numbers to be applied to various non-numerical objects such as polynomials, series, matrices, and functions. A ring is essentially an abelian group paired with another operation. The combined structure must satisfy properties of associativity, being distributive over its primary operation, and have a defining element (although not every author requires this, consult the Notes on Definition for clarification). By extension from integers, this fundamental operation is referred to as addition, while the secondary operation is multiplication. Whether or not a ring behaves commutatively—whether switching the order of multiplying two elements impacts the result or not—has significant implications for its characteristics as an abstract entity. This reality makes Commutative Ring Theory (commonly known as Commutative Algebra) crucial in the field of Ring Theory, greatly influenced by challenges and concepts arising naturally in algebraic number theory and geometry. Some notable commutative rings include the set of integers, equipping addition and multiplication; polynomials with their addition and multiplication defined over them; the coordinate ring of an affine variety; and the integers from a number field. Non-commutative examples are more extensive and complex and cover square matrices of size n where n is two or greater, representation theory's group rings, operator algebras used in functional analysis, rings of differential operators in differential theories, and cohomology rings found in topological spaces. The foundational work for the understanding of rings dates back to the late 19th century. Key contributors such as Dedekind, Hilbert, Fraenkel, and Noether all significantly contributed over several decades until around the 1920s. Their early conceptualizations built upon mathematical generalizations seen in number theory with Dedekind domains, in geometry with polynomial rings, and in invariant theories. Rings proved to be foundational elements in advancing concepts within broader areas of mathematics including algebraic and analytic works.