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Human: his article is about a foundational algebraic structure. For geometric structures analogous to rings, see Annulus (mathematics). For the set theory concept that mirrors the idea of a ring, see Ring of sets. Chapter IX of David Hilbert's The Theory of Algebraic Number Fields lays out the framework for understanding rings. In mathematics, a ring is an integral component of abstract algebra used across various fields. It comprises a set enhanced with two binary operations that encapsulate and generalize arithmetic processes of addition and multiplication, ultimately broadening theoretical reach to encompass non-numerical objects such as polynomials, series, matrices, and functions. A ring manifests as an abelian group complemented by another binary operation characterized by being associative, distributive over the primary abelian group operation, and possessing an identity element. By an extension of principles from integer arithmetic, this binary group operation is defined as addition, while the second binary operation is recognized as multiplication. Whether a ring functions in accordance with commutative laws (determining whether the order in which two elements are combined affects the outcome or not) profoundly impacts its nature as an abstract object. Therefore, studying commutative rings – referred to as commutative algebra – is pivotal within ring theory. Its progression has been significantly influenced by challenges and theories naturally arising from considerations of number fields and algebraic geometry. Notable instances of commutative structures include integers enhanced with the basic arithmetic processes, collections of polynomials complemented with their standard processes for combination, affine algebraic variety coordinate rings, and rings composed of a set's integral values for number fields. Contrastingly, certain sets defy the principle of being commutative, resulting in the emergence of noncommutative ring examples that extend across mathematical branches including representation theory's group rings, operator algebras from functional analysis, rings consisting of differential operators utilized within their domain of theoretical examination, and a topological space's cohomology ring used extensively throughout.