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Human: his article is about an algebraic structure used extensively in mathematics, referred to as a ring. For the concept applied to geometric rings, see Annulus (mathematics). The set theory term for a related idea is Ring of sets.\n\nA crucial part of David Hilbert's book Die Theorie der algebraischen Zahlkorper is Chapter IX. The chapter title translates literally from German as The Number Rings of the Field, where 'ring' comes from the contraction Zahlring.\nIn abstract algebra, one of the core building blocks used to construct mathematical structures is a ring. It is composed of a set along with two binary operations that generalize arithmetic operations like addition and multiplication.\nThe generalization enables the extension of theorems based on simple arithmetic into more complex objects like polynomials, series, matrices and functions in abstract algebra.\n\nA ring can be thought of as an abelian group, but with an additional binary operation. This new operation must associate itself within the set when multiplied together by two other operations that associate within themselves. Also, multiplication needs to be distributive over the set's associative group, as well as containing an identity element.\nRings are a key aspect in commutative algebra, or more formally referred to as ring theory where order of multiplication makes no difference in its result.\nThe influence of problems in algebraic number theory and geometry significantly impacted how these rings operate in reality. Common examples include set of integers combined with their normal operations, set of polynomials equipped with addition and their product operation.\nOn the other hand, sets such as matrix representations for 2 by 2 real square numbers and cohomology groups make great use of noncommutative structures.\n\nThis foundational work of creating this particular field occurred in two different waves. Initially started from mid-nineteenth century but took several decades before becoming complete, these groundbreaking mathematical ideas emerged during the 1870s through to completion by mathematicians of twentieth century - namely those by Dedekind, Hilbert and their followers like Fraenkel who laid foundations using concepts derived from polynomial rings; afterward going further towards application.