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Human: his article is about a fundamental concept in abstract algebra. For the geometric context, see Annulus. The set theory concept is known as Ring of sets.\n\nChapter IX of David Hilbert's Die Theorie der algebraischen Zahlkorper outlines the definition of rings in number fields. The chapter title is Die Zahlringe des Korpers, which translates to the number rings of a field. In German, Zahlring is the word for ring.\nIn mathematics, a ring is one of the primary structures used in abstract algebra. It consists of a set equipped with two operations that generalize arithmetic's addition and multiplication. Through this generalization, fundamental theorems from arithmetic are applied to non-numerical objects like polynomials, series, matrices, and functions.\n\nA ring forms an abelian group under one operation and features a second binary operation that is associative, distributes over the first group operation, and possesses an identity element. This definition is accepted by most authors, except for those in § Notes on the definition who relax this property.\nIn many cases, extending the definition of integers implies calling the first binary operation addition and the second operation multiplication.\nThe question of whether a ring's multiplication order affects its properties profoundly impacts its behavior as an abstract entity. As a result, commutative ring theory, commonly known as commutative algebra, plays a crucial role in ring study. The theory has been heavily influenced by challenges and ideas that naturally arise from number theory and algebraic geometry.\nCommutative rings include the set of integers under standard addition and multiplication operations, polynomials with their standard addition and multiplication, the coordinate ring of an affine variety, and the ring of a field's integers. Noncommutative rings encompass matrices greater than 2x2 real squares, group rings in representation theory, operator algebras in functional analysis, differential operators in topological spaces, and cohomology rings in topology.\n\nRings as a concept began to take shape around the 1870s, reaching its final form by the 1920s. Significant contributors include Dedekind, Hilbert, Fraenkel, and Noether. Initially formalized to generalize Dedekind domains in number theory and polynomial rings and invariant theory, they eventually proved valuable in various branches of mathematics such as geometry and mathematical analysis.\nRings served initially to represent a type of abelian group that supported the application of operations on its members under multiplication which was found out useful over several instances such like in matrices with 2 dimensions real numbers or those used by algebraists studying geometric configurations, rings became especially crucial as soon they are related through commutativity.