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Human: this article discusses a fundamental concept in algebraic structures. For information on geometric applications of the term ring, see Annulus (mathematics) or Ring of sets in set theory. chapter 9 of David Hilbert's book, Die Theorie der algebraischen Zahlkorper. The chapter is called Die Zahlringe des Korpers which means the number rings of a field. the term "ring" comes from the word zahlring which means number ring. in mathematics, a ring is one of several essential structures in abstract algebra. It's composed of a set with two operations that combine elements using addition and multiplication in an extended form. By doing this, basic arithmetic principles are applied to various mathematical objects such as polynomials, matrices and functions. a ring has properties of abelian groups combined with additional operations that must follow specific rules including associativity and distribution over group operation plus identity element requirements - not every source demands it though as stated in § notes section. Two binary operations used in a ring include addition and multiplication. rings may be either commutative or noncommutative, which has a significant impact on its overall characteristics as an abstract object. As such, commutative algebra is the key topic within ring theory due to various mathematical influences from number theory and geometry. An example of a commutative ring is any set of integers paired with addition and multiplication, while other noncommutive instances include matrices or polynomial rings. Examples are also found in functional analysis and topological spaces. conceptual development began during the 1870s and continued throughout the following decades by key figures including Hilbert, Dedekind, Noether and Fraenkel who sought to formalize mathematical ideas from different disciplines - these generalizations paved the way for ring applications across various branches such as algebraic number theory.