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His article discusses an algebraic structure known as a ring. For the geometric definition, see Annulus (mathematics), and for the set theory concept, Ring of sets. Chapter IX of David Hilbert's Die Theorie der algebraischen Zahlkörper delves into the world of number rings in field theory. The term "ring" is derived from the contraction of Zahlring. A ring forms a fundamental component within abstract algebra, being one of the foundational structures used by mathematicians. This structure consists of a set paired with two binary operations that generalize addition and multiplication, both of which are arithmetic operations. Through this broad generalization, mathematical principles based on number theory can be extended to objects such as polynomials, series, matrices, and functions. Mathematicians define a ring as an abelian group accompanied by a second binary operation that is associative and has the properties of being distributive over the group operation in question. This set also contains an identity element, though this requirement varies slightly among authors. In analogy with integer operations, the first operation within this structure is known as addition while the second is called multiplication. Commutative or noncommutative behavior has profound implications for rings and the properties of objects falling into each category vary in their applications. This understanding forms a basis of commutative ring theory commonly known as algebraic ring theory whose development has been heavily influenced by theoretical work on both number theory and geometry, notably through contributions that came about naturally within each field of study. Some examples include polynomials, integers equipped with arithmetic operations, the coordinate ring for any algebraic variety found in geometry, as well as rings that deal specifically with the integers present within an abstract number system. However non-commutative behavior exists in some types including groups represented through their ring and differential operators as well, and notably even when applying certain topological structures of study on particular mathematical objects. Mathematicians initially formalized these properties beginning from work completed in 1870 and finalized by the 1920s through contributions primarily provided by individuals like Noether and Hilbert, who generalized upon key theories that dealt with rings of variables existing within polynomials which served as a direct precursor for number field rings later applied within various studies across mathematics such as geometric or differential analysis.