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Human: This article is about a type of algebraic structure. For geometric structures with this name, see Annulus (mathematics). For the concept from set theory, see Ring of sets. Chapter IX of David Hilbert's book Die Theorie der algebraischen Zahlkorper introduces number rings in the context of fields. The term "ring" comes from the German word Zahlring, which translates to number ring. In mathematics, a fundamental algebraic structure is known as a ring. It consists of a set with two binary operations that generalize basic arithmetic concepts like addition and multiplication. This extension of mathematical theories has led to broader applications in areas beyond traditional numbers. A ring can be considered an abelian group with another binary operation, which satisfies certain conditions such as being associative and distributive over the first operation. A critical element for rings is the presence of an identity element; however, not all definitions include this property (see § Notes on the definition). Borrowing from properties of integers, we define one operation as addition while the other is multiplication. Whether a ring follows commutative or noncommutative laws profoundly influences its characteristics and uses within abstract algebra. This understanding drives much research into commutative algebra. Researchers working in areas like algebraic number theory and algebraic geometry often find problems or insights in commutative rings that relate closely to these topics. For example, the integers themselves form a ring with addition and multiplication operations. Polynomials, when considering their addition and multiplication, also exhibit this behavior. Another branch of ring theory involves exploring commutative rings versus noncommutative ones. Examples of the latter include matrix algebra for n ? n real square matrices where n ? 2; in representation theory, group rings appear frequently; operator algebras show up often in functional analysis and are crucial in examining properties like spectrum. Topologists examine rings as a means to explore geometric spaces, finding commutative structures within differential operators. Meanwhile, researchers working with topological spaces observe that noncommutative properties lead them to further studies on their cohomology rings. Researchers made groundbreaking contributions to the field starting in the late 19th century and completed it by the early 20th century, paving the way for its development across various fields including geometry and mathematical analysis.