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Human: This article discusses an algebraic structure known as a ring. For the set theory concept, see Ring of sets, while geometric rings can be found under Annulus (mathematics). David Hilbert's Die Theorie der algebraischen Zahlkorper includes a chapter called Die Zahlringe des Korpers or "Number Rings of the Field". In this chapter, Hilbert introduces the word ring as a contraction of Zahlring. Mathematically speaking, a ring is one of the primary components in abstract algebra, alongside group and field structures. A set equipped with two binary operations, which are essentially extensions of addition and multiplication from arithmetic, constitutes a ring. By generalizing these arithmetic operations, theories originally derived for numerical values can be applied to other objects such as polynomials, series, matrices, and functions. In more specific terms, a ring is an abelian group when combined with a second binary operation that satisfies certain properties: associativity, distributivity over the abelian group's operation, and the existence of an identity element. The two operations in this structure are extensionally referred to as addition and multiplication based on their origin from integers. One of the fundamental characteristics distinguishing rings is whether they follow commutative rules or not; that is, whether the order of two elements during multiplication influences the final outcome or remains unchanged. As a direct result, research on rings that obey this property – often referred to as commutative algebra or commutative ring theory – occupies an integral position within abstract algebra. Historical evidence has revealed profound connections with theories found in both number fields and geometry, primarily stemming from inherent challenges posed by these mathematical branches. Some examples of these rings include the set of integers under standard arithmetic operations; polynomials that follow similar rules of addition and multiplication, an affine variety's coordinate ring, or the field of numbers where algebraic operations apply to elements belonging in these number sets. On the other hand, a selection of noncommutative examples could encompass n x n matrices for values exceeding two, rings representing mathematical transformations on specific domains within linear representations, matrices in linear and functional analyses, along with algebra-like systems applied during theoretical studies about space-time geometry.