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Human: His article centers on a fundamental concept in abstract algebra, specifically the algebraic structure known as a ring. For geometric rings, see Annulus (mathematics), and for the set theory concept, see Ring of sets.\n\nThe chapter "Die Zahlringe des Korpers" in David Hilbert's book Die Theorie der algebraischen Zahlkorper marks an important milestone in the development of the concept of a ring. This German term, literally meaning number rings of the field, is derived from Zahlring, which translates to number ring.\nIn mathematics, a ring refers to one of several key algebraic structures employed in abstract algebra. It comprises a set enhanced by two binary operations that expand upon arithmetic addition and multiplication, allowing for an extensive generalization of theorems originally derived from these numerical operations. This extension enables applications across non-numerical objects such as polynomials, series, matrices, and functions.\n\nA ring is characterized by its ability to act as an abelian group with a secondary binary operation that is associative, distributes over the abelian group operation, and possesses an identity element. The requirement of having an identity element for this last property varies among authors, as outlined in § Notes on the definition.\nRelying on intuition from the integers, we name the primary operation as addition and refer to the secondary binary operation as multiplication.\nWhether a ring is commutative or not has profound implications for its properties. Consequently, commutative ring theory, commonly referred to as commutative algebra, represents a major aspect of ring theory, influenced significantly by the inherent connections with number theory and algebraic geometry.\nExamples of rings that exhibit commutativity include the set of integers using addition and multiplication operations, the set of polynomials in its simplest form equipped with their respective addition and multiplication rules, and more generally, any coordinate ring associated with an affine variety, along with any integral domains. By contrast, examples of non-commutative rings are comprised of the class of square matrices for which n is less than 2, specific types of group rings observed within representation theory, several types of operator algebras, certain types of differential operators in various theories related to functions and numbers, and cohomology rings in the context of topological spaces.\n\nThe evolution and definition of the concept of a ring progressed over several decades from its initiation in the late 1870s to its culmination around the early part of the 20th century. David Hilbert's work served as a foundational point for further developments along with the contributions from mathematicians such as Fraenkel, Noether, and others whose work laid a strong basis in both number theory and algebraic geometry that were instrumental in establishing ring structures across diverse mathematical disciplines like topology, geometry, and functional analysis.