Marquis Ring 92 3D print model

cgtrader

Human: his article is about an abstract algebra structure known as a ring. For the set theory concept, see Ring of sets.\n\nDavid Hilbert's Die Theorie der algebraischen Zahlkorper has a chapter dedicated to this topic, titled Die Zahlringe des Korpers, which literally translates to number rings of the field.\nIn mathematics, a fundamental structure known as a ring is used in abstract algebra. It consists of a set equipped with two binary operations that generalize addition and multiplication. This generalization extends arithmetic theorems to objects like polynomials, matrices, functions, and series.\n\nA ring is essentially an abelian group with a second binary operation. This second operation must be associative, distributive over the first operation, and have an identity element - although some authors disagree on this last point.\nBy extending from the integers, we refer to addition as the abelian group operation and multiplication as the second binary operation.\n\nWhether a ring is commutative or not has significant implications for its behavior. As a result, commutative algebra, also known as commutative ring theory, is a crucial aspect of ring theory. Its development was heavily influenced by problems in algebraic number theory and algebraic geometry.\nExamples of commutative rings include the set of integers with addition and multiplication, polynomials with their respective operations, and the coordinate ring of an affine variety.\nNoncommutative rings include n ? n real square matrices, group rings in representation theory, operator algebras, and the cohomology ring of a topological space.\n\nThe conceptualization of rings started in the 1870s and was completed by the 1920s. Key contributors to this field include Dedekind, Hilbert, Fraenkel, and Noether.\nRings were initially formalized as a generalization of number theory's Dedekind domains, polynomial rings, and algebraic geometry's invariant rings.\nAfterward, they also found application in geometry and mathematical analysis.