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In mathematics, a ring is a fundamental algebraic structure used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication, which enables extending theorems from arithmetic to non-numerical objects like polynomials, series, matrices, and functions. A ring is an abelian group that has a second binary operation that is associative, distributive over the abelian group operation, and has an identity element. By extension from integers, the abelian group operation is called addition, and the second binary operation is called multiplication. Commutative ring theory, or commutative algebra, is a key topic in ring theory, where problems and ideas from algebraic number theory and algebraic geometry have greatly influenced its development. Examples of commutative rings include the set of integers with their addition and multiplication operations, polynomials equipped with their addition and multiplication, and coordinate rings of affine algebraic varieties. Rings are non-numerical objects used in abstract algebra, which are often defined by sets with binary operations that generalize arithmetic operations. Whether a ring is commutative or not affects its behavior as an abstract object, and therefore studying these structures has far-reaching implications. The ring's conceptualization began in the 1870s and was completed in the 1920s by key contributors like Dedekind, Hilbert, Fraenkel, and Noether. Initially formalized to generalize Dedekind domains, polynomial rings, and rings of invariants in number theory, algebraic geometry, and invariant theory, they eventually found utility in geometry, mathematical analysis, and other areas of mathematics. The generalization of arithmetic operations enables extending theorems from arithmetic to various objects like polynomials and matrices. Ring structures have two binary operations: addition and multiplication. When a ring is commutative, changing the order of multiplying two elements doesn't change the result. However, some rings are not commutative, meaning the order in which you multiply them does affect the result. Ring theory has contributed to mathematics significantly over time, with applications across multiple areas such as algebraic number theory and algebraic geometry. In topology, ring theory applies to the study of differential operators.