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A ring is an integral part of the fundamental algebraic structures used extensively in abstract algebra. A ring consists of a set equipped with two binary operations that effectively generalize the arithmetic operations of addition and multiplication. This generalization enables the extension of theorems from arithmetic to non-numerical objects such as polynomials, series, matrices, and functions. A key feature of a ring is its abelian group nature, which possesses a second binary operation that is associative, distributive over the abelian group operation, and has an identity element. This fundamental characteristic of rings makes them an essential topic in mathematics. The commutative property of a ring plays a crucial role in determining its behavior as an abstract object. As a result, commutative ring theory, commonly known as commutative algebra, is a significant area of study within the field of ring theory. The development of commutative algebra has been greatly influenced by problems and ideas arising naturally in algebraic number theory and algebraic geometry. Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with their addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of n x n real square matrices for n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, and the cohomology ring of a topological space. The concept of rings originated in the late 19th century and was fully developed by the early 20th century. Key contributors to this development include Dedekind, Hilbert, Fraenkel, and Noether. Initially, rings were formalized as a generalization of Dedekind domains that occur in number theory and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Subsequently, they also proved useful in other areas of mathematics such as geometry and mathematical analysis.