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The chapter on ring theory was first outlined by David Hilbert in his book Die Theorie der algebraischen Zahlkorper. Hilbert called it "Die Zahlringe des Korpers," which translates to the number rings of the field, highlighting the importance of this abstract structure in mathematics. In mathematics, a fundamental structure is the ring, an algebraic system used in abstract algebra. It consists of a set equipped with two binary operations that generalize arithmetic addition and multiplication, extending mathematical theorems beyond numerical objects like polynomials, series, matrices, and functions. A ring is a specific type of group with another binary operation that is associative, distributive over the abelian group operation, and has an identity element. Drawing inspiration from the properties of integers, the abelian group operation is called addition and the second binary operation is referred to as multiplication. Whether a ring is commutative or not has significant implications for its behavior as an abstract object, leading to profound differences in the study of this structure. As a result, researchers focus on developing commutative ring theory, commonly known as commutative algebra. Its growth and advancements have been shaped by questions arising naturally from number theory and geometry. Rings are studied using various approaches, including mathematical models that resemble numerical systems like integer rings or polynomial equations with multiple variables. Examples of these abstract objects include integer groups equipped with their respective arithmetic operations and polynomial functions represented as elements in a set equipped with its addition and multiplication rules. These patterns extend into higher-dimensional matrices and sets in abstract geometric environments. Some key individuals who significantly contributed to the development of ring theory were David Hilbert, Richard Dedekind, Abraham Fraenkel, and Emmy Noether, all of whom expanded on each other's ideas between the 1870s and the 1920s. By generalizing mathematical domains and algebraic geometric constructs into abstract groups with binary operations, they uncovered patterns in both commutative and noncommutative contexts that laid a foundation for modern applications across numerous areas of mathematics.