Letter C 103 3D print model

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A Ring is a fundamental Algebraic Structure used in Abstract Algebra. It Consists of a Set Equipped with Two Binary Operations Generalizing the Arithmetic Operations of Addition and Multiplication. This Generalization Extends Theorems from Arithmetic to Non-numerical Objects such as Polynomials, Series, Matrices, and Functions. A Ring is an Abelian Group with a Second Binary Operation that is Associative, Distributive over the Abelian Group Operation, and has an Identity Element. This Property is not required by all Authors, but is Often Assumed. By Extension from the Integers, the Abelian Group Operation is called Addition and the Second Binary Operation is called Multiplication. Whether a Ring is Commutative or Not has Profound Implications on its Behavior as an Abstract Object. Commutative Ring Theory, also known as Commutative Algebra, is a Key Topic in Ring Theory. Its Development has been Greatly Influenced by Problems and Ideas Occurring Naturally in Algebraic Number Theory and Algebraic Geometry. Examples of Commutative Rings include the Set of Integers equipped with 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 with 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 in Topology. The Conceptualization of Rings Began in the 1870s and was Completed in the 1920s. Key Contributors Include Dedekind, Hilbert, Fraenkel, and Noether. Rings were First 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. Afterward, they also Proved to be Useful in other Branches of Mathematics such as Geometry and Mathematical Analysis.