Arithmetic properties

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

The conceptualization of rings started in the 1870s and 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.

A ring is an abelian group with a second binary operation that is associative, is distributive over the abelian group operation, and has an identity element. By extension from the integers, the abelian group operation is called addition and the second binary operation is called multiplication.

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