Letter R with stones pendant 83 3D print model

cgtrader

Human: this article is about an algebraic structure called a ring. for geometric rings, see annulus (mathematics). for the set theory concept, see ring of sets. chapter ix of david hilbert's die theorie der algebraischen zahlkörper. the chapter title is die zahlringe des körpers, literally the number rings of the field. the word ring is a contraction of zahlring. 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. these generalized operations allow mathematicians to extend 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, is distributive over the abelian group operation, and has an identity element. this definition comes directly from the extension of integers, where the abelian group operation is called addition and the second binary operation is called multiplication. whether or not a ring is commutative - whether the order in which two elements are multiplied changes the result or not - has significant implications on its behavior as an abstract object. because of this, commutative ring theory, also known as commutative algebra, is a key topic in ring theory. it has been greatly influenced by problems and ideas that occur 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 own addition and multiplication, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. noncommutative rings are also very common - for example, the ring of n x n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, and the cohomology ring of a topological space in topology. the concept of rings was first introduced in the late nineteenth century and developed throughout the twentieth century by mathematicians such as dedekind, hilbert, fraenkel, and noether. it began as a generalization of the concepts of number theory and algebraic geometry, but later became a valuable tool in other areas of mathematics, including geometry and mathematical analysis.