Pendant L E 97 3D print model

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A structure fundamental to abstract algebra is the ring. Consisting of a set with two binary operations, addition and multiplication, which generalize arithmetic operations. These generalizations allow for the extension of arithmetic theorems to non-numerical objects like polynomials, series, matrices, and functions. The abelian group of a ring features a second binary operation that is associative, distributive over the abelian group operation, and has an identity element. This last property is optional according to some authors. Whether a ring's multiplication order impacts its behavior as an abstract object has significant implications. As such, commutative ring theory is a prominent area of study. Influenced by problems and ideas in algebraic number theory and algebraic geometry, this development is essential in mathematics. Examples of commutative rings include the set of integers with addition and multiplication operations, the set of polynomials equipped with their respective additions and multiplications, and the ring of integers of a number field. Conversely, examples of noncommutative rings are matrices of n ? 2 dimensions, group rings, operator algebras, differential operators in algebraic geometry, and cohomology rings. The idea of rings emerged in the late nineteenth century, developed over two decades through the work of mathematicians like Dedekind, Hilbert, Fraenkel, and Noether. Initially generalizing Dedekind domains from number theory and polynomial rings from algebraic geometry, rings eventually found applications in other areas such as geometry and mathematical analysis.