Man ring 71 3D print model

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A ring is one of the fundamental algebraic structures used extensively in abstract algebra. It comprises 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 like polynomials, series, matrices, and functions. An abelian group is at the heart of any ring, characterized by a second binary operation that is associative, distributive over the abelian group operation, and has an identity element. Some authors require only a portion of these properties, see section on definition notes. By extension from integers, the abelian group operation is called addition and the second binary operation is called multiplication. Whether a ring is commutative or not significantly impacts its behavior as an abstract object. This distinction gives rise to two distinct branches: commutative ring theory, commonly referred to as commutative algebra, is a pivotal aspect of ring theory. Its development has been substantially influenced by problems and ideas that naturally arise in algebraic number theory and algebraic geometry. Examples of commutative rings include the set of integers 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. On the other hand, noncommutative rings are found in areas such as matrix theory, group representation, functional analysis, differential operator theory, and topological space cohomology. The concept of rings took shape in the 1870s and reached completion in the 1920s. Key figures include Dedekind, Hilbert, Fraenkel, and Noether. Rings were initially formulated as a generalization of number fields, polynomial rings, and invariant theory constructs that appeared in algebraic geometry. Eventually, they found utility in other branches of mathematics such as geometry and mathematical analysis.