Beautiful ring with stones 113 3D print model

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The word ring originates from the German language, with Zahlring literally translating to number ring. This chapter title is from David Hilbert's book "Die Theorie der algebraischen Zahlkorper", a seminal work in abstract algebra that explores the fundamental properties of algebraic structures. A ring is a fundamental concept in mathematics that consists of a set and two binary operations, generalizing arithmetic addition and multiplication. By applying these operations to non-numerical objects like polynomials, series, matrices, and functions, mathematicians have been able to extend numerous theorems from arithmetic. In this sense, a ring is an abelian group with another operation that satisfies associativity, distributes over the group's operation, and features an identity element. From integers, we borrow the terms addition for the first binary operation and multiplication for the second one. If a ring exhibits commutative properties - that its elements behave equally regardless of order - this influences how it functions as an abstract object. Consequently, theories related to commutative rings are extremely important in mathematical discussions and form a crucial branch of algebra known as commutative algebra or ring theory. Notable topics under discussion have drawn significantly from algebraic geometry and number theory, while other examples include matrices with 2 elements or more, the set of integers in various contexts, and rings found in topology and representation theory. As abstract objects with certain unique characteristics, mathematicians discovered and utilized this type of algebraic structure to help analyze a wide range of problems - starting as far back as the nineteenth century when it first caught people's attention through ideas from Dedekind. Important individuals have played significant roles in further defining rings and extending their use across multiple mathematical disciplines over the following century, particularly David Hilbert, Abraham Fraenkel, and Emmy Noether.