Internal ring 75 3D print model

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The concept of a ring has its roots in the 1870s, and by the 1920s, it had taken shape in mathematics. This abstract algebraic structure is fundamental to many areas of math, including number theory, algebraic geometry, and representation theory. In mathematics, a ring consists of a set with two binary operations that generalize addition and multiplication. By extension from integers, one operation is called addition and the other is called multiplication. This generalization allows theorems from arithmetic to be extended to non-numerical objects like polynomials and matrices. A key characteristic of a ring is being an abelian group under the first operation, with a second binary operation that is associative and distributes over the group operation. Rings also have identity elements in one or both operations. Commutative ring theory, often called commutative algebra, has a significant impact on the study of rings. Its development was influenced by ideas from number theory and geometry, and it remains a vital part of ring theory to this day. Some examples of commutative rings include the set of integers with addition and multiplication, polynomial rings, and the coordinate ring of an affine algebraic variety. Noncommutative rings, on the other hand, include square matrices and group rings in representation theory. Rings were formalized as a generalization of Dedekind domains and polynomial rings, and later found application in geometry and mathematical analysis.