BVLG RI ring 72 3D print model

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In mathematics, a fundamental building block is the ring. A ring consists of a set, two binary operations that generalize arithmetic addition and multiplication. Through this generalization, theorems from arithmetic are extended to objects like polynomials, series, matrices, and functions. A ring is an abelian group with another operation that is associative, distributed over the group's operation, and has an identity element. This last property isn't required by all authors. By extension, the abelian group operation is called addition and the other operation is called multiplication. Commutative or not, a ring's behavior changes profoundly. As a result, commutative ring theory is crucial in this area of study. Its development has been influenced greatly by algebraic number theory and geometry problems that occur naturally. The set of integers, equipped with the addition and multiplication operations, is an example of a commutative ring. So are polynomials with their respective additions and multiplications, coordinate rings of varieties, and integer rings of number fields. Examples of noncommutative rings include matrix sets with square elements greater than or equal to 2, group rings in representation theory, operator algebras in analysis, differential operator rings in the theory of operators, and topological space cohomology rings.