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Human: this article focuses on a fundamental concept in abstract algebra - the ring. for geometric rings, see annulus (mathematics). for set theory, see ring of sets. chapter ix of david hilbert's die theorie der algebraischen zahlkorper explores the number rings of fields. "zahlring" is an older contraction for this concept. in mathematics, a ring serves as one of several essential algebraic structures. it consists of a collection paired with two binary operations that generalize arithmetic addition and multiplication. through this broadening, arithmetic principles can be extended to objects beyond numerical domains - such as polynomials, series, matrices, and functions. a ring is an abelian group with another binary operation characterized by being associative, distributive over the group operation, and featuring an identity element (this final attribute isn't required in every instance). this setup extends naturally from the integers: addition becomes the primary group operation and multiplication, the secondary binary operation. whether or not a ring is commutative profoundly affects its nature as an abstract entity. commutative algebra - otherwise known as commutative ring theory - represents a core area of focus within the field of ring theory, heavily influenced by occurrences in both number theory and geometry. prime examples include sets equipped with addition and multiplication (as seen in the set of integers), polynomials and their related operations, the coordinate rings associated with algebraic varieties, and fields where only the integral numbers are used. examples also abound of noncommutative rings: those consisting of square matrices (provided n exceeds two), group representations in theoretical settings, function operators within analysis contexts, operator algebras used to study spaces with topological characteristics, and cohomology groups used in space analyses.