 Examples of commutative rings include the set of integers equipped with the 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.Examples of noncommutative rings include the ring of "n" × "n" real square matrices with "n" ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, and the cohomology ring of a topological space in topology.The conceptualization of rings began in the 1870s and was completed in the 1920s.