ring, in mathematics, system consisting of a set R of elements and two binary operations, such that addition makes R a commutative group and multiplication is associative and distributes over addition (see commutative law; associative law; distributive law). A commutative ring is one in which the commutative law also holds for multiplication. Examples of commutative rings are the sets of integers (see number) and real numbers. Square matrices (see matrix) furnish examples of non-communtative rings.