quotient ring, or residue class ring, n. the ring, denoted R/K, the members of which are the cosets of an ideal K in a ring R; these cosets are called residue classes, and are equivalence classes the members of which differ by a member of K. Since K is an ideal, the sums and products of elements of the factor ring are unique, and the cosets of a sum or product of ring elements equals the sum or product (in the same order) of their individual cosets. K is the zero of the factor ring, and it has a unit element if R has. See also factor space.