factor ring

A quotient ring (also called a residue-class ring) is a ring that is the quotient of a ring A and one of its ideals a, denoted A/a. For example, when the ring A is Z (the integers) and the ideal is 6Z (multiples of 6), the quotient ring is Z_6 = Z/6Z. In general, a quotient ring is a set of equivalence classes where [x] = [y] iff x - y element a. The quotient ring is an integral domain iff the ideal a is prime. A stronger condition occurs when the quotient ring is a field, which corresponds to when the ideal a is maximal.

field | ideal | integer | integral domain | maximal ideal | module | prime ideal | residue field | ring