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Definition: 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] if and only if x-y is in a.