Quotient Ring

A quotient ring (also called a residue-class ring) is a ring that is the quotient of a ring and one of its ideals , denoted . For example, when the ring is (the integers) and the ideal is (multiples of 6), the quotient ring is .

In general, a quotient ring is a set of equivalence classes where iff .

The quotient ring is an integral domain iff the ideal is prime. A stronger condition occurs when the quotient ring is a field, which corresponds to when the ideal is maximal.

The ideals in a quotient ring are in a one-to-one correspondence with ideals in which contain the ideal . In particular, the zero ideal in corresponds to in . In the example above from the integers, the ideal of even integers contains the ideal of the multiples of 6. In the quotient ring, the evens correspond to the ideal in .