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Quotient 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 in 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.

The ideals in a quotient ring A/a are in a one-to-one correspondence with ideals in A which contain the ideal a. In particular, the zero ideal in A/a corresponds to a in A. 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 {0,2,4} in Z_6=Z/6Z.


See also

Field, Ideal, Integer, Integral Domain, Maximal Ideal, Module, Prime Ideal, Residue Field, Ring

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Quotient Ring." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/QuotientRing.html

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