A **quotient ring** is, for a given ideal I in a ring R, the set of cosets of I in R with addition and multiplication defined as:

To check that these operations are well-defined, let and be two representations of the same coset , so for some . And let and be two representations of the same coset , so for some .

Then . Since , due to the closure of an ideal subring, this means , so addition is well-defined.

Also, . Now due to I being an ideal, and due to I being a closed ideal subring, so , meaning , so multiplication is well-defined.

It is also possible to verify that this is indeed a ring - the operations are both closed by the above argument, and associativity, commutativity and distributivity follow from the operations in the ring . The additive identity is , ie the coset , and the additive inverse of is , or .