Example

The additive group of Q is commutative. Therefore, the subgroup Z is a normal subgroup of Q. The cosets of Z in Q are the sets of the form a/b + Z, where a, b in Z and b 0. For example, 1/2 + Z. Computing in the quotient Q/Z comes down to `computing modulo integers'. For example

(3/4 + Z) + (5/6 + Z) = 7/12 + Z.

So far, we have only used the additive structure; Z is not an ideal in the ring Q and so we cannot speak of the quotient ring Q/Z.