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:

${\displaystyle \begin{align}&(I+x)+(I+y)=I+(x+y)\\&(I+x)(I+y)=I+xy\\&\forall x,y\in R\end{align}}$

To check that these operations are well-defined, let ${\displaystyle I+x_1}$ and ${\displaystyle I+x_2}$ be two representations of the same coset ${\displaystyle I+x}$ , so ${\displaystyle x_1=x_2+a}$ for some ${\displaystyle a\in I}$ . And let ${\displaystyle I+y_1}$ and ${\displaystyle I+y_2}$ be two representations of the same coset ${\displaystyle I+y}$ , so ${\displaystyle y_1=y_2+b}$ for some ${\displaystyle b\in I}$ .

Then ${\displaystyle x_1+y_1=(x_2+a)+(y_2+b)=(x_2+y_2)+(a+b)}$ . Since ${\displaystyle a+b\in I}$ , due to the closure of an ideal subring, this means ${\displaystyle I+(x_1+y_1)=I+(x_2+y_2)}$ , so addition is well-defined.

Also, ${\displaystyle x_1\cdot y_1=(x_2+a)\cdot(y_2+b)=x_2\cdot y_2+(x_2\cdot b+a\cdot y_2+a\cdot b)}$ . Now ${\displaystyle x_2\cdot b,a\cdot y_2\in I}$ due to I being an ideal, and ${\displaystyle a\cdot b\in I}$ due to I being a closed ideal subring, so ${\displaystyle x_2\cdot b+a\cdot y_2+a\cdot b\in I}$ , meaning ${\displaystyle I+x_1\cdot y_1=I+x_2\cdot y_2}$ , 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 ${\displaystyle R}$ . The additive identity is ${\displaystyle I+0}$ , ie the coset ${\displaystyle I}$ , and the additive inverse of ${\displaystyle I+x}$ is ${\displaystyle I+(-x)}$ , or ${\displaystyle I-x}$ .