Quotient ring

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:

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

To check that these operations are well-defined, let $I+x_{1}$ and $I+x_{2}$ be two representations of the same coset $I+x$ , so $x_{1}=x_{2}+a$ for some $a\in I$ . And let $I+y_{1}$ and $I+y_{2}$ be two representations of the same coset $I+y$ , so $y_{1}=y_{2}+b$ for some $b\in I$ .

Then $x_{1}+y_{1}=(x_{2}+a)+(y_{2}+b)=(x_{2}+y_{2})+(a+b)$ . Since $a+b\in I$ , due to the closure of an ideal subring, this means $I+(x_{1}+y_{1})=I+(x_{2}+y_{2})$ , so addition is well-defined.

Also, $x_{1}\cdot y_{1}=(x_{2}+a)\cdot (y_{2}+b)=x_{2}\cdot y_{2}+(x_{2}\cdot b+a\cdot y_{2}+a\cdot b)$ . Now $x_{2}\cdot b,a\cdot y_{2}\in I$ due to I being an ideal, and $a\cdot b\in I$ due to I being a closed ideal subring, so $x_{2}\cdot b+a\cdot y_{2}+a\cdot b\in I$ , meaning $I+x_{1}\cdot y_{1}=I+x_{2}\cdot y_{2}$ , 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 $R$ . The additive identity is $I+0$ , ie the coset $I$ , and the additive inverse of $I+x$ is $I+(-x)$ , or $I-x$ .