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{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 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 .

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