Commutative ring/Residue class ring/Definition

Residue class ring

Let ${}R$ be a commutative ring, and let ${}I\subseteq R$ be an ideal in ${}R$. Then the residue class ring ${}R/I$ (R modulo I) is a commutative ring that is determined by the following data.

1. As a set, ${}R/I$ is the set of all cosets to ${}I$.
2. An addition of cosets is defined by

   ${}(a+I)+(b+I):=(a+b+I)\,.$

3. A multiplication of cosets is defined by

   ${}(a+I)\cdot (b+I):=(a\cdot b+I)\,.$

4. ${}{\bar {0}}=0+I=I$ is the neutral element of the addition (the zero class).
5. ${}{\bar {1}}=1+I$ is the neutral element of the multiplication (the unit class).