05 - Quotients

Quotient

Let $I$ be an ideal in ring $R$

Quotient Group $(R/I,+)$ is defined through

Equivalence Relation $r\sim s$ if $r-s\in I$ with equivalence class cosets

$$\{r+I:r\in R\}$$

Operation

$$\begin{array}{rl}(r_1+I)+(r_2+I)& =(r_1+r_2)+I\\ (r_1+I)\times (r_2+I)& =r_1.r_2+I\end{array}$$