Originally Posted by

**kalagota** so you mean, if $\displaystyle y_1 , y_2 \in R/N$ such that $\displaystyle y_1 = y_2$, then i have to show that $\displaystyle \phi_*(y_1) =\phi_*(y_2)$

ok.. if $\displaystyle y_1 = x_1N$ and $\displaystyle y_2 = x_2N$ for some $\displaystyle x_1, x_2 \in R$..

$\displaystyle \phi_*(y_1) = \phi(x_1)N'$ and $\displaystyle \phi_*(y_2) = \phi(x_2)N'$

$\displaystyle \implies \phi(x_1)N' = \phi(x_2)N' \Longleftrightarrow \phi(x_1)\phi(x_2)^{-1} = \phi(x_1)\phi(x_2^{-1}) = \phi(x_1x_2^{-1}) \in N'$

i this right??