Let $R = \Bbb C[x]/J$. Then the ideals of $R$ are in 1-1-correspondence with the ideals of $\Bbb C[x]$ that contain $J$. Since $\Bbb C[x]$ is a PID then every ideal is principal. Let $I=\langle f(x)\rangle$ be a principal ideal that contains $J$. Then this corresponds to the ideal $I/J$ in $R$.
My question is:
Is the ideal $I/J$ in $R$ the same as the principal ideal in $R$ written as $\langle f(x) + J\rangle$? Why or why not?
To be a bit more specific: Let $I=\langle x \rangle$ be an ideal in $\Bbb C[x]$ that contains $J$. Is $\langle x + J \rangle$ the corresponding ideal in $R = \Bbb C[x]/J$? Or should that be written $\langle x \rangle/J?$