Quotient ring being isomorphic to the initial ring This may be a very basic question. 

Let $R$ be a commutative ring with unity (may be assumed to be an integral domain, if necessary) and $I$ be an ideal of $R$. Assume that $R/I$ is isomorphic to $R$. Does this mean $I=(0)$ ?


Counterexample:
$R=\Bbb Q [X_1, X_2, \dots , X_n, \dots]$. Define the ring morphism $f: R \to R$ by
$$\begin{cases}X_i \mapsto X_{i-1} & i>1 \\ X_1 \mapsto 0
\end{cases}
$$
and call $I= \ker f$.
Then $$R/I \cong \mathrm{Im} f= R$$
