If $S$ is a quotient ring of $R$, then either $char(S)$ divides $char(R)$, or $char(S)=0$

Question

[True/False] If $S$ is a quotient ring of $R$, then either $char(S)$ divides $char(R)$, or $char(S)=0$

$char(R)$ is notation to denote characteristic of a ring $R$

Efforts,

Let $f : R \to R'$ be a ring homomorphism where $\mathrm{char}(R) > 0$.We can show that

$$0 < \mathrm{char}(f(R)) \leq \mathrm{char}(R).$$ and it follows from Euclidean Algorithm that $char(f(R))$ divided $char(R)$

Solution: Ring homomorphism: $0 < \mathrm{char}(f(R)) \leq \mathrm{char}(R)$

So the statetement should be True. Right? The book says answer is False. Am I missing something here?

Thanks.