If $\phi(a)$ is invertible then $a$ is invertible

Question

Let $R$ and $S$ be rings and $\phi: R \to S$ is a surjective ring homomorphism.

If $\phi(a)$ is invertible then $a$ is invertible

Is this correct? We know $\phi(1_R)=1_S$ but I am confused thats what I tried:

Let $\phi(a)$ be invertible and its inverse be the $\phi(b)$. Then:

$\phi(a)\phi(b)=1_S$

$\phi(ab)=1_S$

I am stuck here. Can we say $ab=1_R$? I think we cant but I couldnt find any counterexample.

Answer 1

Hint: Let $p$ be a prime number and $\mathbb{Z}\rightarrow \mathbb{Z}/p$ the quotient, for example take $p=3$, $2$ is not invertible, but $p(2)$ is invertible.