If $\varphi:R\to S$ be a surjective ring homomorphism then $\varphi(R^*) $is not always $S^*$

Question

I don't know any example. Whatever examples I know the statement holds true

Answer 1

Any (unital) ring homomorphism $f:R\to S$ maps invertible elements to invertible elements, i.e. $f(R^*)\subseteq S^*$. So you are looking for an example where that inclusion is proper.

For that consider the ring of integers $\mathbb{Z}$ and let $p\in\mathbb{Z}$, $p>3$ be a prime number. Finally consider the quotient map $q:\mathbb{Z}\to\mathbb{Z}/(p)$ and note that $\mathbb{Z}/(p)$ is a field and so it has $p-1$ invertible elements (which is at least four) while $\mathbb{Z}$ has only two invertible elements: $-1$ and $1$. So $f(\mathbb{Z}^*)\subsetneq\big(\mathbb{Z}/(p)\big)^*$.