unit under ring-homomorphism

Let \(R\) be a ring, \(r \in R\) a unit and \(\varphi: R \rightarrow S\) a ring-homomorphism for another ring \(S\). Then \(\varphi(r)\) is also a unit

By assumption, there exists an \(r^{-1}\). Thus

\begin{align*} 1 =& \varphi(1) \\ =& \varphi(r \cdot r^{-1}) \\ =& \varphi(r) \cdot \varphi(r^{-1}) \end{align*}

and hence \(\varphi(r)\) is invertible

left-inverse analoguosly