Ring homomorphism and units homomorphism

Question

I am trying to show that given $\phi: R\rightarrow R'$ is a ring homomorphism $R, R'$ are rings, there exist a homomorphism between the group of units that is $\phi': R^\times\rightarrow R'^\times$ is a group homomorphism. I was able to show that the the ring homomorphism maps units to units but I am not sure how to check the homomorphism condition explicitly.

Answer 1

(I assume you're using $\phi'=\phi\rvert_{R^\times}$, that is, the restriction of $\phi$ to $R^\times$.)

Then, the statement follows trivially from the fact that $\phi$ is a ring homomorphism: From $$\forall x,y\in R.\quad \phi(x\cdot y)=\phi(x)\cdot\phi(y) $$ and $R^\times\subseteq R$, it is obvious that $$\forall x,y\in R^\times.\quad \phi'(x\cdot y)=\phi'(x)\cdot\phi'(y) \text.$$