Probably this is easiest, but as I am somehow stuck I would be pleased about some comments.
What I give myself is a rule $f$ which does the following:
To every commutative ring $A$ with $1$ the rule $f$ assigns a unit $f(A)\in A^*$, and this assignment shall satisfy the following property:
If $\varphi: A\rightarrow B$ is any ring homomorphism, then $\varphi(f(A)) =f(B)$.
I am convinced that the only rules which can do this are the following two:
(i) $f(A)=1$ for each ring $A$ simultaneously,
(ii) $f(A)=-1$ for each ring $A$ simultaneously.
How could one show this (if it is right)?