If $A$ and $B$ are rings with unity, prove that the group of units $U(A \times B)$ of the direct product $A \times B$ is isomorphic to the direct product $U(A) \times U(B)$ of the respective groups of units.
I have a problem when I try to define the function, because I don't understand how it works.
I think $f\colon U(A \times B) \to U(A) \times U(B)$ should be $f((a,b)) = (a,b)$, where $(a,b) \in U(A \times B)$.