= Solution
Multiplication in a <direct product of rings> is coordinatewise. Thus $(r_1,\ldots,r_n)$ has a two-sided inverse exactly when each $r_j$ has a two-sided inverse in $R_j$, and then
$$
(r_1,\ldots,r_n)^{-1}=(r_1^{-1},\ldots,r_n^{-1}).
$$
Consequently its <unit group> is the <direct product of groups>
$$
\boxed{U(R_1\times\cdots\times R_n)=U(R_1)\times\cdots\times U(R_n)}.
$$
The identification preserves multiplication because both sides use coordinatewise multiplication. This is the <unit group of a direct product of rings> formula.
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