In the base-three number system, $a = 2012_3$ and $b = 201_3.$ What is the product $ab$ expressed in the base-three number system?
We may multiply in base $3$ just as in base $10:$ anytime we get a number greater than $3,$ we record the remainder when the number is divided by $3$ and carry the quotient. \[
\begin{array}{r}
2012 \\
\times 201 \\ \hline
2012 \\
11101\hphantom{00} \\ \hline
1112112
\end{array}
\] Therefore, the product is $\boxed{1112112_3}.$