1. Consider the matrix $\mathbf{e}=\left(\begin{array}{11}0 & 2 1 5 & 1\end{array} ight) \in M_{2}\left(\mathbb{Z}_{10} ight)$. (i) Is e an idempotent in $M_{2}\left( \mathbb{Z}_{10} ight) ?$ Justify your answer. [2 marks] (ii) Use e to write $\mathbb{Z}_{10}$ as the direct sum of two non- trivial modules $M$ and $N$. List all elements of $M$ and $N$. [8 Marks] CS.VS. 1463