1. Consider the matrix $\mathbf{e}=\left(\begin{array}{11}0 & 2 \\ 5 & 1\end{array} ight) \in M_{2}\left(\mathbb{Z}_{10} ight)$. (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] 2. Let $R=M_{2}(\mathbb{Z})$ and $M=\left\{\left(\begin{array}{11}a & b 10 & O\end{array} ight): a, b \in \mathbb{Z} ight\}$. (i) Show that, under the usual matrix multiplication, $M$ is a right $R$-module, but that $M$ is not a left $R$-module. [4 Marks] (ii) Is $M$ a free right $R$-module? Justify your answer. [6 Marks ] (iii) Is $M$ a finitely generated projective right $R$-module? Justify your answer. CS.VS. 1467