Let $V=M_{2 \times 1}(k)$ where $k$ is a field, and consider $V$ like a module over a ring $R=M_{2 \times 2}(k)$ over adition and multiplication of matrix. Prove that $V$ is a cyclic $R$-module and that every $\begin{pmatrix}1\\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ generate $V$ as $R$-module. Calculate the ideals order of each of these generators and compare them.
Can you guys please help me, I know that these elements generate the module, but I don't understand the meaning about order ideal or ideal order.
Thanks for the help in advance.