My question is kinda long, so please bear with me... And I only need hints to get me started.
So, I'm working on the ring $R =\left( \begin{matrix} \mathbb{Z} & \mathbb{Q} \\ 0 & \mathbb{Q} \end{matrix} \right)$ .
$\textbf{1.}$ I consider the left ideal $I_1= \left( \begin{matrix} 0& \mathbb{Q} \\ 0 & 0 \end{matrix} \right)$ ; I am trying to show that it is not projective, any help with that ?
$\textbf{2.} $ I also wanna show that if $M$ is a projective right $R$-module then any submodule of $M$ is also projective. I also need a nudge in the right direction to prove this claim.
Thanks