Assume (ii), and write the ring decomposition as
$$
\mathcal OG\cong\operatorname{End}_{\mathcal O}(W)\times B
\cong M_n(\mathcal O)\times B.
$$
The natural column module $W\cong\mathcal O^n$ is projective over $M_n(\mathcal O)$ by <Morita equivalence>; extending it by zero across $B$ makes it projective over $\mathcal OG$. After extending scalars to $K$, the action factors through $M_n(K)$ on its natural module $K^n$, which is simple. Therefore
$$
\boxed{\text{(ii)}\Longrightarrow\text{(iii)}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 138 / 6 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-138/6/a/ii/solution