= Solution
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
$$
\boxed{\text{(ii)}\Longrightarrow\text{(iii)}.}
$$
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