Full and elementary nets over the quotient field of a principal ideal ring

Let $K$ be the quotient field of a principal ideal ring $R$, and $\sigma=(\sigma_{ij})$ be a full (elementary) net of order $n\geq2$ (respectively, $n\geq3$) over $K$ such that the additive subgroups $\sigma_{ij}$ are nonzero $R$-modules. It is proved that, up to conjugation by diagonal matrix, all $\sigma_{ij}$ are ideals of a fixed intermediate subring $P$, $R\subseteq P\subseteq K$.