19.48 (2018)

A system of additive subgroups $\sigma_{ij}$, $1 \leqslant i, j \leqslant n$, of a field $K$ is called a net (or a carpet) of order $n$ if $\sigma_{ir}\sigma_{rj} \subseteq \sigma_{ij}$ for all $i, r, j$. A net that does not contain the diagonal is called an elementary net. A net $\sigma = (\sigma_{ij})$ is said to be irreducible if all $\sigma_{ij}$ are nontrivial. A net $\sigma$ is said to be closed if the elementary net subgroup $E(\sigma)$ does not contain additional elementary transvections.
Let $R$ be a principal ideal domain with $1 \in R$, let $k$ be the field of fractions of $R$, and $K$ an algebraic extension of the field $k$.

Let $\sigma = (\sigma_{ij})$ be an irreducible elementary net over $K$ such that all $\sigma_{ij}$ are $R$-modules. Is the net $\sigma$ closed?