Nets associated with the elementary nets

For an elementary net (i.e. a net without diagonal) $\sigma=(\sigma_{ij})$ of additive subgroups $\sigma_{ij}$, $i\ne j$, of a commutative ring $R$ with 1 two nets are constructed: the net $\omega_\sigma$ associated with $\sigma$ and the net $\Omega^\sigma$ associated with the elementary group $E(\sigma)$, and (on the off-diagonal positions) we have $\omega_\sigma\subseteq\sigma\subseteq\Omega^\sigma$.