Geodesic
Nets associated with the elementary nets
Vladikavkazskij matematičeskij žurnal, Tome 12 (2010) no. 4, pp. 39-43

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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 $\omega_\sigma\subseteq\sigma\subseteq\Omega^\sigma$. 
@article{VMJ_2010_12_4_a5,
     author = {V. A. Koibaev},
     title = {Nets associated with the elementary nets},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {39--43},
     publisher = {mathdoc},
     volume = {12},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2010_12_4_a5/}
}
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V. A. Koibaev. Nets associated with the elementary nets. Vladikavkazskij matematičeskij žurnal, Tome 12 (2010) no. 4, pp. 39-43. http://geodesic.mathdoc.fr/item/VMJ_2010_12_4_a5/