Cyclical elementary nets

Let R be a commutative ring with the unit and n∈N. A set σ=(σij), 1≤i,j≤n, of additive subgroups of the ring R is a net over R of order n, if σirσrj⊆σij for all 1≤i,r,j≤n. A net which doesn't contain the diagonal is called an elementary net. An elementary net σ=(σij),1≤i≠j≤n, is complemented, if for some additive subgroups σii of R the set σ=(σij),1≤i,j≤n is a full net. An elementary net σ is called closed, if the elementary group E(σ)=⟨tij(α):α∈σij,1≤i≠j≤n⟩ doesn't contain elementary transvections. It is proved that the cyclic elementary odd-order nets are complemented. In particular, all such nets are closed. It is also shown that for every odd n∈N there exists an elementary cyclic net which is not complemented.