An elementary net associated with the elementary group

Let R be an arbitrary commutative ring with identity, n be a positive integer, n≥2. The set σ=(σij), 1≤i,j≤n, of additive subgroups of the ring R is called a net (or carpet) over the ring R of order n, if the inclusions σirσrj⊆σij hold for all i, r, j. The net without the diagonal, is called an elementary net. The elementary net σ=(σij), 1≤i≠j≤n, is called complemented, if for some additive subgroups σii of the ring R the set σ=(σij), 1≤i,j≤n is a (full) net. The elementary net σ=(σij) is complemented if and only if the inclusions σijσjiσij⊆σij hold for any i≠j. Some examples of not complemented elementary nets are well known. With every net σ can be associated a group G(σ) called a net group. This groups are important for the investigation of different classes of groups. It is proved in this work that for every elementary net σ there exists another elementary net Ω associated with the elementary group E(σ). It is also proved that an elementary net Ω associated with the elementary group E(σ) is the smallest elementary net that contains the elementary net σ.