Embedding of Elementary Net into Gap of Nets

Let R be a commutative unital ring and n ∈ ℕ, n ≥ 2. A system σ = (σ_ij), 1 ≤ i, j ≤ n, of additive subgroups σ_ij of R is called a net or carpet over R of order n if σ_irσ_rj ⊆ σ_ij for all i, r, and j. A net without diagonal is called an elementary net or elementary carpet. Let n ≥ 3. Consider a matrix ω = (ω_ij) of additive subgroups ω_ij of R, defined for i ≠ j as follows: ${\omega}_{ij}=\sum \limits_{k=1}^n{\sigma}_{ik}{\sigma}_{kj}$, k ≠ i, j. The set ω = (ω_ij) of elementary subgroups ω_ij of R is an elementary net ω and is called an elementary derived net. The diagonal of the derived net ω is defined by the formula ${\omega}_{ii}=\sum \limits_{k\ne s}{\sigma}_{ik}{\sigma}_{ks}{\sigma}_{si}$, 1 ≤ i ≤ n, where the sum is taken over all 1 ≤ k ≠ s ≤ n. It is proved that an elementary net σ induces the derived net ω = (ω_ij) and the net Ω = (Ω_ij) associated with the elementary group E(σ), where ω ⊆ σ ⊆ Ω, ω_irΩ_rj ⊆ ω_ij and Ω_irω_rj ⊆ ω_ij (1 ≤ i, r, j ≤ n). In particular, the matrix ring M(ω) is a two-sided ideal of the ring M(Ω). For the nets of order n = 3, we establish a more precise result.