An embedding theorem for an elementary net

Let Λ be a commutative unital ring and n∈N, n≥2. A set σ=(σij), 1≤i,j≤n, of additive subgroups σij of Λ is said to be a net or a carpet of order n over the ring Λ if σirσrj⊆σij for all i, r, j. A net without diagonal is called an elementary net. An elementary net σ=(σij), 1≤i≠j≤n, is said to be complemented (to a full net), if for some additive subgroups (subrings) σii of Λ the matrix (with the diagonal) σ=(σij), 1≤i,j≤n is a full net. Assume that σ=(σij) is an elementary net over the ring Λ of the order n. Consider a set ω=(ωij) of additive subgroups ωij of the ring Λ, where i≠j defined by the rule ωij=∑nk=1σikσkj, k≠i; k≠j. The set ω=(ωij) of elementary subgroups ωij of the ring Λ is an elementary net called an elementary derived net.} An elementary net ω can be completed to a full net by the standard way. In this article we propose a second way to complete an elementary net to a full net. The notion of a net Ω=(Ωij) associated with an elementary group E(σ) is also introduced. The following theorem is the main result of the paper: An elementary net σ generates an elementary derived net ω=(ωij) and a net Ω=(Ωij) associated with the elementary group E(σ) such that ω⊆σ⊆Ω. If ω=(ωij) is completed with a diagonal to the full net in the standard way, then for all r and i≠j we have ωirΩrj⊆ωij and Ωirωrj⊆ωij. If ω=(ωij) ic completed with a diagonal to the full net in the second way then the inclusions are valid for all i, r, j.