EMBEDDING OF ELEMENTARY NET INTO GAP OF NETS

摘要

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_(ir)σ_(ij) (≤) 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:ω_(ij) = ∑_(k=1)~n σ_(ik)σ_(kj),k ≠ i,j. The set ω = (ω_(ij)) of elementary subgroups ω_(ij) of R is an elementarynet ω and is called an elementary derived net. The diagonal of the derived net us is defined by the formula ω_(ii) = ∑_(k≠s) σ_(ik)σ_(ks)σ_(si), 1 ≤ i ≤ n, where the sum is taken over all 1 ≤ k ≤ s ≤ n.It is proved that an elementary net a 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_(ij)Ω_(rj) (≤) ω_(ij) and Ω_(ir)ω_(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.