Abstract:
Elementary net (carpet) σ=(σij) is called closed (admissible) if the elementary net (carpet) group E(σ) does not contain a new elementary transvections. The work is related to the question of V. M. Levchuk 15.46 from the Kourovka notebook( closedness (admissibility) of the elementary net (carpet)over a field). Let R be a discrete valuation ring, K be the field of fractions of R, σ=(σij) be an elementary net of order n over R, ω=(ωij) be a derivative net for σ, and ωij is ideals of the ring R. It is proved that if K is a field of odd characteristic, then for the closedness (admissibility) of the net σ, the closedness (admissibility) of each pair (σij,σji) is sufficient for all i≠j.