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We study the structure of nets over quadratic fields. Let be a quadratic field, and let be the ring of integers of . A set of additive subgroups of is a net (carpet) of order over if for all values of the indices , , , . A net is irreducible if all additive subgroups are nonzero. A net is a -net if , . Let be an irreducible -net of order over , where are -modules. We prove that, up to conjugation by a diagonal matrix, all are fractional ideals of a fixed intermediate subring , , and all diagonal rings coincide with ; i.e., , where are integer ideals of for all , and if . Furthermore, for all and . [ABSTRACT FROM AUTHOR]

Under study are the questions of closedness of an elementary net (carpet) over a field. Every complete net is closed (admissible). We construct an example of a closed elementary net over a field of characteristic 0 which cannot be supplemented to a complete net. [ABSTRACT FROM AUTHOR]

Let R be a commutative unital ring and n ∈ ℕ, n ≥ 2. A system σ = (σij), 1 ≤ i, j ≤ n, of additive subgroups σij of R is called a net or carpet over R of order n if σirσrj ⊆ σij 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 = (ωij) of additive subgroups ωij of R, defined for i ≠ j as follows: ω ij = ∑ k = 1 n σ ik σ kj , k ≠ i, j. The set ω = (ωij) of elementary subgroups ωij 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 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 σ induces the derived net ω = (ωij) and the net Ω = (Ωij) associated with the elementary group E(σ), where ω ⊆ σ ⊆ Ω, ωirΩrj ⊆ ωij and Ωirωrj ⊆ ωij (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. [ABSTRACT FROM AUTHOR]

A set of additive subgroups σ = (σij), 1 ≤ i, j ≤ n, of a field (or ring) K is called a net of order n over K if σirσrj ⊆ σij for all values of the indices i, r, j. The same system, but without diagonal, is called an elementary net. A full or elementary net σ = (σij) is said to be irreducible if all the additive subgroups σij are different from zero. An elementary net σ is closed if the subgroup E(σ) does not contain new elementary transvections. The present paper is related to a question posed by Y. N. Nuzhin in connection with V. M. Levchuk's question No. 15.46 from the Kourovka notebook about the admissibility (closure) of elementary net (carpet) σ = (σij) over a field K. Let J be an arbitrary subset of {1,... , n}, n ≥ 3, and the cardinality m of J satisfies the condition 2 ≤ m ≤ n − 1. Let R be a commutative integral domain (non-field) with identity, and let K be the quotient field of R. An example of a net σ = (σij) of order n over K, for which the group E(σ) ∩ 〈tij(K) : i, j ∈ J〉 is not contained in the group 〈tij(σij) : i, j ∈ J〉, is constructed. [ABSTRACT FROM AUTHOR]

It is proved that every elementary carpet of nonzero additive subgroups which is associated with a Chevalley group of a Lie rank exceeding one over a locally finite field coincides, up to conjugation by a diagonal element, with a carpetwhose additive subgroups are equal to some chosen subfield of the ground field. A similar result is obtained for a full matrix carpet (a full net). [ABSTRACT FROM AUTHOR]

Let K be an algebraic extension of a field k, let σ = ( σ ) be an irreducible full (elementary) net of order n ≥ 2 (respectively, n ≥ 3) over K, while the additive subgroups σ are k-subspaces of K. We prove that all σij coincide with an intermediate subfield P, k ⊆ P ⊆ K, up to conjugation by a diagonal matrix. [ABSTRACT FROM AUTHOR]

Let σ = (σ ) be an elementary net (elementary carpet) of additive subgroups of a commutative ring (in other words, a net without diagonal), n the order of σ, ω = (ω ) the derived net with respect to σ, and Ω = (Ω ) the net associated with the elementary group E(σ). It is assumed that ω ⊆ σ ⊆ Ω and Ω is the smallest (complemented) net containing σ. The main result consists in finding the decomposition of any elementary transvection t(α) into the product of two matrices M ∈ 〈 t ( σ ), t ( σ )〉 and M ∈ G(τ), where $$ \uptau =\left(\begin{array}{ll}{\varOmega}_{11}\hfill & {\upomega}_{12}\hfill \\ {}{\upomega}_{21}\hfill & {\varOmega}_{22}\hfill \end{array}\right) $$ . [ABSTRACT FROM AUTHOR]

In this paper, the normalizer N(σ) of the elementary net group E(σ) associated with a nonsplit maximal torus T (d) in the general linear group GL(n, k) over a field k of odd characteristic is computed. The nonsplit maximal torus T = T (d) is determined by the radical extension $$ k\left(\sqrt[n]{d}\right) $$ of degree n of the ground field k (minisotropic torus). [ABSTRACT FROM AUTHOR]

The paper provides an overview of the authors' results on nets, carpets, elementary net subgroups, carpet subgroups, and carpet Lie rings. Some open questions are also formulated. [ABSTRACT FROM AUTHOR]

The paper deals with the structure of intermediate subgroups of the general linear group GL( n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let $$ K = k\left( {\sqrt[n]{d}} \right) $$ be a radical degree-n extension of a field k of odd characteristic, and let T =( d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL( n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T ( d) generate a subring R( d) in the field k.Let R be an intermediate subring, R( d) ⊆ R ⊆ k, d ∈ R. Let σ denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σ denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σ) be the subgroup generated by all transvections from the net group G(σ). In the paper it is proved that the product TE(σ) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σ,then TE(σ) ≤ H ≤ N(σ),where N(σ) is the normalizer of the elementary net group E(σ) in G. For the normalizer N(σ),the formula N(σ)= TG(σ) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

The paper belongs to the line of inquiry concerning the description of overgroups of nonsplit maximal tori. It deals with the case where the ground field is the field of fractions of a Dedekind domain. Its principal result is a construction of maximal subgroups of the above-mentioned kind. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

In the present paper, a group G(n) acting on the set of all triangular nets is determined. Bibliography: 2 titles. [ABSTRACT FROM AUTHOR]