On the structure of nets over quadratic fields. (Russian. English summary) Zbl 07620410

Ikaev, S. S.; Koĭbaev, V. A.; Likhacheva, A. O.

Vladikavkaz. Mat. Zh. 24, No. 3, 87-95 (2022); translation in Sib. Math. J. 64, No. 3, 725-730 (2023).

Summary: The structure of nets over quadratic fields is studied. Let \(K=\mathbb{Q} (\sqrt{d})\) be a quadratic field, \( \mathfrak{D}\) the ring of integers of the quadratic field \(K\). A set of additive subgroups \(\sigma=(\sigma_{ij})\), \(1\leq i,j\leq n\), of a field \(K\) is called a net of order \(n\) over \(K\) if \(\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}}\) for all values of the index \(i, r, j\). A net \(\sigma=(\sigma_{ij})\) is called irreducible if all additive subgroups \(\sigma_{ij}\) are different from zero. A net \(\sigma = (\sigma_{ij})\) is called a \(D\)-net if \(1 \in\tau_{ii}, 1\leq i\leq n\). Let \(\sigma = (\sigma_{ij})\) be an irreducible \(D\)-net of order \(n\geq 2\) over \(K\), where \(\sigma_{ij}\) are \(\mathfrak{D} \)-modules. We prove that, up to conjugation diagonal matrix, all \(\sigma_{ij}\) are fractional ideals of a fixed intermediate subring \(P\), \(\mathfrak{D}\subseteq P \subseteq K\), and all diagonal rings coincide with \(P: \sigma_{11}=\sigma_{22}=\ldots =\sigma_{nn}=P,\) where \(\sigma_{ij}\subseteq P\) are integer ideals of the ring \(P\) for any \(i<j\), if \(i>j\), then \(P\subseteq\sigma_{ij} \). For any \(i, j\) we have \(\sigma_{1j}\subseteq\sigma_{ij} \).

MSC:

16S99	Associative rings and algebras arising under various constructions
20H25	Other matrix groups over rings
20G15	Linear algebraic groups over arbitrary fields

Keywords:

nets; carpets; algebraic number field; quadratic field

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