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

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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### References:

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