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E-mail: rio@smath.ru  DOI: 10.46698/h3104-8810-6070-x

# On the Structure of Elementary Nets Over Quadratic Field

Koibaev, V. A.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 4.
Abstract:
The structure of elementary nets over quadratic fields is studied. A set of additive subgroups $$\sigma=(\sigma_{ij})$$, $$1\leq i,j\leq n$$, of a ring $$R$$ is called a net of order $$n$$ over $$R$$ if $$\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}}$$ for all $$i$$, $$r$$, $$j$$. The same system, but without the diagonal, is called elementary ne (elementary carpet). An elementary net $$\sigma=(\sigma_{ij})$$ is called irreducible if all additive subgroups $$\sigma_{ij}$$ are different from zero. Let $$K=\mathbb{Q} (\sqrt{d}\,)$$ be a quadratic field, $$D$$ a ring of integers of the quadratic field $$K$$, $$\sigma = (\sigma_{ij})$$ an irreducible elementary net of order $$n\geq 3$$ over $$K$$, and $$\sigma_{ij}$$ a $$D$$-modules. If the integer $$d$$ takes one of the following values (22 fields): $$-1$$, $$-2$$, $$-3$$, $$-7$$, $$-11$$, $$-19$$, $$2$$, $$3$$, $$5$$, $$6$$, $$7$$, $$11$$, $$13$$, $$17$$, $$19$$, $$21$$, $$29$$, $$33$$, $$37$$, $$41$$, $$57$$, $$73$$, then for some intermediate subring $$P$$, $$D\subseteq P\subseteq K$$, the net $$\sigma$$ is conjugated by a diagonal matrix of $$D(n, K)$$ with an elementary net of ideals of the ring $$P$$.
Keywords: net, carpet, elementary net, closed net, algebraic number field, quadratic field
Language: Russian Download the full text For citation: Koibaev, V. A. On the Structure of Elementary Nets Over Quadratic Fields, Vladikavkaz Math. J., 2020, vol. 22, no. 4, pp.87-91 (in Russian). DOI 10.46698/h3104-8810-6070-x

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