×

Full and elementary nets over the quotient field of a principal ideal ring. (English. Russian original) Zbl 1401.16031

J. Math. Sci., New York 234, No. 2, 141-147 (2018); translation from Zap. Nauchn. Semin. POMI 455, 42-51 (2017).
Summary: Let \(K\) be the quotient field of a principal ideal ring \(R\), and let \(\sigma = (\sigma_{ij})\) be a full (respectively, elementary) net of order \(n\geq2\) (respectively, \(n\geq3\)) over \(K\) such that the additive subgroups \(\sigma_{ij}\) are nonzero \(R\)-modules. It is proved that, up to conjugation by a diagonal matrix, all \(\sigma_{ij}\) are ideals of a fixed intermediate subring \(P\), \(R\subseteq P\subseteq K\).

MSC:

16S50 Endomorphism rings; matrix rings
16S34 Group rings
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
13F10 Principal ideal rings
PDFBibTeX XMLCite
Full Text: DOI MNR

References:

[1] Borevich, ZI, Subgroups of linear groups rich in transvections, Zap. Nauchn. Semin. POMI, 75, 22-31, (1978) · Zbl 0446.20026
[2] Koibaev, VA, Nets associated with the elementary nets, Vladikavkaz. Mat. Zh., 12, 39-43, (2010) · Zbl 1218.20035
[3] Koibaev, VA, Elementary nets in linear groups, Trudy Inst. Mat. Mekh. UrO RAN, 17, 134-141, (2011)
[4] Koibaev, VA; Nuzhin, YN, Subgroups of the Chevalley groups and Lie rings definable by a collection of additive subgroups of the initial ring, Fundam. Prikl. Matem., 18, 75-84, (2013) · Zbl 1311.20048
[5] Kuklina, SK; Likhacheva, AO; Nuzhin, YN, On closeness of carpets of Lie type over commutative rings, Trudy Inst. Mat. Mekh. UrO RAN, 21, 192-196, (2015)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.