RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Zap. Nauchn. Sem. POMI:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Zap. Nauchn. Sem. LOMI, 1982, Volume 114, Pages 37–49 (Mi znsl1765)  

Determinants in net subgroups

Z. I. Borevich, N. A. Vavilov


Abstract: Suppose $R$ is a commutative ring with 1, $\sigma=(\sigma_{ij})$ is a fixed $D$-net of ideals of $R$ of order $n$, and $G(\sigma)$ is the corresponding net subgroup of the general linear group $GL(n,R)$. There is constructed for $\sigma$ a homomorphism $\det_\sigma$ of the subgroup $G(\sigma)$ into a certain Abelian group $\Phi(\sigma)$. Let $I$ be the index set $\{1,…,n\}$. For each subset $\alpha\subseteq I$ let $\sigma(\alpha)=\sum\sigma_{ij}\sigma_{ji}$, where $i$, ranges over all indices in $\alpha$ and $j$ independently over the indices in the complement $I\backslash\alpha$ ($\sigma(I)$ is the zero ideal). Let $\det_\alpha(a)$ denote the principal minor of order $|\alpha|\leqslant n$ of the matrix $a\in G(\sigma)$ corresponding to the indices in $\alpha$, and let $\Phi(\sigma)$ be the Cartesian product of the multiplicative groups of the quotient rings $R/\sigma(\alpha)$ over all subsets $\alpha\subseteq I$. The homomorphism $\det_\sigma$ is defined as follows:
$$ \det_\sigma(a)=(\det_\alpha(a)\mod\sigma(\alpha))_\alpha\in\Phi(\sigma). $$
It is proved that if $R$ is a semilocal commutative Bezout ring, then the kernel $\operatorname{Ker}\det_\sigma$ coincides with the subgroup $E(\sigma)$ generated by all transvections in $G(\sigma)$. For these $R$ is also defined $\operatorname{Im}\det_\sigma$.

Full text: PDF file (1112 kB)

English version:
Journal of Soviet Mathematics, 1984, 27:4, 2855–2865

Bibliographic databases:

UDC: 519.46

Citation: Z. I. Borevich, N. A. Vavilov, “Determinants in net subgroups”, Modules and algebraic groups, Zap. Nauchn. Sem. LOMI, 114, "Nauka", Leningrad. Otdel., Leningrad, 1982, 37–49; J. Soviet Math., 27:4 (1984), 2855–2865

Citation in format AMSBIB
\Bibitem{BorVav82}
\by Z.~I.~Borevich, N.~A.~Vavilov
\paper Determinants in net subgroups
\inbook Modules and algebraic groups
\serial Zap. Nauchn. Sem. LOMI
\yr 1982
\vol 114
\pages 37--49
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl1765}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=669558}
\zmath{https://zbmath.org/?q=an:0552.15002|0496.15008}
\transl
\jour J. Soviet Math.
\yr 1984
\vol 27
\issue 4
\pages 2855--2865
\crossref{https://doi.org/10.1007/BF01410739}


Linking options:
  • http://mi.mathnet.ru/eng/znsl1765
  • http://mi.mathnet.ru/eng/znsl/v114/p37

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Записки научных семинаров ПОМИ
    Number of views:
    This page:112
    Full text:33

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2017