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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2013, Volume 16, Issue 2, Pages 171–187 (Mi adm445)

RESEARCH ARTICLE

Reducibility and irreducibility of monomial matrices over commutative rings

V. M. Bondarenkoa, M. Yu. Bortosb, R. F. Dinisc, A. A. Tylyshchakb

a Institute of Mathematics, Tereshchenkivska 3, 01601 Kyiv, Ukraine
b Faculty of Mathematics, Uzhgorod National Univ., Universytetsyka str., 14, 88000 Uzhgorod, Ukraine
c Faculty of Mechanics and Mathematics, Kyiv National Taras Shevchenko Univ., Volodymyrska str., 64, 01033 Kyiv, Ukraine

Abstract: Let $R$ be a local ring with nonzero Jacobson radical. We study monomial matrices over $R$ of the form
$$( \begin{smallmatrix} 0&\ldots&0&t^{s_n} t^{s_1}&\ldots&0&0 \vdots&\ddots&\vdots&\vdots 0&\ldots&t^{s_{n-1}}&0 \end{smallmatrix} ),$$
and give a criterion for such matrices to be reducible when $n\leq 6$, $s_1\ldots,s_n\in\{0,1\}$ and the radical is a principal ideal with generator $t$. We also show that the criterion does not hold for $n=7$.

Keywords: irreducible matrix, similarity, local ring, Jacobson radical.

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Bibliographic databases:
MSC: 15B33, 15A30
Revised: 20.10.2013
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Citation: V. M. Bondarenko, M. Yu. Bortos, R. F. Dinis, A. A. Tylyshchak, “Reducibility and irreducibility of monomial matrices over commutative rings”, Algebra Discrete Math., 16:2 (2013), 171–187

Citation in format AMSBIB
\Bibitem{BonBorDin13} \by V.~M.~Bondarenko, M.~Yu.~Bortos, R.~F.~Dinis, A.~A.~Tylyshchak \paper Reducibility and irreducibility of monomial matrices over commutative rings \jour Algebra Discrete Math. \yr 2013 \vol 16 \issue 2 \pages 171--187 \mathnet{http://mi.mathnet.ru/adm445} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3186082} 

1. Vitaliy M. Bondarenko, Maria Yu. Bortos, Ruslana F. Dinis, Alexander A. Tylyshchak, “Indecomposable and irreducible $t$-monomial matrices over commutative rings”, Algebra Discrete Math., 22:1 (2016), 11–20