This article is cited in 4 papers

RESEARCH ARTICLE

Reducibility and irreducibility of monomial matrices over commutative rings

V. M. Bondarenko^a, M. Yu. Bortos^b, R. F. Dinis^c, A. A. Tylyshchak<sup>b

a</sup> 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
$$ \left( \begin{smallmatrix} 0&\ldots&0&t^{s_n}\\ t^{s_1}&\ldots&0&0\\ \vdots&\ddots&\vdots&\vdots\\ 0&\ldots&t^{s_{n-1}}&0\\ \end{smallmatrix} \right), $$
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.

MSC: 15B33, 15A30

Received: 20.10.2013
Revised: 20.10.2013

Language: English

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