Geodesic
Reducibility and irreducibility of monomial matrices over commutative rings
Algebra and discrete mathematics, Tome 16 (2013) no. 2, pp. 171-187

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Let $R$ be a local ring with nonzero Jacobson radical. We study monomial matrices over $R$ of the form $$ \left( \begin{smallmatrix} 0\ldots0^{s_n}\\ t^{s_1}\ldots00\\ \vdots\ddots\vdots\vdots\\ 0\ldots^{s_{n-1}}0\\ \end{smallmatrix} \right), $$ 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. 
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     author = {V. M. Bondarenko and M. Yu. Bortos and R. F. Dinis and A. A. Tylyshchak},
     title = {Reducibility and irreducibility of monomial matrices over commutative rings},
     journal = {Algebra and discrete mathematics},
     pages = {171--187},
     publisher = {mathdoc},
     volume = {16},
     number = {2},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2013_16_2_a2/}
}
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V. M. Bondarenko; M. Yu. Bortos; R. F. Dinis; A. A. Tylyshchak. Reducibility and irreducibility of monomial matrices over commutative rings. Algebra and discrete mathematics, Tome 16 (2013) no. 2, pp. 171-187. http://geodesic.mathdoc.fr/item/ADM_2013_16_2_a2/