Reducibility and irreducibility of monomial matrices over commutative rings

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 \(n=7\).