On hereditary reducibility of 2-monomial matrices over commutative rings

Abstract

A 2-monomial matrix over a commutative ring \(R\) is by definition any matrix of the form \(M(t,k,n)=\Phi\left(\begin{smallmatrix}I_k&0\\0&tI_{n-k}\end{smallmatrix}\right)\), \(0<k<n\), where \(t\) is a non-invertible element of \(R\), \(\Phi\) the companion matrix to \(\lambda^n-1\) and \(I_k\)  the identity \(k\times k\)-matrix. In this paper we introduce the notion of hereditary reducibility (for these matrices) and indicate one general condition of the introduced reducibility.