\(N\)-radicals of rings of matrices of the Rees type (Q1311195)

By a Rees matrix ring \({\mathcal M}(T;I,\Lambda;P)\), where \(T\) is an arbitrary ring, \(I\) and \(\Lambda\) are arbitrary sets, and \(P\) is a fixed \(\Lambda \times I\) matrix over \(T\), we mean the ring of \(I \times \Lambda\) matrices over \(T\) with a finite number of non-zero entries, where addition is as usual and multiplication \(\circ\) is defined by the rule \(A \circ B = APB\). An \(N\)-radical is a supernilpotent radical which is left strong and left hereditary. The main result of the paper states that, for an \(N\)-radical \(\pi\) and a Rees matrix ring \(R = {\mathcal M}(T;I,\Lambda;P)\), a matrix \(X \in R\) belongs to \(\pi(R)\) if and only if all entries of \(PXP\) are from \(\pi(T)\).