Universal covering rings and their radicals (Q752150)

Let \(\phi\) be an automorphism on a skew field \(\Delta\) such that \(\phi^{n-1}=\text{id}_{\Delta}\) \((n\geq 2)\) and \(c\) be a central element of \(\Delta\). In the ring \(\Delta_ m\) of all \(m\times m\) matrices over \(\Delta\) we define an \(n\)-ary multiplication as follows: \(x_ 1x_ 2\dots x_{n-1}x_ n=cx_ 1\phi^{n-2}(x_ 2)\cdots\phi(x_{n-1})x_ n\) for \(x_ 1,x_ 2,\dots,x_ n\in\Delta_ m\). By \(\Delta^ n_ m(c,\phi)\) we denote the \((2,n)\)-ring of all matrices of \(\Delta_ m\) with ordinary addition and \(n\)-ary multiplication. The author considers a universal covering ring [see \textit{G. Čupona}, Bull. Soc. Math. Phys. Macédoine 16, 5-10 (1965; Zbl 0144.25503)] for the \((2,n)\)-ring \(\Delta^ n_ m(c,\phi)\).