Distributive multiplication rings (Q1203468)

A ring \(R\) is said to be a left \(n\)-distributive multiplication ring if \(aa_ 1\dots a_ n=aa_ 1aa_ 2\dots aa_ n\) for all \(a,a_ 1,\dots,a_ n\in R\) (\(n\geq 2\)). If this is so, then the set \(N\) of nilpotent elements is an ideal of \(R\), \(N^{n+1}=0\) and \(R/N\) is a semiprime ring satisfying \(x^ n=x\). If, moreover, \(R/N\) possesses a unit element, then \(R\) is isomorphic to \(N\oplus R/N\).