Characterization of higher commutators (Q2118941)

For any ring \(R\), the higher commutators of \(R\) are: \(R\), a higher commutator of \(R\) with length \(1\), and \([U,V]\), a higher commutator of \(R\) with length \(s+t\), where \(U\) and \(V\) are higher commutators of \(R\) with lengths \(s\) and \(t\), respectively. Here, \([U,V]\) denotes the additive subgroup of \(R\) generated by the commutators of the form \([u,v]=uv-vu\) (\(u\in U\) and \(v\in V\)). Let \(R\) be a ring with a further property that \textit{every proper ideal of \(R\) is contained in a maximal ideal of \(R\) } (for example, when \(1\in R\)). Let \(L=R\) or \(L=[R,V]\) for some higher commutator \(V\). The main result of the paper states:\textit{ If \(L\) generates \(R\) as an ideal, then \(R=L+L^{2}\). }\\ Some consequences are \begin{itemize} \item If \(1\in[R,R]\) or \(R\) is a noncommutative simple ring, then \(R=[R,R]+[R,R]^{2}\). \item If \(L=[R,V]\) is finite, where \(V\) is a higher commutator of \(R\), and \(L\) generates \(R\) as an ideal, then \(R=L+L^{2}\) and \(R\) is a finite ring. \end{itemize}