A note on \(k\)-th commutators in an associative ring
Set \([x,y]=[x,y]_1=xy-yx\), and for \(k>1\), \([x,y]_{k+1}=[[x,y]_k,y]\). The main theorem in this paper shows that if \(k>0\) is fixed, \(R\) is a 2-torsion free ring containing no nonzero nil right ideal, and for all \(a,b,c,d\in R\), \(f(a,b,c,d)=[[a,b],[c,d]]_k\) is nilpotent, then \(R\) is commutative. Consequences of this result are that if \(f(a,b,c,d)^n\) is in the center of \(R\) with \(n=n(a,b,c,d)\), then \(R\) embeds in \(M_2(C)\) for \(C\) a commutative ring, and if instead \(f(a,b,c,d)\) is either nilpotent or regular then \(R\) is a domain, is commutative, or is an order in \(M_2(F)\) for \(F\) a field. A corollary extends the main theorem to the case when \(a,b,c,d\in U\), a noncentral Lie ideal of \(R\).
scientific article; zbMATH DE number 1172077