Radical rings with Engel conditions
In a ring \(R\) the Lie multiplication and the Lie commutators are defined inductively by \([r,s]=rs-sr\), and \([r_1,\dots,r_{n+1}]=[[r_1,\dots,r_n],r_{n+1}]\). A ring \(R\) is \(n\)-Engel, if \([r,s,\dots,s]=0\) where \(s\) appears exactly \(n\) times. \(n\)-Engel groups are defined correspondingly where the usual group commutator replaces the Lie commutator. It is proved that for a Jacobson radical ring \(R\) the adjoint group \(R^*\) is \(n\)-Engel if and only if \(R\) is \(m\)-Engel where \(m\) depends on \(n\). The proof is achieved by several Lemmas, some of technical character, others of interest on their own right (e.g. the factor ring \(R/L\) of an \(n\)-Engel ring \(R\) by its Levitzki radical \(L\) is commutative).
scientific article; zbMATH DE number 1525277