Semilocal rings with Engel conditions. (Q2431759)

The relation between the Engel structure of a semilocal ring and its multiplicative group is investigated. Recall that an associative ring \(R\) with unit element is local, if its Jacobson factor ring \(R/J(R)\) is a division ring; \(R\) is semilocal if \(R/J(R)\) is left Artinian and thus a direct sum of rings each of which is isomorphic with a ring of square matrices over a division ring. The author shows how a statement about the (precise) Engel structure of a local ring can be extended to semilocal rings. Suppose that every local ring, whose multiplicative group is an \(m\)-Engel group, is an \(f(m)\)-Engel ring for some function \(f\). Then this statement even holds for semilocal rings, which are generated by their multiplicative group. Here, the latter condition cannot be omitted.