Radical Rings with Engel Conditions

An associative ring R without unity is called radical if it coincides with its Jacobson radical, which means that the set of all elements of R forms a group denoted by R under the circle operation r s = r + s + rs on R. It is proved that, for a radical ring R, the group R satisfies an n-Engel condition for some positive integer n if and only if R is m-Engel as a Lie ring for some positive integer m depending only on n.