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 degrees under the circle operation r circle s = r + s + rs on R. It is proved that, for a radical ring R, the group R degrees 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 In depending only on n, (C) 2000 Academic Press.