Superderivations taking nilpotent values on homogeneous elements in semiprime superalgebras

Let [Formula: see text] be a semiprime associative superalgebra over a commutative unital ring [Formula: see text] with [Formula: see text] and [Formula: see text] be a homogeneous superderivation of degree [Formula: see text]. If [Formula: see text] takes nilpotent values of fixed indices on homogeneous elements in a two-sided graded ideal [Formula: see text] of [Formula: see text] (the indices depending on the parity of the elements), then we prove that [Formula: see text] in case [Formula: see text] and, under a torsionfreeness assumption, [Formula: see text] in case [Formula: see text].