  Given a finite dimensional algebra $A$ over an algebraically closed field we
study the relationship between the powers of the radical of a morphism in the
module category of the algebra $A$ and the induced morphism in the module
category of the endomorphism algebra of a tilting $A$-module. We compare the
nilpotency indices of the radical of the mentioned module categories. We find
an upper bound for the nilpotency index of the radical of the module category
of iterated tilted algebras of Dynkin type.
