When is every linear transformation a sum of an idempotent one and a locally nilpotent one?

For a semisimple module M over a ring R with R/J(R) Boolean, every endomorphism of M is a sum of an idempotent endomorphism and a locally nilpotent endomorphism. As a consequence, it is proved that, for a vector space V over a division ring D, every linear transformation of V is a sum of an idempotent linear transformation and a locally nilpotent linear transformation if and only if D≅F2. This can be seen as an answer to the “local” version of a question raised by Breaz et al. in [1] on nil-cleanness of the ring of linear transformations of an infinite dimensional vector space.