Direct sums of distributive modules

A module is said to be distributive if the lattice of all its submodules is distributive. A direct sum of distributive modules is called a semidistributive module. It is proved that the prime radical of the ring of endomorphisms of a finite direct sum of distributive modules contains all one-sided nilideals of the ring of endomorphisms of this module. A semiprime ring with the maximal condition for right annihilators that decomposes into a direct sum of distributive right ideals is a finite direct product of prime rings.