The Shulchan Aruch (Y.D. 257:22) rules that whatever a slave has belongs to his master, even if the master or anybody else gives him something on the express condition that his master does not acquire it.

Consequently, a slave cannot own a slave any more than he cannot own anything else.

In fact, the only thing a slave can ever acquire for himself, is his document of freedom, which he acquires simultaneously with his freedom.

Be'virchas Kol Tuv

Eliezer Chrysler