A morphism f: M→ N of left R-modules is called an n-phantom morphism if the induced
morphism Tor n R (A, f)= 0 for every (finitely presented) right R-module A. Analogously, a
morphism g: X→ Y of left R-modules is said to be an n-Ext-phantom morphism if the induced
morphism Ext R n (B, g)= 0 for every finitely presented left R-module B. We study the
properties of n-phantom and n-Ext-phantom morphisms. Some applications are given.