Abstract: Let n be a positive integer and R be an associative ring with identity. Firstly, the definition of n-Tor-pair over the ring R is introduced. Secondly, some properties of n-Tor-pairs are given, and hereditary n-Tor-pairs are discussed. Finally, the equivalent characterization of the direct product of modules in X having finite X-projective dimension is given when (X,Y) is an n-Tor-pair.

Key words: n-Tor-pair, hereditary n-Tor-pair, X-projective dimension