The Study Of Small Finitistic Projective Dimension Of Rings

Let R be a ring and FR the class of left R-modules with a finitely projective resolution.A left R-module N is said to be FR-injective provided that ExtR1(F,N)= 0 for any F ∈ FR.And a right R-module M is said to be FR-flat provide that Tor1R(M,F)＝ 0 for any F ∈ FR.We obtain some basic properties of these modules.It is proved that the class of left FR-injective(right FR-flat)modules is both preenveloping and covering.The derived functors Extn(-,-)and Torn(-,-)are obtained by left and right FRI-resolution of left R-moduels and left and right FRF-resolution of right R-modules.Some new ways are given to calculate the left small finitistic projective dimension.