NOTES ON FP-PROJECTIVE MODULES AND FP-INJECTIVE MODULES

Throughout this paper, all rings are associative with identity and all modules are unitary. We write MR (RM) to indicate a right (left) R-module, and freely use the terminology and notations of [1, 4, 9]. A right R-module M is called FP -injective [11] if ExtR(N, M) = 0 for all finitely presented right R-modules N . The concepts of FP -projective dimensions of modules and rings were introduced and studied in [5]. For a right R-module M , the FP -projective dimension fpdR(M) of M is defined to be the smallest integer n ≥ 0 such that Ext R (M, N) = 0 for any FP -injective right R-module N . If no such n exists, set fpdR(M) = ∞. M is called FP -projective if fpdR(M) = 0. We note that the concept of FP -projective modules coincides with that of finitely covered modules introduced by J. Trlifaj (see [12, Definition 3.3 and Theorem 3.4]). It is clear that fpdR(M) measures how far away a right R-module M is from being FP -projective. The right FP -projective dimension rfpD(R) of a ring R is defined as sup{fpdR(M) : M is a finitely