In this paper, we first study RD -projective and RD -injective modules using, among other things, covers and envelopes. Some new characterizations for them are obtained. Then we introduce the RD -projective and RD -injective dimensions for modules and rings. | Turk J Math 35 (2011) , 187 – 205. ¨ ITAK ˙ c TUB doi: Properties of RD -projective and RD -injective modules Lixin Mao Abstract In this paper, we first study RD -projective and RD -injective modules using, among other things, covers and envelopes. Some new characterizations for them are obtained. Then we introduce the RD -projective and RD -injective dimensions for modules and rings. The relations between the RD -homological dimensions and other homological dimensions are also investigated. Key word and phrases: RD -projective module, RD -injective module, RD -flat module, RD -projective dimension, RD -injective dimension, (pre)envelope, (pre)cover. 1. Introduction Following [20], an exact sequence 0 → A → B → C → 0 of left R -modules is called RD -exact if for every a ∈ R , the sequence Hom(R/Ra, B) → Hom(R/Ra, C) → 0 is exact, or equivalently, the sequence 0 → (R/aR) ⊗ A → (R/aR) ⊗ B is exact. A left R -module M is said to be RD -projective if for every RD exact sequence 0 → A → B → C → 0 of left R -modules, the sequence 0 → Hom(M, A) → Hom(M, B) → Hom(M, C) → 0 is exact. A left R -module N is called RD -injective if for every RD -exact sequence 0 → A → B → C → 0 of left R -modules, the sequence 0 → Hom(C, N ) → Hom(B, N ) → Hom(A, N ) → 0 is exact. According to [3], a right R -module F is called RD -flat if for every RD -exact sequence 0 → A → B → C → 0 of left R -modules, the sequence 0 → F ⊗ A → F ⊗ B → F ⊗ C → 0 is exact. For more details about RD -projective, RD -injective and RD -flat modules, we refer the reader to [2, 3, 6, 15, 16, 19, 20]. Though the RD -property is most important and well known in the commutative case, so far not much is known about the RD -property in the theory of modules over non-commutative rings. In this paper, we will establish several basic results for RD -projective, RD -injective and RD -flat modules over a general ring. In Section 2 of this paper, we obtain some properties of RD -projective and RD .
