Envelopes and covers by modules of finite injective and projective dimensions

In this paper, we study the existence of L-perpendicular to-envelopes, L-en velopes, D-perpendicular to-envelopes, D-covers, and L-covers where L and D denote the classes of modules of injective and projective dimension less t han or equal to a natural number n, respectively. We prove that over any ri ng R, special D-perpendicular to-preenvelopes and special D-precovers alway s exist. If the ring is noetherian, the same holds for L-perpendicular to-e nvelopes, and for D-perpendicular to-envelopes and D-covers when the ring i s perfect. When inj.dim R less than or equal to n then L-covers exist, and if R is such that a given class of homomorphisms is closed under well order ed direct limits then L-envelopes exist. (C) 2001 Academic Press.