Envelopes and Covers by Modules of Finite Injective and Projective Dimensions

In this paper, we study the existence of -envelopes, -envelopes, -envelopes, -covers, and -covers where and denote the classes of modules of injective and projective dimension less than or equal to a natural number n, respectively. We prove that over any ring R, special -preenvelopes and special -precovers always exist. If the ring is noetherian, the same holds for - envelopes, and for -envelopes and -covers when the ring is perfect. When inj.dimR ≤ n then -covers exist, and if R is such that a given class of homomorphisms is closed under well ordered direct limits then -envelopes exist.