On Envelopes and Covers in the Category of Short Exact Sequences

We investigate preenvelopes and precovers in the category RE of short exact sequences of left R-modules. Let C be a class of left R-modules. We prove that: (1) A1.d1 B1 is a C-preenvelope in the category R-Mod of left R-modules if and only if any exact sequence 0. A1. A2. A3. 0 has a C L-preenvelope (0. A1. A2. A3. 0) (d1,.d2,d3) (0. B1. B2. B3. 0) in RE, where C L = {0. X. Y. Z. 0. RE : X. C}; (2) If C is closed under pure submodules and direct products, then C LM = {0. X. Y. Z. 0. RE : X. C, Y. C} is a preenveloping class in RE; (3) If C is a coresolving class and an enveloping class in R-Mod, then C LM is an enveloping class in RE; (4) If C is closed under finite direct sums, then C is a preenveloping (resp. enveloping) class if and only if every short exact sequence in RE satisfying the functor Hom(-, C) leaves it exact has an C SE-preenvelope (resp. C SE-envelope), where CSE = {split exact sequences 0. X. Y. Z. 0. RE : X. C, Y. C, Z. C}. Similarly, we obtain the connections between precovers in R-Mod and precovers in RE. As applications, we characterize many rings such as perfect rings, coherent rings and Noetherian rings in terms of preenvelopes and precovers by some special short exact sequences.