On Envelopes and Covers in the Category of Short Exact Sequences

We investigate preenvelopes and precovers in the category RE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{R}{\mathcal {E}}$$\end{document} of short exact sequences of left R-modules. Let C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {C}}$$\end{document} be a class of left R-modules. We prove that: (1) A1→δ1B1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{1}{\mathop {\rightarrow }\limits ^{\delta _{1}}}B_{1}$$\end{document} is a C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {C}}$$\end{document}-preenvelope in the category R-Mod of left R-modules if and only if any exact sequence 0→A1→A2→A3→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\rightarrow A_{1}\rightarrow A_{2}\rightarrow A_{3}\rightarrow 0$$\end{document} has a CL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\mathfrak {C}}}{\mathcal {L}}$$\end{document}-preenvelope (0→A1→A2→A3→0)→(δ1,δ2,δ3)(0→B1→B2→B3→0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0\rightarrow A_{1}\rightarrow A_{2}\rightarrow A_{3}\rightarrow 0){\mathop {\rightarrow }\limits ^{(\delta _{1},\delta _{2},\delta _{3})}}(0\rightarrow B_{1}\rightarrow B_{2}\rightarrow B_{3}\rightarrow 0)$$\end{document} in RE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{R}{\mathcal {E}}$$\end{document}, where CL={0→X→Y→Z→0∈RE:X∈C}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\mathfrak {C}}}{\mathcal {L}} = \{0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0\in \ _{R}{\mathcal {E}}: X\in {\mathfrak {C}}\}$$\end{document}; (2) If C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {C}}$$\end{document} is closed under pure submodules and direct products, then CLM={0→X→Y→Z→0∈\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\mathfrak {C}}}\mathcal {LM} = \{0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0\in $$\end{document}RE:X∈C,Y∈C}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{R}{\mathcal {E}}: X\in {\mathfrak {C}}, Y\in {\mathfrak {C}}\}$$\end{document} is a preenveloping class in RE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{R}{\mathcal {E}}$$\end{document}; (3) If C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {C}}$$\end{document} is a coresolving class and an enveloping class in R-Mod, then CLM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\mathfrak {C}}}\mathcal {LM}$$\end{document} is an enveloping class in RE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{R}{\mathcal {E}}$$\end{document}; (4) If C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {C}}$$\end{document} is closed under finite direct sums, then C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {C}}$$\end{document} is a preenveloping (resp. enveloping) class if and only if every short exact sequence in RE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{R}{\mathcal {E}}$$\end{document} satisfying the functor Hom(-,C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ Hom }(-,{\mathfrak {C}})$$\end{document} leaves it exact has an CSE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\mathfrak {C}}}\mathcal {SE}$$\end{document}-preenvelope (resp. CSE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\mathfrak {C}}}\mathcal {SE}$$\end{document}-envelope), where CSE={splitexactsequences0→X→Y→Z→0∈RE:X∈C,Y∈C,Z∈C}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{{\mathfrak {C}}}\mathcal {SE} = \{\mathrm{split \ exact \ sequences} \ 0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0\in \ _{R}{\mathcal {E}}: X\in {\mathfrak {C}}, Y\in {\mathfrak {C}}, Z\in {\mathfrak {C}}\}$$\end{document}. Similarly, we obtain the connections between precovers in R-Mod and precovers in RE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{R}{\mathcal {E}}$$\end{document}. 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.