On $n$-phantom and $n$-Ext-phantom Morphisms

This paper extends many conclusions based on phantom envelopes and Ext-phantom covers of modules, and we find that many important properties still hold after replacing phantom and Ext-phantom with $n$-phantom and $n$-Ext-phantom respectively. In addition, we also obtain some extra results. Specifically, we give a characterization of the weak dimensions of rings in terms of $n$-phantom envelopes and $n$-Ext-phantom covers of modules with the unique mapping property respectively. We show that $\operatorname{wD}(R) \leq 2n$ whenever every right $R$-module has an $n$-phantom envelope with the unique mapping property or every left $R$-module has an $n$-Ext-phantom cover with the unique mapping property over left coherent rings.

[1]  R. Damiano.  Coflat rings and modules. , 1979 .

[2]  A. Neeman.  The Brown representability theorem and phantomless triangulated categories , 1992 .

[3]  I. Herzog.  The phantom cover of a module , 2007 .

[4]  I. Herzog,et al.  Ideal approximation theory , 2013 .

[5]  Nanqing Ding,et al.  Envelopes and Covers by Modules of Finite FP-Injective and Flat Dimensions , 2007 .

[6]  Jianlong Chen,et al.  Relative coherence and preenvelopes , 1993 .

[7]  L. Mao.  Higher Phantom Morphisms with Respect to a Subfunctor of Ext , 2019 .

[8]  Katharina Weiss,et al.  Lectures On Modules And Rings , 2016 .

[9]  Robert Warfield,et al.  Purity and algebraic compactness for modules , 1969 .

[10]  Overtoun M. G. Jenda,et al.  Relative homological algebra , 1956 .

[11]  Saroj Jain.  Flat and FP-injectivity , 1973 .

[12]  I. Herzog.  Contravariant functors on the category of finitely presented modules , 2008 .

[13]  X. Fu,et al.  Powers of the phantom ideal , 2013, 1312.5348.

[14]  C. Weibel,et al.  An Introduction to Homological Algebra: References , 1960 .

[15]  Jianlong Chen,et al.  Coherence and Generalized Morphic Property of Triangular Matrix Rings , 2014 .

[16]  D. Benson,et al.  Phantom maps and purity in modular representation theory, I , 2002 .

[17]  L. Mao.  Higher phantom and Ext-phantom morphisms , 2018 .

[18]  L. Mao.  On Covers and Envelopes in Some Functor Categories , 2013 .

[19]  E. Enochs.  Injective and flat covers, envelopes and resolvents , 1981 .

[20]  B. Stenström.  Coherent Rings and Fp-Injective Modules , 1970 .

[21]  Sven Strauss,et al.  Theory of categories , 1965 .