$(n,d)$-$\mathcal{X}_R$-phantom and $(n,d)$-$_R\mathcal{X}$-cophantom morphisms

Abstract

Several authors have been interested in some like phantom morphisms such as $d$-phantoms, $d$-$\operatorname{Ext}$-phantoms, neat-phantom morphisms, clean- cophantom morphisms, $RD$-phantom morphisms and $RD$-$\operatorname{Ext}$-phantom morphisms. In this paper, we prove that these notions can be unified. We are mainly interested in proving that the majority of the existing results hold true in our general framework.

Keywords

• $(n,d)$-$\mathcal{X}_R$-phantom morphism
• $(n,d)$-$_R\mathcal{X}$-cophantom morphism
• $n$-$\mathcal{X}$-coherent ring

References

1. M. Auslander and O. Solberg, Relative homology and representation theory I, relative homology and homologically finite categories, Comm. Algebra, 21(9) (1993), 2995-3031.
2. D. Bennis, $n$-$\mathcal{X}$-coherent rings, Int. Electron. J. Algebra, 7 (2010), 128-139.
3. J. Chen and N. Ding, On $n$-coherent rings, Comm. Algebra, 24(10) (1996), 3211-216.
4. S. Crivei, M. Prest and B. Torrecillas, Covers in finitely accessible categories, Proc. Amer. Math. Soc., 138(4) (2010), 1213-1221.
5. X. H. Fu, P. A. Guil Asensio, I. Herzog and B. Torrecillas, Ideal approximation theory, Adv. Math., 244 (2013), 750-790.
6. I. Herzog, The phantom cover of a module, Adv. Math., 215 (2007), 220-249.
7. I. Herzog, Contravariant functors on the category of finitely presented modules, Israel J. Math., 167 (2008), 347-410.
8. K. Lan and B. Lu, On $n$-phantom and $n$-Ext-phantom morphisms, Taiwanese J. Math., 25 (2021), 941-957.

9. L. X. Mao, On covers and envelopes in some functor categories, Comm. Algebra, 41(5) (2013), 1655-1684.
10. L. X. Mao, Precovers and preenvelopes by phantom and Ext-phantom morphisms, Comm. Algebra, 44(4) (2016), 1704-1721.
11. L. X. Mao, RD-phantom and RD-Ext-phantom morphisms, Filomat, 32(8) (2018), 2883-2895.
12. L. X. Mao, Higher phantom and Ext-phantom morphisms, J. Algebra Appl., 17(1) (2018), 1850012 (15 pp).
13. L. X. Mao, Higher phantom morphisms with respect to a subfunctor of Ext, Algebr. Represent. Theory, 22(2) (2019), 407-424.
14. L. X. Mao, Neat-phantom and clean-cophantom morphisms, J. Algebra Appl., 20(9) (2021), 2150172 (24 pp).
15. L. X. Mao and N. Q. Ding, Envelopes and covers by modules of finite FP-injective and flat dimensions, Comm. Algebra, 35(3) (2007), 833-849.
16. C. A. McGibbon, Phantom maps, in Handboook of Algebraic Topology, North-Holland, Amsterdam, (1995), 1209-1257.
17. J. J. Rotman, An Introduction to Homological Algebra, Pure and Applied Mathematics, 85, Academic Press, Inc., New York-London, 1979.