C-PURE SUBMODULES AND C-FLAT MODULES

Abstract

Let R be a ring. A right R-module A is said to be C-flat if the kernel of any epimorphism B → A is C-pure in B, i.e. the induced map Hom(C,B) → Hom(C,A) is surjective for any cyclic right R-module C. Projective modules are C-flat and C-flat modules are weakly-flat and neat-flat. In this article, it is discussed the connections between C-flat, weakly-flat, neat-flat and singly flat modules. It is shown that C-flat modules coincide with singly-projective modules over arbitrary rings. Next, several characterizations of certain classes of rings and modules via C-purity are considered. We prove that, every C-flat module is injective if and only if R is a QF ring. Moreover, we show that R is a CF ring if and only if every FP-injective right R-module is C-flat.

Keywords

• Singly projective modules
• C-pure submodules
• CPS rings

References

1. Y. Alagöz, E. Büyükaşık, On max-flat and max-cotorsion modules. AAECC 32, (2021), 195–215.
2. F. W. Anderson, K. R. Fuller, Rings and Categories of Modules. 2nd ed. Grad. Texts in Math., Vol. 13. Berlin: Springer-Verlag. (1992).
3. G. Azumaya, Finite splitness and finite projectivity. J. Algebra, 106, (1987), 114-134.
4. E. Büyükaşık, Y. Durğun, Neat-flat modules. Comm. Algebra, 44(1), (2016), 416-428.
5. J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules. Frontiers in Mathematics, Birkhauser Verlag, Basel, 2006.
6. S. Crivei, Neat and coneat submodules of modules over commutative rings. Bull. Aust. Math. Soc. 89(2), (2014), 343-352.
7. A. Moradzadeh-Dehkordi and S. H. Shojaee, Singly injective modules, J. Algebra Appl. 18(1), (2019), 1950007, 20.
8. N. V. Dung, D.V. Huynh, P. F. Smith and R. Wisbauer, Extending modules, Putman Research Notes in Mathematics Series, Longman, Harlow, 1994.

9. Y. Durğun, Projectivity relative to closed (neat) submodules. J. Algebra Appl., (2021) DOI: 10.1142/S0219498822501146.
10. C. Faith, Algebra. II, Springer-Verlag, Berlin-New York, 1976. Ring theory, Grundlehren der Mathematischen Wissenschaften, No. 191.
11. V. A. Hiremath, S. S. Gramopadhye, Cyclic Pure Submodules. Int. J. Alg., 3(3), (2009), 125-135.
12. D. V. Huynh and P. Dan, On rings with restricted minimum condition. Arch. Math.(Basel), 51, (1988), 313-326.
13. B.H. Maddox, Absolutely pure modules, Proc. Amer. Math. Soc. 18, (1967), 155–158.
14. L. Mao and N. Ding, New characterizations of pseudo-coherent rings, Forum Math. 22, (2010), 993–1008.
15. L. Mao, A generalization of noetherian rings. Taiwanese J. Math., 12(2), (2008), 501-512.
16. G. Renault, Etude de certains anneaux a lies aux sous-modules complements dun a-module, C. R. Acad. Sci. Paris, 259, (1964), 4203-4205.
17. S. T. Rizvi and M. F. Yousif, On continuous and singular modules, Noncommutative ring theory (Athens, OH, 1989), 1990, pp. 116-124.
18. J. J. Rotman, An Introduction to Homological Algebra, in Pure Appl.Math., Vol. 85, Academic Press, New York, (1979).
19. G. Simmons, Cyclic-purity: a generalization of purity for modules. Houston J. Math., 13(1), (1987), 135-150.
20. Sklyarenko, E. G. 1978. Relative Homological Algebra in Categories of Modules. Russian Math. Surveys 33(3):97-137. Translated from Russian from Uspehi Mat. Nauk 33(201):85-120.
21. B. Stenström, High submodules and purity. Arkiv för Matematik, 7(11), (1967), 173-176.
22. R.B. Warfield Jr., Purity and algebraic compactness for modules, Pacific J. Math. 28, (1969), 699-719.
23. H. Zöschinger, Schwach-Flache moduln. Comm. Algebra, 41 (12), (2013) 4393-4407.