C-PURE SUBMODULES AND C-FLAT MODULES

Year 2021, Volume: 4 Issue: 2, 222 - 229, 31.07.2021

Yusuf Alagöz

https://doi.org/10.33773/jum.952394

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

• Y. Alagöz, E. Büyükaşık, On max-flat and max-cotorsion modules. AAECC 32, (2021), 195–215.
• F. W. Anderson, K. R. Fuller, Rings and Categories of Modules. 2nd ed. Grad. Texts in Math., Vol. 13. Berlin: Springer-Verlag. (1992).
• G. Azumaya, Finite splitness and finite projectivity. J. Algebra, 106, (1987), 114-134.
• E. Büyükaşık, Y. Durğun, Neat-flat modules. Comm. Algebra, 44(1), (2016), 416-428.
• J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules. Frontiers in Mathematics, Birkhauser Verlag, Basel, 2006.
• S. Crivei, Neat and coneat submodules of modules over commutative rings. Bull. Aust. Math. Soc. 89(2), (2014), 343-352.
• A. Moradzadeh-Dehkordi and S. H. Shojaee, Singly injective modules, J. Algebra Appl. 18(1), (2019), 1950007, 20.
• N. V. Dung, D.V. Huynh, P. F. Smith and R. Wisbauer, Extending modules, Putman Research Notes in Mathematics Series, Longman, Harlow, 1994.
• Y. Durğun, Projectivity relative to closed (neat) submodules. J. Algebra Appl., (2021) DOI: 10.1142/S0219498822501146.
• C. Faith, Algebra. II, Springer-Verlag, Berlin-New York, 1976. Ring theory, Grundlehren der Mathematischen Wissenschaften, No. 191.
• V. A. Hiremath, S. S. Gramopadhye, Cyclic Pure Submodules. Int. J. Alg., 3(3), (2009), 125-135.
• D. V. Huynh and P. Dan, On rings with restricted minimum condition. Arch. Math.(Basel), 51, (1988), 313-326.
• B.H. Maddox, Absolutely pure modules, Proc. Amer. Math. Soc. 18, (1967), 155–158.
• L. Mao and N. Ding, New characterizations of pseudo-coherent rings, Forum Math. 22, (2010), 993–1008.
• L. Mao, A generalization of noetherian rings. Taiwanese J. Math., 12(2), (2008), 501-512.
• G. Renault, Etude de certains anneaux a lies aux sous-modules complements dun a-module, C. R. Acad. Sci. Paris, 259, (1964), 4203-4205.
• S. T. Rizvi and M. F. Yousif, On continuous and singular modules, Noncommutative ring theory (Athens, OH, 1989), 1990, pp. 116-124.
• J. J. Rotman, An Introduction to Homological Algebra, in Pure Appl.Math., Vol. 85, Academic Press, New York, (1979).
• G. Simmons, Cyclic-purity: a generalization of purity for modules. Houston J. Math., 13(1), (1987), 135-150.
• 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.
• B. Stenström, High submodules and purity. Arkiv för Matematik, 7(11), (1967), 173-176.
• R.B. Warfield Jr., Purity and algebraic compactness for modules, Pacific J. Math. 28, (1969), 699-719.
• H. Zöschinger, Schwach-Flache moduln. Comm. Algebra, 41 (12), (2013) 4393-4407.