On MF-projective modules

Year 2021, Volume: 50 Issue: 2, 471 - 482, 11.04.2021

Yusuf Alagöz

https://doi.org/10.15672/hujms.731098

Cited By: 2

Abstract

In this paper, we study the left orthogonal class of max-flat modules which are the homological objects related to s-pure exact sequences of modules and module homomorphisms. Namely, a right module $A$ is called MF-projective if ${Ext}^{1}_{R}(A,B)=0$ for any max-flat right $R$-module $B$, and $A$ is called strongly MF-projective if ${Ext}^{i}_{R}(A,B)=0$ for all max-flat right $R$-modules $B$ and all $i\geq 1$. Firstly, we give some properties of $MF$-projective modules and SMF-projective modules. Then we introduce and study MF-projective dimensions for modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. We characterize some classes of rings such as perfect rings, $QF$ rings and max-hereditary rings by $(S)MF$-projective modules. We also study the rings whose right ideals are MF-projective. Finally, we characterize the rings whose $MF$-projective modules are projective.

Keywords

(Max-)flat modules , MF-projective modules , Max-hereditary rings

References

• [1] Y. Alagöz, On m-injective and m-projective modules, Math. Sci. Appl. E-Notes, 8, 46–50, 2020.
• [2] E. Büyükaşık and Y. Durğun, Absolutely s-pure modules and neat-flat modules, Comm. Algebra, 43 (2), 384–399, 2015.
• [3] E. Büyükaşık and Y. Durğun, Neat-flat modules. Comm. Algebra 44 (1), 416-428, 2016.
• [4] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, Frontiers in Math- ematics, Birkh¨auser Verlag, Basel, 2006.
• [5] I. Crivei, s-pure submodules, Int. J. Math. Math. Sci. 4, 491–497, 2005.
• [6] S. Crivei, Neat and coneat submodules of modules over commutative rings, Bull. Aust. Math. Soc. 89 (2), 343–352, 2014.
• [7] Y. Durğun, On some generalizations of closed submodules, Bull. Korean Math. Soc. 52 (5), 1549–1557, 2015.
• [8] E.E. Enochs and O.M.G Jenda, Relative homological algebra, de Gruyter, Berlin, 2000.
• [9] E.E. Enochs, O.M.G. Jenda and J.A. Lopez-Ramos, The existence of Gorenstein flat covers, Math. Scand. 94 (1), 46–62, 2004.
• [10] C. Faith, Algebra. II, Springer-Verlag, Berlin-New York, 1976. Ring theory, Grundlehren der Mathematischen Wissenschaften, No. 191.
• [11] X. Fu, H. Zhu and N. Ding, On Copure Projective Modules and Copure Projective Dimensions, Comm. Algebra, 40 (1), 343–359, 2012.
• [12] L. Fuchs, Neat submodules over integral domains, Period. Math. Hungar. 64 (2), 131–143, 2012.
• [13] M.F. Hamid, Coneat injective modules, Missouri J. Math. Sci. 31 (2), 201–211, 2019.
• [14] K. Honda, Realism in the theory of abelian groups I, Comment. Math. Univ. St. Pauli 5, 37-75, 1956.
• [15] H. Holm and P. Jorgensen, Covers, precovers, and purity, Illinois J. Math. 52 (2), 691–703, 2008.
• [16] C.U. Jensen and D. Simon, Purity and generalized chain conditions, J. Pure Appl. Algebra 14, 297-305, 1979.
• [17] T.Y. Lam, Lectures on modules and rings, Springer-Verlag, New York, 1999.
• [18] L. Mao, When does every simple module have a projective envelope?, Comm. Algebra, 35 (5), 1505–1516, 2007.
• [19] E. Mermut and Z. Türkoğlu, Neat submodules over commutative rings, Comm. Alge- bra, 48 (3), 1231–1248, 2020.
• [20] W.K. Nicholson and J.F. Watters, Rings with projective socle, Proc. Amer. Math. Soc. 102, 443-450, 1988.
• [21] J. Rada and M. Saorin, Rings characterized by (pre)envelopes and (pre)covers of their modules, Comm. Algebra, 26 (3), 899–912, 1998.
• [22] V.S. Ramamurthi, On the injectivity and flatness of certain cyclic modules, Proc. Amer. Math. Soc. 48, 21–25, 1975.
• [23] J.J. Rotman, An Introduction to Homological Algebra, in Pure Appl.Math., Vol. 85, Academic Press, New York, 1979.
• [24] P.F. Smith, Injective modules and prime ideals. Comm. Algebra, 9 (9), 989–999, 1981.
• [25] M.Y. Wang, Frobenius structure in algebra (chinese). Science Press, Beijing, 2005.
• [26] M.Y. Wang and G. Zhao, On maximal injectivity, Acta Math. Sin. (Engl. Ser.) 21 (6), 1451–1458, 2005.
• [27] Y. Xiang, Max-injective, max-flat modules and max-coherent rings, Bull. Korean Math. Soc. 47 (3), 611–622, 2010.