KaydedilenlerDerginizi ekleyin

Dergiler / Hacettepe Journal of Mathematics and Statistics / 2021 / Cilt: 50 - Sayı: 2

On MF-projective modules

Yusuf Alagöz
Sayfa
471–482
Yayın Tarihi
2022-10-18
DOI
—
🏛️Bu makalenin bibliyografik verileri ve tam metin bağlantısı DergiPark üzerinden sağlanmıştır.
DergiPark Kaydı ↗
✨

Yapay Zeka Hızlı Bakış & Çıkarımlar

AI Insight
⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

Finally, we characterize the rings whose $MF$-projective modules are projective.

📌 3 Maddede Bu Makale

🎯
Amaç & Hipotez: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.
📊
Temel Bulgular & Veri: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 $i\geq 1$.
💡
Sonuç & Katkı:Finally, we characterize the rings whose $MF$-projective modules are projective.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:(Max-)flat modules · MF-projective modules · Max-hereditary rings

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.

🔖

Bu Makaleye Atıf Yap

Alagöz, Y. (2022). On MF-projective modules. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 2, 471–482. https://ajindex.com/articles/529530

📄 Tam Metin PDF Okuyucu

📄

PDF görüntüleyici hazırlanıyor...

Kaynakça (23)

  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, A