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Homological objects of min-pure exact sequences

Year 2024, , 342 - 355, 23.04.2024
https://doi.org/10.15672/hujms.1186239

Abstract

In a recent paper, Mao has studied min-pure injective modules to investigate the existence of min-injective covers. A min-pure injective module is one that is injective relative only to min-pure exact sequences. In this paper, we study the notion of min-pure projective modules which is the projective objects of min-pure exact sequences. Various ring characterizations and examples of both classes of modules are obtained. Along this way, we give conditions which guarantee that each min-pure projective module is either injective or projective. Also, the rings whose injective objects are min-pure projective are considered. The commutative rings over which all injective modules are min-pure projective are exactly quasi-Frobenius. Finally, we are interested with the rings all of its modules are min-pure projective. We obtain that a ring $R$ is two-sided K\"othe if all right $R$-modules are min-pure projective. Also, a commutative ring over which all modules are min-pure projective is quasi-Frobenius serial. As consequence, over a commutative indecomposable ring with $J(R)^{2}=0$, it is proven that all $R$-modules are min-pure projective if and only if $R$ is either a field or a quasi-Frobenius ring of composition length $2$.

Supporting Institution

SCHOOL OF MATHEMATICS, INSTITUTE FOR RESEARCH IN FUNDAMENTAL SCIENCES (IPM), TEHRAN, IRAN

Project Number

1401160414

References

  • [1] Y. Alagöz and E. Büyükasık, On max-flat and max-cotorsion modules, AAECC 32, 195-215, 2021.
  • [2] Y. Alagöz, S. Göral Benli and E. Büyükasık, On simple-injective modules, J. Algebra Appl, 2022. https://doi.org/10.1142/S0219498823501384.
  • [3] M. Arabi-Kakavand, Sh. Asgari and Y. Tolooei, Noetherian rings with almost injective simple modules, Comm. Algebra, 45 (8), 3619-3626, 2017.
  • [4] M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S.H. Shojaee, On left Köthe rings and a generalization of Köthe-Cohen-Kaplansky Theorem, Proc. Amer. Math. Soc. 142, 2625–2631, 2014.
  • [5] M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S.H. Shojaee, On FCPurity and I-Purity of Modules and Köthe Rings, Comm. Algebra, 42 (5), 2061–2081, 2014.
  • [6] M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S.H. Shojaee, C-Pure Projective Modules, Comm. Algebra, 41, 4559–4575, 2013.
  • [7] J.E. Björk, Rings satisfying certain chain conditions, J. Reine Angew Math. 245, 63–73, 1970.
  • [8] E. Büyükasık and Y. Durgun, Absolutely s-pure modules and neat-flat modules Comm. Algebra, 43 (2), 384–399, 2015.
  • [9] I.S. Cohen and I. Kaplansky, Rings for which every module is a direct sum of cyclic modules Math. Z. 54, 97–101, 1951.
  • [10] P.M. Cohn, On the free product of associative rings, Math. Z. 71, 380–398, 1959.
  • [11] F. Couchot, RD-flatness and RD-injectivity, Comm. Algebra, 34, 3675–3689, 2006.
  • [12] R.R. Colby, Rings which have flat injective modules, J. Algebra 35, 239–252, 1975.
  • [13] K. Divaani-Aazar, M.A. Esmkhani and M. Tousi, A criterion for rings which are locally valuation rings, Colloq. Math. 116, 153–164, 2009.
  • [14] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules, Pitman Research Notes in Mathematics Series, vol. 313, Longman Scientific and Technical, Harlow, 1994.
  • [15] E.E. Enochs and O.M.G. Jenda, Relative homological algebra, Berlin: Walter de Gruyter, 2000.
  • [16] C. Faith, Algebra. II, Springer-Verlag, Berlin-New York, 1976.
  • [17] C. Faith and E.A. Walker, Direct sum representation of injective modules, J. Algebra, 5 (2), 203–221, 1967.
  • [18] A. Facchini, Module Theory, Birkhauser Verlag-Basel, 1998.
  • [19] A.I. Generalov, Weak and $\omega$-high purities in the category of modules, Mat. Sb. (N.S.) 34 (3), 345–356, 1978.
  • [20] K.R. Goodearl and R.B. Warfield, An Introduction to Noncommutative Noetherian Rings 2nd ed. Cambridge: Cambridge University Press, 2004.
  • [21] M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Algebra Appl. 3 (3), 1–26, 2004.
  • [22] M. Harada, Self mini-injective rings, Osaka J. Math. 19 (2), 587–597, 1982.
  • [23] H. Holm and P. Jorgensen, Covers, precovers, and purity, Illinois J. Math. 52 (2), 691–703, 2008.
  • [24] T. Honold, Characterization of finite Frobenius rings, Arch. Math. 76 (6), 406–415, 2001.
  • [25] G. Köthe, Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring, (German). Math. Z. 39, 31–44, 1935.
  • [26] T.Y. Lam, Lectures on modules and rings Springer-Verlag, New York, 1999.
  • [27] Z.K. Liu, Rings with flat left socle, Comm. Algebra, 23 (6), 1645–1656, 1995.
  • [28] L. Mao, On mininjective and min-flat modules, Publ. Math. Debrecen 72 (3-4), 347–358, 2008.
  • [29] L. Mao, Min-flat modules and min-coherent rings, Comm. Algebra, 35 (2), 635–650, 2007.
  • [30] A.R. Mehdi, Purity relative to classes of finitely presented modules, J. Algebra Appl. 12 (8), 1350050, 2013.
  • [31] A. Moradzadeh-Dehkordi and F. Couchot, RD-flatness and RD-injectivity of simple modules, J.Pure Appl. Algebra 226, 107034, 2022.
  • [32] W.K. Nicholson and J.F. Watters, Rings with projective socle, Proc. Amer. Math. Soc. 102, 443–450, 1988.
  • [33] W.K. Nicholson and M.F. Yousif, Mininjective rings, J. Algebra 187, 548–578, 1997.
  • [34] G. Puninski, M. Prest and P. Rothmaler, Rings described by various purities, Comm. Algebra, 27, 2127–2162, 1999.
  • [35] B. Stenström, Pure submodules, Ark. Mat. 7, 159–171, 1967.
  • [36] R.B. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28, 699–719, 1969.
  • [37] R. Wisbauer, Foundations of Module and Ring Theory, New York: Gordon and- Breach, 1991.
  • [38] J.A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (3), 555–575, 1999.
Year 2024, , 342 - 355, 23.04.2024
https://doi.org/10.15672/hujms.1186239

Abstract

Project Number

1401160414

References

  • [1] Y. Alagöz and E. Büyükasık, On max-flat and max-cotorsion modules, AAECC 32, 195-215, 2021.
  • [2] Y. Alagöz, S. Göral Benli and E. Büyükasık, On simple-injective modules, J. Algebra Appl, 2022. https://doi.org/10.1142/S0219498823501384.
  • [3] M. Arabi-Kakavand, Sh. Asgari and Y. Tolooei, Noetherian rings with almost injective simple modules, Comm. Algebra, 45 (8), 3619-3626, 2017.
  • [4] M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S.H. Shojaee, On left Köthe rings and a generalization of Köthe-Cohen-Kaplansky Theorem, Proc. Amer. Math. Soc. 142, 2625–2631, 2014.
  • [5] M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S.H. Shojaee, On FCPurity and I-Purity of Modules and Köthe Rings, Comm. Algebra, 42 (5), 2061–2081, 2014.
  • [6] M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S.H. Shojaee, C-Pure Projective Modules, Comm. Algebra, 41, 4559–4575, 2013.
  • [7] J.E. Björk, Rings satisfying certain chain conditions, J. Reine Angew Math. 245, 63–73, 1970.
  • [8] E. Büyükasık and Y. Durgun, Absolutely s-pure modules and neat-flat modules Comm. Algebra, 43 (2), 384–399, 2015.
  • [9] I.S. Cohen and I. Kaplansky, Rings for which every module is a direct sum of cyclic modules Math. Z. 54, 97–101, 1951.
  • [10] P.M. Cohn, On the free product of associative rings, Math. Z. 71, 380–398, 1959.
  • [11] F. Couchot, RD-flatness and RD-injectivity, Comm. Algebra, 34, 3675–3689, 2006.
  • [12] R.R. Colby, Rings which have flat injective modules, J. Algebra 35, 239–252, 1975.
  • [13] K. Divaani-Aazar, M.A. Esmkhani and M. Tousi, A criterion for rings which are locally valuation rings, Colloq. Math. 116, 153–164, 2009.
  • [14] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules, Pitman Research Notes in Mathematics Series, vol. 313, Longman Scientific and Technical, Harlow, 1994.
  • [15] E.E. Enochs and O.M.G. Jenda, Relative homological algebra, Berlin: Walter de Gruyter, 2000.
  • [16] C. Faith, Algebra. II, Springer-Verlag, Berlin-New York, 1976.
  • [17] C. Faith and E.A. Walker, Direct sum representation of injective modules, J. Algebra, 5 (2), 203–221, 1967.
  • [18] A. Facchini, Module Theory, Birkhauser Verlag-Basel, 1998.
  • [19] A.I. Generalov, Weak and $\omega$-high purities in the category of modules, Mat. Sb. (N.S.) 34 (3), 345–356, 1978.
  • [20] K.R. Goodearl and R.B. Warfield, An Introduction to Noncommutative Noetherian Rings 2nd ed. Cambridge: Cambridge University Press, 2004.
  • [21] M. Greferath, A. Nechaev and R. Wisbauer, Finite quasi-Frobenius modules and linear codes, J. Algebra Appl. 3 (3), 1–26, 2004.
  • [22] M. Harada, Self mini-injective rings, Osaka J. Math. 19 (2), 587–597, 1982.
  • [23] H. Holm and P. Jorgensen, Covers, precovers, and purity, Illinois J. Math. 52 (2), 691–703, 2008.
  • [24] T. Honold, Characterization of finite Frobenius rings, Arch. Math. 76 (6), 406–415, 2001.
  • [25] G. Köthe, Verallgemeinerte Abelsche Gruppen mit hyperkomplexem Operatorenring, (German). Math. Z. 39, 31–44, 1935.
  • [26] T.Y. Lam, Lectures on modules and rings Springer-Verlag, New York, 1999.
  • [27] Z.K. Liu, Rings with flat left socle, Comm. Algebra, 23 (6), 1645–1656, 1995.
  • [28] L. Mao, On mininjective and min-flat modules, Publ. Math. Debrecen 72 (3-4), 347–358, 2008.
  • [29] L. Mao, Min-flat modules and min-coherent rings, Comm. Algebra, 35 (2), 635–650, 2007.
  • [30] A.R. Mehdi, Purity relative to classes of finitely presented modules, J. Algebra Appl. 12 (8), 1350050, 2013.
  • [31] A. Moradzadeh-Dehkordi and F. Couchot, RD-flatness and RD-injectivity of simple modules, J.Pure Appl. Algebra 226, 107034, 2022.
  • [32] W.K. Nicholson and J.F. Watters, Rings with projective socle, Proc. Amer. Math. Soc. 102, 443–450, 1988.
  • [33] W.K. Nicholson and M.F. Yousif, Mininjective rings, J. Algebra 187, 548–578, 1997.
  • [34] G. Puninski, M. Prest and P. Rothmaler, Rings described by various purities, Comm. Algebra, 27, 2127–2162, 1999.
  • [35] B. Stenström, Pure submodules, Ark. Mat. 7, 159–171, 1967.
  • [36] R.B. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28, 699–719, 1969.
  • [37] R. Wisbauer, Foundations of Module and Ring Theory, New York: Gordon and- Breach, 1991.
  • [38] J.A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (3), 555–575, 1999.
There are 38 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Yusuf Alagöz 0000-0002-2535-4679

Ali Moradzadeh-dehkordı 0000-0002-1794-8688

Project Number 1401160414
Early Pub Date August 15, 2023
Publication Date April 23, 2024
Published in Issue Year 2024

Cite

APA Alagöz, Y., & Moradzadeh-dehkordı, A. (2024). Homological objects of min-pure exact sequences. Hacettepe Journal of Mathematics and Statistics, 53(2), 342-355. https://doi.org/10.15672/hujms.1186239
AMA Alagöz Y, Moradzadeh-dehkordı A. Homological objects of min-pure exact sequences. Hacettepe Journal of Mathematics and Statistics. April 2024;53(2):342-355. doi:10.15672/hujms.1186239
Chicago Alagöz, Yusuf, and Ali Moradzadeh-dehkordı. “Homological Objects of Min-Pure Exact Sequences”. Hacettepe Journal of Mathematics and Statistics 53, no. 2 (April 2024): 342-55. https://doi.org/10.15672/hujms.1186239.
EndNote Alagöz Y, Moradzadeh-dehkordı A (April 1, 2024) Homological objects of min-pure exact sequences. Hacettepe Journal of Mathematics and Statistics 53 2 342–355.
IEEE Y. Alagöz and A. Moradzadeh-dehkordı, “Homological objects of min-pure exact sequences”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, pp. 342–355, 2024, doi: 10.15672/hujms.1186239.
ISNAD Alagöz, Yusuf - Moradzadeh-dehkordı, Ali. “Homological Objects of Min-Pure Exact Sequences”. Hacettepe Journal of Mathematics and Statistics 53/2 (April 2024), 342-355. https://doi.org/10.15672/hujms.1186239.
JAMA Alagöz Y, Moradzadeh-dehkordı A. Homological objects of min-pure exact sequences. Hacettepe Journal of Mathematics and Statistics. 2024;53:342–355.
MLA Alagöz, Yusuf and Ali Moradzadeh-dehkordı. “Homological Objects of Min-Pure Exact Sequences”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 2, 2024, pp. 342-55, doi:10.15672/hujms.1186239.
Vancouver Alagöz Y, Moradzadeh-dehkordı A. Homological objects of min-pure exact sequences. Hacettepe Journal of Mathematics and Statistics. 2024;53(2):342-55.