Homological objects of min-pure exact sequences

Year 2024, Volume: 53 Issue: 2, 342 - 355, 23.04.2024

Yusuf Alagöz , Ali Moradzadeh-dehkordı

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 $2$.

Keywords

(min-)purity, Köthe rings, universally mininjective rings, quasi-Frobenius rings

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.