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RD-projective module whose subprojectivity domain is minimal

Year 2022, Volume: 51 Issue: 2, 373 - 382, 01.04.2022

Yilmaz Durğun

https://doi.org/10.15672/hujms.776933

Abstract

A p-indigent module is one that is subprojective only to projective modules. An RD-projective module is subprojective to any torsionfree (and flat) module. An RD-projective module $T$ is called rdp-indigent if it is subprojective only to torsionfree modules. In this work, we consider the structure of SRDP rings whose (simple) RD-projective right $R$-modules are rdp-indigent or torsionfree. Moreover, new characterizations of P-coherent rings and torsionfree rings are presented by subprojectivity domains.

Keywords

RD-projective module subprojectivity domain rdp-indigent module torsionfree module flat module

Supporting Institution

TUBITAK

Project Number

118F311

Thanks

This research was supported by TUBITAK 1002-Short Term R\&D Funding Program.

References

  • [1] Y. Alagöz and Y. Durğun, An alternative perspective on pure-projectivity of modules, São Paulo J. Math. Sci. 14 (2), 631–650, 2020.
  • [2] A.N. Alahmadi, M. Alkan and S.R. López-Permouth, Poor modules: The opposite of injectivity, Glas- gow Math. J. 52A, 7–17, 2010.
  • [3] U. Albrecht, J. Dauns and L. Fuchs, Torsion-freeness and non-singularity over right p.p.-rings, J. Algebra, 285 (1), 98–119, 2005.
  • [4] R. Alizade and Y. Durğun, Test modules for flatness, Rend. Semin. Mat. Univ. Padova, 137, 75–91, 2017.
  • [5] P. Aydoğdu and S. R. López-Permouth, An alternative perspective on injectivity of modules, J. Algebra, 338, 207–219, 2011.
  • [6] P. Aydoğdu and B. Saraç, On artinian rings with restricted class of injectivity domains, J. Algebra, 377, 49–65, 2013.
  • [7] M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S.H. Shojaee, C-pure projective modules, Comm. Algebra, 41 (12), 4559–4575, 2013.
  • [8] A.W. Chatters and C.R. Hajarnavis, Rings with chain conditions, Pitman, Boston, Mass.-London, 1980.
  • [9] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, Birkhäuser Verlag, Basel, 2006.
  • [10] F. Couchot, RD-flatness and RD-injectivity, Comm. Algebra, 34 (10), 3675–3689, 2006.
  • [11] J. Dauns and L. Fuchs, Torsion-freeness for rings with zero divisor, J. Algebra Appl. 3 (3), 221–237, 2004.
  • [12] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules, Longman Scientific & Technical, 1994.
  • [13] Y. Durğun, Rings whose modules have maximal or minimal subprojectivity domain, J. Algebra Appl. 14 (6), 1550083, 2015.
  • [14] Y. Durğun, An alternative perspective on flatness of modules, J. Algebra Appl. 15 (8) 1650145, 2016.
  • [15] Y. Durğun, Subprojectivity domains of pure-projective modules, J. Algebra Appl. 19 (5), 2050091, 2020.
  • [16] E. E. Enochs and O. M. G. Jenda Relative homological algebra, Walter de Gruyter & Co., 2000.
  • [17] A. Facchini and A. Moradzadeh-Dehkordi, Rings over which every RD-projective module is a direct sums of cyclically presented modules, J. Algebra 401, 179–200, 2014.
  • [18] A. Hattori, A foundation of torsion theory for modules over general rings, Nagoya Math. J. 17, 147– 158, 1960.
  • [19] C. Holston, S.R. López-Permouth and N.O. Ertaş, Rings whose modules have maximal or minimal projectivity domain, J. Pure Appl. Algebra, 216 (3), 673–678, 2012.
  • [20] C. Holston, S.R. López-Permouth, J. Mastromatteo and J.E. Simental-Rodriguez, An alternative per- spective on projectivity of modules, Glasgow Math. J. 57 (1), 83–99, 2015.
  • [21] K. Honda, Realism in the theory of abelian groups. I, Comment. Math. Univ. St. Paul. 5, 37–75, 1956.
  • [22] T.Y. Lam, Lectures on modules and rings, Springer-Verlag, 1999.
  • [23] S.R. López-Permouth, J. Mastromatteo, Y. Tolooei and B. Ungor, Pure-injectivity from a different perspective, Glasg. Math. J. 60 (1), 135–151, 2018.
  • [24] L. Mao, Properties of RD-projective and RD-injective modules, Turkish J. Math. 35 (2), 187–205, 2011.
  • [25] L. Mao and N. Ding, On divisible and torsionfree modules, Comm. Algebra, 36 (2), 708–731, 2008.
  • [26] C. Megibben, Absolutely pure modules, Proc. Amer. Math. Soc. 26, 561–566, 1970.
  • [27] P. Rothmaler, Torsion-free, divisible, and Mittag-Leffler modules, Comm. Algebra 43 (8), 3342–3364, 2015.
  • [28] J. Rotman, An introduction to homological algebra, Academic Press, 1979.
  • [29] F. L. Sandomierski, Nonsingular rings, Proc. Amer. Math. Soc. 19, 225–230, 1968.
  • [30] J. Trlifaj, Whitehead test modules, Trans. Amer. Math. Soc. 348 (4), 1521–1554, 1996.
  • [31] R. B. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28, 699–719, 1969.

Year 2022, Volume: 51 Issue: 2, 373 - 382, 01.04.2022

Yilmaz Durğun

https://doi.org/10.15672/hujms.776933

Abstract

Project Number

118F311

References

  • [1] Y. Alagöz and Y. Durğun, An alternative perspective on pure-projectivity of modules, São Paulo J. Math. Sci. 14 (2), 631–650, 2020.
  • [2] A.N. Alahmadi, M. Alkan and S.R. López-Permouth, Poor modules: The opposite of injectivity, Glas- gow Math. J. 52A, 7–17, 2010.
  • [3] U. Albrecht, J. Dauns and L. Fuchs, Torsion-freeness and non-singularity over right p.p.-rings, J. Algebra, 285 (1), 98–119, 2005.
  • [4] R. Alizade and Y. Durğun, Test modules for flatness, Rend. Semin. Mat. Univ. Padova, 137, 75–91, 2017.
  • [5] P. Aydoğdu and S. R. López-Permouth, An alternative perspective on injectivity of modules, J. Algebra, 338, 207–219, 2011.
  • [6] P. Aydoğdu and B. Saraç, On artinian rings with restricted class of injectivity domains, J. Algebra, 377, 49–65, 2013.
  • [7] M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi and S.H. Shojaee, C-pure projective modules, Comm. Algebra, 41 (12), 4559–4575, 2013.
  • [8] A.W. Chatters and C.R. Hajarnavis, Rings with chain conditions, Pitman, Boston, Mass.-London, 1980.
  • [9] J. Clark, C. Lomp, N. Vanaja and R. Wisbauer, Lifting modules, Birkhäuser Verlag, Basel, 2006.
  • [10] F. Couchot, RD-flatness and RD-injectivity, Comm. Algebra, 34 (10), 3675–3689, 2006.
  • [11] J. Dauns and L. Fuchs, Torsion-freeness for rings with zero divisor, J. Algebra Appl. 3 (3), 221–237, 2004.
  • [12] N.V. Dung, D.V. Huynh, P.F. Smith and R. Wisbauer, Extending modules, Longman Scientific & Technical, 1994.
  • [13] Y. Durğun, Rings whose modules have maximal or minimal subprojectivity domain, J. Algebra Appl. 14 (6), 1550083, 2015.
  • [14] Y. Durğun, An alternative perspective on flatness of modules, J. Algebra Appl. 15 (8) 1650145, 2016.
  • [15] Y. Durğun, Subprojectivity domains of pure-projective modules, J. Algebra Appl. 19 (5), 2050091, 2020.
  • [16] E. E. Enochs and O. M. G. Jenda Relative homological algebra, Walter de Gruyter & Co., 2000.
  • [17] A. Facchini and A. Moradzadeh-Dehkordi, Rings over which every RD-projective module is a direct sums of cyclically presented modules, J. Algebra 401, 179–200, 2014.
  • [18] A. Hattori, A foundation of torsion theory for modules over general rings, Nagoya Math. J. 17, 147– 158, 1960.
  • [19] C. Holston, S.R. López-Permouth and N.O. Ertaş, Rings whose modules have maximal or minimal projectivity domain, J. Pure Appl. Algebra, 216 (3), 673–678, 2012.
  • [20] C. Holston, S.R. López-Permouth, J. Mastromatteo and J.E. Simental-Rodriguez, An alternative per- spective on projectivity of modules, Glasgow Math. J. 57 (1), 83–99, 2015.
  • [21] K. Honda, Realism in the theory of abelian groups. I, Comment. Math. Univ. St. Paul. 5, 37–75, 1956.
  • [22] T.Y. Lam, Lectures on modules and rings, Springer-Verlag, 1999.
  • [23] S.R. López-Permouth, J. Mastromatteo, Y. Tolooei and B. Ungor, Pure-injectivity from a different perspective, Glasg. Math. J. 60 (1), 135–151, 2018.
  • [24] L. Mao, Properties of RD-projective and RD-injective modules, Turkish J. Math. 35 (2), 187–205, 2011.
  • [25] L. Mao and N. Ding, On divisible and torsionfree modules, Comm. Algebra, 36 (2), 708–731, 2008.
  • [26] C. Megibben, Absolutely pure modules, Proc. Amer. Math. Soc. 26, 561–566, 1970.
  • [27] P. Rothmaler, Torsion-free, divisible, and Mittag-Leffler modules, Comm. Algebra 43 (8), 3342–3364, 2015.
  • [28] J. Rotman, An introduction to homological algebra, Academic Press, 1979.
  • [29] F. L. Sandomierski, Nonsingular rings, Proc. Amer. Math. Soc. 19, 225–230, 1968.
  • [30] J. Trlifaj, Whitehead test modules, Trans. Amer. Math. Soc. 348 (4), 1521–1554, 1996.
  • [31] R. B. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28, 699–719, 1969.
There are 31 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Yilmaz Durğun 0000-0002-1230-8964

Project Number 118F311
Publication Date April 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 2

Cite

APA Durğun, Y. (2022). RD-projective module whose subprojectivity domain is minimal. Hacettepe Journal of Mathematics and Statistics, 51(2), 373-382. https://doi.org/10.15672/hujms.776933
AMA Durğun Y. RD-projective module whose subprojectivity domain is minimal. Hacettepe Journal of Mathematics and Statistics. April 2022;51(2):373-382. doi:10.15672/hujms.776933
Chicago Durğun, Yilmaz. “RD-Projective Module Whose Subprojectivity Domain Is Minimal”. Hacettepe Journal of Mathematics and Statistics 51, no. 2 (April 2022): 373-82. https://doi.org/10.15672/hujms.776933.
EndNote Durğun Y (April 1, 2022) RD-projective module whose subprojectivity domain is minimal. Hacettepe Journal of Mathematics and Statistics 51 2 373–382.
IEEE Y. Durğun, “RD-projective module whose subprojectivity domain is minimal”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 2, pp. 373–382, 2022, doi: 10.15672/hujms.776933.
ISNAD Durğun, Yilmaz. “RD-Projective Module Whose Subprojectivity Domain Is Minimal”. Hacettepe Journal of Mathematics and Statistics 51/2 (April 2022), 373-382. https://doi.org/10.15672/hujms.776933.
JAMA Durğun Y. RD-projective module whose subprojectivity domain is minimal. Hacettepe Journal of Mathematics and Statistics. 2022;51:373–382.
MLA Durğun, Yilmaz. “RD-Projective Module Whose Subprojectivity Domain Is Minimal”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 2, 2022, pp. 373-82, doi:10.15672/hujms.776933.
Vancouver Durğun Y. RD-projective module whose subprojectivity domain is minimal. Hacettepe Journal of Mathematics and Statistics. 2022;51(2):373-82.

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