Research Article
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Year 2024, Early Access, 1 - 32

Weiqing Li

https://doi.org/10.24330/ieja.1502064

Abstract

References

  • S. T. Aldrich, E. E. Enochs, O. M. G. Jenda and L. Oyonarte, Envelopes and covers by modules of finite injective and projective dimensions, J. Algebra, 242(2) (2001), 447-459.
  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics, 13, Springer-Verlag, New York, 1992.
  • L. Angeleri Hugel and F. U. Coelho, Infinitely generated tilting modules of finite projective dimension, Forum Math., 13(2) (2001), 239-250.
  • L. Angeleri Hugel, D. Herbera and J. Trlifaj, Tilting modules and Gorenstein rings, Forum Math., 18(2) (2006), 211-229.
  • L. Angeleri Hugel and O. Mendoza Hernandez, Homological dimensions in cotorsion pairs, Illinois J. Math., 53(1) (2009), 251-263.
  • I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebras, Vol. 1, Techniques of Representation Theory, London Mathematical Society Student Texts, 65, Cambridge University Press, Cambridge, 2006.
  • D. Bennis and N. Mahdou, Global Gorenstein dimensions of polynomial rings and of direct products of rings, Houston J. Math., 35(4) (2009), 1019-1028.
  • D. Bennis and N. Mahdou, Global Gorenstein dimensions, Proc. Amer. Math. Soc., 138(2) (2010), 461-465.
  • I. Bican, R. El Bashir and E. E. Enochs, All modules have flat covers, Bull. London Math. Soc., 33(4) (2001), 385-390.
  • D. Bravo, J. Gillespie and M. Hovey, The stable module category of a general ring, arXiv:1405.5768v1 [math.RA] (2014).
  • D. Bravo and C. E. Parra, Torsion pairs over n-hereditary rings, Comm. Algebra, 47(5) (2019), 1892-1907.
  • D. Bravo and M. A. Perez, Finiteness conditions and cotorsion pairs, J. Pure Appl. Algebra, 221(6) (2017), 1249-1267.
  • J. W. Brewer, E. A. Rutter and J. J. Watkins, Coherence and weak global dimension of $R$[[$X$]] when $R$ is von Neumann regular, J. Algebra, 46(1) (1977), 278-289.
  • J. L. Chen and N. Q. Ding, On n-coherent rings, Comm. Algebra, 24(10) (1996), 3211-3216.
  • L. W. Christensen, S. Estrada and P. Thompson, Gorenstein weak global dimension is symmetric, Math. Nachr., 294(11) (2021), 2121-2128.
  • D. L. Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra, 22(10) (1994), 3997-4011.
  • G. C. Dai and N. Q. Ding, Coherent rings and absolutely pure precovers, Comm. Algebra, 47(11) (2019), 4743-4748.
  • P. C. Eklof and J. Trlifaj, How to make Ext vanish, Bull. London Math. Soc., 33(1) (2001), 41-51.
  • E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math., 39(3) (1981), 189-209.
  • E. E. Enochs, A. Iacob and O. M. G. Jenda, Closure under transfinite extensions, Illinois J. Math., 51(2) (2007), 561-569.
  • E. E. Enochs and O. M. G. Jenda, Copure injective modules, Quaestiones Math., 14(4) (1991), 401-409.
  • E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z., 220(4) (1995), 611-633.
  • E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter & Co., Berlin, 2000.
  • E. E. Enochs, O. M. G. Jenda and B. Torrecillas, Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan, 10(1) (1993), 1-9.
  • E. E. Enochs and L. Oyonarte, Covers, Envelopes and Cotorsion Theories, Nova Science Publishers, Inc., New York, 2002.
  • Z. H. Gao and F. G. Wang, Weak injective and weak flat modules, Comm. Algebra, 43(9) (2015), 3857-3868.
  • J. Gillespie, On the homotopy category of AC-injective complexes, Front. Math. China, 12(1) (2017), 97-115.
  • R. Gobel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, De Gruyter Expositions in Mathematics, 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2006.
  • H. Holm and P. J{\o}rgensen, Covers, precovers, and purity, Illinois J. Math., 52(2) (2008), 691-703.
  • H. Holm and P. J{\o}rgensen, Cotorsion pairs induced by duality pairs, J. Commut. Algebra, 1(4) (2009), 621-633.
  • M. Hrbek, One-tilting classes and modules over commutative rings, J. Algebra, 462 (2016), 1-22.
  • W. Q. Li, J. C. Guan and B. Y. Ouyang, Strongly FP-injective modules, Comm. Algebra, 45(9) (2017), 3816-3824.
  • W. Q. Li and D. Liu, A non-commutative analogue of Costa's first conjecture, J. Algebra Appl., 19(1) (2020), 2050007 (6 pp).
  • W. Q. Li and B. Y. Ouyang, (n, d)-Injective covers, n-coherent rings, and (n, d)-rings, Czechoslovak Math. J., 64 (2014), 289-304.
  • W. Q. Li, L. Yan and B. Y. Ouyang, On global C -dimensions, Rocky Mountain J. Math., 49(2) (2019), 557-577.
  • W. Q. Li, L. Yan and D. Zhang, Some results on Gorenstein (weak) global dimension of rings, Comm. Algebra, 51(1) (2023), 264-275.
  • N. Mahdou and K. Ouarghi, On G-(n, d)-rings, Rocky Mountain J. Math., 42(3) (2012), 999-1013.
  • L. X. Mao and N. Q. Ding, FP-projective dimensions, Comm. Algebra, 33(4) (2005), 1153-1170.
  • L. X. Mao and N. Q. Ding, Relative projective modules and relative injective modules, Comm. Algebra, 34(7) (2006), 2403-2418.
  • B. Y. Ouyang, L. L. Duan and W. Q. Li, Relative projective dimensions, Bull. Malays. Math. Sci. Soc. (2), 37(3) (2014), 865-879.
  • W. H. Rant, Minimally generated modules, Canad. Math. Bull., 23(1) (1980), 103-105.
  • F. Richman, Flat dimension, constructivity, and the Hilbert syzygy theorem, New Zealand J. Math., 26(2) (1997), 263-273.
  • J. Saroch and J. Stovicek, Singular compactness and definability for $\sum$-cotorsion and Gorenstein modules, Selecta Math. (N.S.), 26(2) (2020), 23 (40 pp).
  • J. Stovicek, On purity and applications to coderived and singularity categories, arXiv:1412.1615v1 [math.CT] (2014).
  • W. V. Vasconcelos, The Rings of Dimension Two, Lecture Notes in Pure and Applied Mathematics, 22, Marcel Dekker, Inc., New York-Basel, 1976.
  • J. P. Wang, Z. K. Liu and X. Y. Yang, A negative answer to a question of Gillespie (in Chinese), Sci. Sin. Math., 48(9) (2018), 1121-1130.
  • F. Zareh-Khoshchehreh and K. Divaani-Aazar, The existence of relative pure injective envelopes, Colloq. Math., 130 (2013), 251-264.
  • D. D. Zhang and B. Y. Ouyang, On n-coherent rings and (n, d)-injective modules, Algebra Colloq., 22 (2015), 349-360.
  • T. Zhao, Homological properties of modules with finite weak injective and weak flat dimensions, Bull. Malays. Math. Sci. Soc., 41(2) (2018), 779-805.
  • D. X. Zhou, On n-coherent rings and (n, d)-rings, Comm. Algebra, 32(6) (2004), 2425-2441.

On $G$-$(n,d)$-rings and $n$-coherent rings

Year 2024, Early Access, 1 - 32

Weiqing Li

https://doi.org/10.24330/ieja.1502064

Abstract

Let $n$ and $d$ be non-negative integers. We introduce the concept of $strongly$ $(n,d)$-$injective$ modules to characterize $n$-coherent rings. For a right perfect ring $R$, it is shown that $R$ is right $n$-coherent if and only if every right $R$-module has a strongly $(n,d)$-injective (pre)cover for some non-negative integer $d \leq n$. We also provide equivalent conditions for an $(n,d)$-ring being $n$-coherent. Then we investigate the so-called $right$ $G$-$(n,d)$-$rings$, over which every $n$-presented right module has Gorenstein projective dimension at most $d$. Finally, we prove a Gorenstein analogue of Costa's first conjecture.

Keywords

$n$-Coherent ring G-(n,d)-ring strongly (n,d) injective (flat) module cotorsion theory

References

  • S. T. Aldrich, E. E. Enochs, O. M. G. Jenda and L. Oyonarte, Envelopes and covers by modules of finite injective and projective dimensions, J. Algebra, 242(2) (2001), 447-459.
  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics, 13, Springer-Verlag, New York, 1992.
  • L. Angeleri Hugel and F. U. Coelho, Infinitely generated tilting modules of finite projective dimension, Forum Math., 13(2) (2001), 239-250.
  • L. Angeleri Hugel, D. Herbera and J. Trlifaj, Tilting modules and Gorenstein rings, Forum Math., 18(2) (2006), 211-229.
  • L. Angeleri Hugel and O. Mendoza Hernandez, Homological dimensions in cotorsion pairs, Illinois J. Math., 53(1) (2009), 251-263.
  • I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebras, Vol. 1, Techniques of Representation Theory, London Mathematical Society Student Texts, 65, Cambridge University Press, Cambridge, 2006.
  • D. Bennis and N. Mahdou, Global Gorenstein dimensions of polynomial rings and of direct products of rings, Houston J. Math., 35(4) (2009), 1019-1028.
  • D. Bennis and N. Mahdou, Global Gorenstein dimensions, Proc. Amer. Math. Soc., 138(2) (2010), 461-465.
  • I. Bican, R. El Bashir and E. E. Enochs, All modules have flat covers, Bull. London Math. Soc., 33(4) (2001), 385-390.
  • D. Bravo, J. Gillespie and M. Hovey, The stable module category of a general ring, arXiv:1405.5768v1 [math.RA] (2014).
  • D. Bravo and C. E. Parra, Torsion pairs over n-hereditary rings, Comm. Algebra, 47(5) (2019), 1892-1907.
  • D. Bravo and M. A. Perez, Finiteness conditions and cotorsion pairs, J. Pure Appl. Algebra, 221(6) (2017), 1249-1267.
  • J. W. Brewer, E. A. Rutter and J. J. Watkins, Coherence and weak global dimension of $R$[[$X$]] when $R$ is von Neumann regular, J. Algebra, 46(1) (1977), 278-289.
  • J. L. Chen and N. Q. Ding, On n-coherent rings, Comm. Algebra, 24(10) (1996), 3211-3216.
  • L. W. Christensen, S. Estrada and P. Thompson, Gorenstein weak global dimension is symmetric, Math. Nachr., 294(11) (2021), 2121-2128.
  • D. L. Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra, 22(10) (1994), 3997-4011.
  • G. C. Dai and N. Q. Ding, Coherent rings and absolutely pure precovers, Comm. Algebra, 47(11) (2019), 4743-4748.
  • P. C. Eklof and J. Trlifaj, How to make Ext vanish, Bull. London Math. Soc., 33(1) (2001), 41-51.
  • E. E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math., 39(3) (1981), 189-209.
  • E. E. Enochs, A. Iacob and O. M. G. Jenda, Closure under transfinite extensions, Illinois J. Math., 51(2) (2007), 561-569.
  • E. E. Enochs and O. M. G. Jenda, Copure injective modules, Quaestiones Math., 14(4) (1991), 401-409.
  • E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z., 220(4) (1995), 611-633.
  • E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter & Co., Berlin, 2000.
  • E. E. Enochs, O. M. G. Jenda and B. Torrecillas, Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan, 10(1) (1993), 1-9.
  • E. E. Enochs and L. Oyonarte, Covers, Envelopes and Cotorsion Theories, Nova Science Publishers, Inc., New York, 2002.
  • Z. H. Gao and F. G. Wang, Weak injective and weak flat modules, Comm. Algebra, 43(9) (2015), 3857-3868.
  • J. Gillespie, On the homotopy category of AC-injective complexes, Front. Math. China, 12(1) (2017), 97-115.
  • R. Gobel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules, De Gruyter Expositions in Mathematics, 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2006.
  • H. Holm and P. J{\o}rgensen, Covers, precovers, and purity, Illinois J. Math., 52(2) (2008), 691-703.
  • H. Holm and P. J{\o}rgensen, Cotorsion pairs induced by duality pairs, J. Commut. Algebra, 1(4) (2009), 621-633.
  • M. Hrbek, One-tilting classes and modules over commutative rings, J. Algebra, 462 (2016), 1-22.
  • W. Q. Li, J. C. Guan and B. Y. Ouyang, Strongly FP-injective modules, Comm. Algebra, 45(9) (2017), 3816-3824.
  • W. Q. Li and D. Liu, A non-commutative analogue of Costa's first conjecture, J. Algebra Appl., 19(1) (2020), 2050007 (6 pp).
  • W. Q. Li and B. Y. Ouyang, (n, d)-Injective covers, n-coherent rings, and (n, d)-rings, Czechoslovak Math. J., 64 (2014), 289-304.
  • W. Q. Li, L. Yan and B. Y. Ouyang, On global C -dimensions, Rocky Mountain J. Math., 49(2) (2019), 557-577.
  • W. Q. Li, L. Yan and D. Zhang, Some results on Gorenstein (weak) global dimension of rings, Comm. Algebra, 51(1) (2023), 264-275.
  • N. Mahdou and K. Ouarghi, On G-(n, d)-rings, Rocky Mountain J. Math., 42(3) (2012), 999-1013.
  • L. X. Mao and N. Q. Ding, FP-projective dimensions, Comm. Algebra, 33(4) (2005), 1153-1170.
  • L. X. Mao and N. Q. Ding, Relative projective modules and relative injective modules, Comm. Algebra, 34(7) (2006), 2403-2418.
  • B. Y. Ouyang, L. L. Duan and W. Q. Li, Relative projective dimensions, Bull. Malays. Math. Sci. Soc. (2), 37(3) (2014), 865-879.
  • W. H. Rant, Minimally generated modules, Canad. Math. Bull., 23(1) (1980), 103-105.
  • F. Richman, Flat dimension, constructivity, and the Hilbert syzygy theorem, New Zealand J. Math., 26(2) (1997), 263-273.
  • J. Saroch and J. Stovicek, Singular compactness and definability for $\sum$-cotorsion and Gorenstein modules, Selecta Math. (N.S.), 26(2) (2020), 23 (40 pp).
  • J. Stovicek, On purity and applications to coderived and singularity categories, arXiv:1412.1615v1 [math.CT] (2014).
  • W. V. Vasconcelos, The Rings of Dimension Two, Lecture Notes in Pure and Applied Mathematics, 22, Marcel Dekker, Inc., New York-Basel, 1976.
  • J. P. Wang, Z. K. Liu and X. Y. Yang, A negative answer to a question of Gillespie (in Chinese), Sci. Sin. Math., 48(9) (2018), 1121-1130.
  • F. Zareh-Khoshchehreh and K. Divaani-Aazar, The existence of relative pure injective envelopes, Colloq. Math., 130 (2013), 251-264.
  • D. D. Zhang and B. Y. Ouyang, On n-coherent rings and (n, d)-injective modules, Algebra Colloq., 22 (2015), 349-360.
  • T. Zhao, Homological properties of modules with finite weak injective and weak flat dimensions, Bull. Malays. Math. Sci. Soc., 41(2) (2018), 779-805.
  • D. X. Zhou, On n-coherent rings and (n, d)-rings, Comm. Algebra, 32(6) (2004), 2425-2441.
There are 50 citations in total.

Details

Primary Language English
Subjects Algebra and Number Theory
Journal Section Articles
Authors

Weiqing Li This is me

Early Pub Date June 18, 2024
Publication Date
Published in Issue Year 2024 Early Access

Cite

APA Li, W. (2024). On $G$-$(n,d)$-rings and $n$-coherent rings. International Electronic Journal of Algebra1-32. https://doi.org/10.24330/ieja.1502064
AMA Li W. On $G$-$(n,d)$-rings and $n$-coherent rings. IEJA. Published online June 1, 2024:1-32. doi:10.24330/ieja.1502064
Chicago Li, Weiqing. “On $G$-$(n,d)$-Rings and $n$-Coherent Rings”. International Electronic Journal of Algebra, June (June 2024), 1-32. https://doi.org/10.24330/ieja.1502064.
EndNote Li W (June 1, 2024) On $G$-$(n,d)$-rings and $n$-coherent rings. International Electronic Journal of Algebra 1–32.
IEEE W. Li, “On $G$-$(n,d)$-rings and $n$-coherent rings”, IEJA, pp. 1–32, June 2024, doi: 10.24330/ieja.1502064.
ISNAD Li, Weiqing. “On $G$-$(n,d)$-Rings and $n$-Coherent Rings”. International Electronic Journal of Algebra. June 2024. 1-32. https://doi.org/10.24330/ieja.1502064.
JAMA Li W. On $G$-$(n,d)$-rings and $n$-coherent rings. IEJA. 2024;:1–32.
MLA Li, Weiqing. “On $G$-$(n,d)$-Rings and $n$-Coherent Rings”. International Electronic Journal of Algebra, 2024, pp. 1-32, doi:10.24330/ieja.1502064.
Vancouver Li W. On $G$-$(n,d)$-rings and $n$-coherent rings. IEJA. 2024:1-32.