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Year 2020, Volume: 49 Issue: 2, 808 - 821, 02.04.2020
https://doi.org/10.15672/hujms.624000

Abstract

References

  • [1] S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 , 457-473, 1960.
  • [2] J. L. Chen and N. Q. Ding, On n-coherent rings, Comm. Algebra 24 (3), 3211-3216, 1996.
  • [3] R. R. Colby, Rings which have flat injective modules, J. Algebra 35, 239-252, 1975.
  • [4] D. L. Costa, Parameterizing families of non-noetherian rings, Comm. Algebra 22 (10), 3997-4011, 1994.
  • [5] N. Q. Ding and J. L. Chen, On envelopes with the unique mapping property, Comm. Algebra 24 (4), 1459-1470, 1996.
  • [6] D. E. Dobbs, S. Kabbaj, and N. Mahdou, n-coherent rings and modules, Lecture Notes in Pure and Appl. Math. 185, 269-281, 1997
  • [7] E. Enochs, A note on absolutely pure modules, Canad. Math. Bull. 19 (3), 361-362, 1976.
  • [8] E. E. Enochs and O. M. G. Jenda, Copure injective resolutions, flat resolutions and dimensions, Comment. Math. Univ. Carolin 34 (2), 203-211, 1993.
  • [9] E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter. Berlin-New York, 2000.
  • [10] 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.
  • [11] S. Jøndrup, p.p.rings and finitely generated flat ideals, Proc. Amer. Math. Soc. 28 (2), 431-435, 1971.
  • [12] Z. K. Liu, Rings with flat left socles, Comm. Algebra 23 (5), 1645-1656, 1995.
  • [13] L. X. Mao and N. Q. Ding, FI-injective and FI-flat modules, J. Algebra 309 (1), 367-385, 2007.
  • [14] L. X. Mao and N. Q. Ding, On divisible and torsionfree modules, Comm. Algebra 36 (2), 708-731, 2008.
  • [15] L. X. Mao, On mininjective and min-flat modules, Publ. Math. Debrecen 72 (3-4), 347-358, 2008.
  • [16] W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra 174 (1), 77-93, 1995.
  • [17] J. J. Rotman, An Introduction to Homological Algebra, Academic press, New Yock, 1979.
  • [18] E. A. Rutter, Rings with the principle extension property, Comm. Algebra 3 (3), 203-212, 1975.
  • [19] B. Stenström, Coherent rings and FP-injective modules, J. London Math. Soc. 2, 323-329, 1970.
  • [20] J. Trlifaj, Cover, envelopes, and cotorsion theories, in:Homological Methods in Module Theory. Lecture notes for the workshop, Cortona, 10-16, 2000.
  • [21] R. Wisbauer, Foundations of Module and Ring Theory, London-Tokyo: Gordon and Breach 1991.
  • [22] Y. F. Xiao, Rings with flat socles, Proc. Amer. Math. Soc. 123 (8), 2391-2395, 1995.
  • [23] J. Z. Xu, Flat Covers of Modules, Lecture Note in Math. Springer-Verlag, Berlin- Heidelberg-New York, 1634, 1996.
  • [24] X. X. Zhang , J. L. Chen and J. Zhang, On (m, n)-injective modules and (m, n)- coherent rings, Algebra Colloq. 12 (1) , 149-160, 2005.
  • [25] X. X. Zhang and J. L. Chen, On n-semihereditary and n-coherent rings, Int. Electron. J. Algebra 1, 1-10, 2007.
  • [26] Z. M. Zhu, On n-coherent rings, n-hereditary rings and n-regular rings, Bull. Iranian Math. Soc. 37 (4), 251-267, 2011.
  • [27] Z. M. Zhu, C -coherent rings, C -semihereditary rings and C -regular rings, Studia Sci. Math. Hungar 50 (4), 491-508, 2013.
  • [28] Z. M. Zhu, Strongly C -coherent rings, Math. Rep. 19 (4), 367-380, 2017.
  • [29] Z. M. Zhu, On Π-coherence of rings, Hacet. J. Math. Stat. 46 (5), 875-886, 2017.

On $\mathscr{C}$-coherent rings, strongly $\mathscr{C}$-coherent rings and $\mathscr{C}$-semihereditary rings

Year 2020, Volume: 49 Issue: 2, 808 - 821, 02.04.2020
https://doi.org/10.15672/hujms.624000

Abstract

Let $R$ be a ring and $\mathscr{C}$ be a class of some finitely presented left $R$-modules. A left $R$-module $M$ is called $\mathscr{C}$-injective if Ext$^1_R(C, M)=0$ for every $C\in \mathscr{C}$; a left $R$-module $M$ is called $\mathscr{C}$-projective if ${\rm Ext}^1_R(M, E)=0$ for any $\mathscr{C}$-injective module $E$. $R$ is called left $\mathscr{C}$-coherent if every $C\in \mathscr{C}$ is 2-presented; $R$ is called left strongly $\mathscr{C}$-coherent, if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C\in \mathscr{C}$ and $P$ is finitely generated projective, then $K$ is $\mathscr{C}$-projective; a ring $R$ is called left $\mathscr{C}$-semihereditary, if whenever $0\rightarrow K\rightarrow P\rightarrow C\rightarrow 0$ is exact, where $C\in \mathscr{C}$ , $P$ is finitely generated projective, then $K$ is projective. In this paper, we give some new characterizations and properties of left $\mathscr{C}$-coherent rings, left strongly $\mathscr{C}$-coherent rings and left $\mathscr{C}$-semihereditary rings.

References

  • [1] S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 , 457-473, 1960.
  • [2] J. L. Chen and N. Q. Ding, On n-coherent rings, Comm. Algebra 24 (3), 3211-3216, 1996.
  • [3] R. R. Colby, Rings which have flat injective modules, J. Algebra 35, 239-252, 1975.
  • [4] D. L. Costa, Parameterizing families of non-noetherian rings, Comm. Algebra 22 (10), 3997-4011, 1994.
  • [5] N. Q. Ding and J. L. Chen, On envelopes with the unique mapping property, Comm. Algebra 24 (4), 1459-1470, 1996.
  • [6] D. E. Dobbs, S. Kabbaj, and N. Mahdou, n-coherent rings and modules, Lecture Notes in Pure and Appl. Math. 185, 269-281, 1997
  • [7] E. Enochs, A note on absolutely pure modules, Canad. Math. Bull. 19 (3), 361-362, 1976.
  • [8] E. E. Enochs and O. M. G. Jenda, Copure injective resolutions, flat resolutions and dimensions, Comment. Math. Univ. Carolin 34 (2), 203-211, 1993.
  • [9] E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Walter de Gruyter. Berlin-New York, 2000.
  • [10] 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.
  • [11] S. Jøndrup, p.p.rings and finitely generated flat ideals, Proc. Amer. Math. Soc. 28 (2), 431-435, 1971.
  • [12] Z. K. Liu, Rings with flat left socles, Comm. Algebra 23 (5), 1645-1656, 1995.
  • [13] L. X. Mao and N. Q. Ding, FI-injective and FI-flat modules, J. Algebra 309 (1), 367-385, 2007.
  • [14] L. X. Mao and N. Q. Ding, On divisible and torsionfree modules, Comm. Algebra 36 (2), 708-731, 2008.
  • [15] L. X. Mao, On mininjective and min-flat modules, Publ. Math. Debrecen 72 (3-4), 347-358, 2008.
  • [16] W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebra 174 (1), 77-93, 1995.
  • [17] J. J. Rotman, An Introduction to Homological Algebra, Academic press, New Yock, 1979.
  • [18] E. A. Rutter, Rings with the principle extension property, Comm. Algebra 3 (3), 203-212, 1975.
  • [19] B. Stenström, Coherent rings and FP-injective modules, J. London Math. Soc. 2, 323-329, 1970.
  • [20] J. Trlifaj, Cover, envelopes, and cotorsion theories, in:Homological Methods in Module Theory. Lecture notes for the workshop, Cortona, 10-16, 2000.
  • [21] R. Wisbauer, Foundations of Module and Ring Theory, London-Tokyo: Gordon and Breach 1991.
  • [22] Y. F. Xiao, Rings with flat socles, Proc. Amer. Math. Soc. 123 (8), 2391-2395, 1995.
  • [23] J. Z. Xu, Flat Covers of Modules, Lecture Note in Math. Springer-Verlag, Berlin- Heidelberg-New York, 1634, 1996.
  • [24] X. X. Zhang , J. L. Chen and J. Zhang, On (m, n)-injective modules and (m, n)- coherent rings, Algebra Colloq. 12 (1) , 149-160, 2005.
  • [25] X. X. Zhang and J. L. Chen, On n-semihereditary and n-coherent rings, Int. Electron. J. Algebra 1, 1-10, 2007.
  • [26] Z. M. Zhu, On n-coherent rings, n-hereditary rings and n-regular rings, Bull. Iranian Math. Soc. 37 (4), 251-267, 2011.
  • [27] Z. M. Zhu, C -coherent rings, C -semihereditary rings and C -regular rings, Studia Sci. Math. Hungar 50 (4), 491-508, 2013.
  • [28] Z. M. Zhu, Strongly C -coherent rings, Math. Rep. 19 (4), 367-380, 2017.
  • [29] Z. M. Zhu, On Π-coherence of rings, Hacet. J. Math. Stat. 46 (5), 875-886, 2017.
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Zhu Zhanmin This is me 0000-0002-3131-3865

Publication Date April 2, 2020
Published in Issue Year 2020 Volume: 49 Issue: 2

Cite

APA Zhanmin, Z. (2020). On $\mathscr{C}$-coherent rings, strongly $\mathscr{C}$-coherent rings and $\mathscr{C}$-semihereditary rings. Hacettepe Journal of Mathematics and Statistics, 49(2), 808-821. https://doi.org/10.15672/hujms.624000
AMA Zhanmin Z. On $\mathscr{C}$-coherent rings, strongly $\mathscr{C}$-coherent rings and $\mathscr{C}$-semihereditary rings. Hacettepe Journal of Mathematics and Statistics. April 2020;49(2):808-821. doi:10.15672/hujms.624000
Chicago Zhanmin, Zhu. “On $\mathscr{C}$-Coherent Rings, Strongly $\mathscr{C}$-Coherent Rings and $\mathscr{C}$-Semihereditary Rings”. Hacettepe Journal of Mathematics and Statistics 49, no. 2 (April 2020): 808-21. https://doi.org/10.15672/hujms.624000.
EndNote Zhanmin Z (April 1, 2020) On $\mathscr{C}$-coherent rings, strongly $\mathscr{C}$-coherent rings and $\mathscr{C}$-semihereditary rings. Hacettepe Journal of Mathematics and Statistics 49 2 808–821.
IEEE Z. Zhanmin, “On $\mathscr{C}$-coherent rings, strongly $\mathscr{C}$-coherent rings and $\mathscr{C}$-semihereditary rings”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 2, pp. 808–821, 2020, doi: 10.15672/hujms.624000.
ISNAD Zhanmin, Zhu. “On $\mathscr{C}$-Coherent Rings, Strongly $\mathscr{C}$-Coherent Rings and $\mathscr{C}$-Semihereditary Rings”. Hacettepe Journal of Mathematics and Statistics 49/2 (April 2020), 808-821. https://doi.org/10.15672/hujms.624000.
JAMA Zhanmin Z. On $\mathscr{C}$-coherent rings, strongly $\mathscr{C}$-coherent rings and $\mathscr{C}$-semihereditary rings. Hacettepe Journal of Mathematics and Statistics. 2020;49:808–821.
MLA Zhanmin, Zhu. “On $\mathscr{C}$-Coherent Rings, Strongly $\mathscr{C}$-Coherent Rings and $\mathscr{C}$-Semihereditary Rings”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 2, 2020, pp. 808-21, doi:10.15672/hujms.624000.
Vancouver Zhanmin Z. On $\mathscr{C}$-coherent rings, strongly $\mathscr{C}$-coherent rings and $\mathscr{C}$-semihereditary rings. Hacettepe Journal of Mathematics and Statistics. 2020;49(2):808-21.