Research Article
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Gorenstein semihereditary rings and Gorenstein Prüfer domains

Year 2017, Volume: 22 Issue: 22, 45 - 61, 11.07.2017

Tao Xiong

https://doi.org/10.24330/ieja.325922

Abstract

We investigate the Gorenstein semihereditary rings and Gorenstein
Prüfer domains in terms of the notion of the copure
flat dimension $cfD(R)$ of a ring $R$ which is defined in  [X. H.
Fu and  N. Q. Ding, Comm. Algebra, 38(12) (2010), 4531-4544].

Keywords

Gorenstein semihereditary ring Gorenstein Prüfer domain

References

  • J. Abuhlail and M. Jarrar, Tilting modules over almost perfect domains, J. Pure Appl. Algebra, 215(8) (2011), 2024-2033.
  • H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc., 95 (1960), 466-488.
  • D. Bennis, Rings over which the class of Gorenstein at modules is closed under extensions, Comm. Algebra, 37(3) (2009), 855-868.
  • D. Bennis, A note on Gorenstein at dimension, Algebra Colloq., 18(1) (2011), 155-161.
  • J. L. Chen and X. X. Zhang, Coherent Rings and FP-injective Rings, Science Press, Beijing, 2014.
  • E. E. Enochs and O. M. G. Jenda, Copure injective resolutions, at resolvents and dimensions, Comment. Math. Univ. Carolin., 34(2) (1993), 203-211.
  • E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter & Co., Berlin, 2000.
  • X. H. Fu and N. Q. Ding, On strongly copure at modules and copure at dimensions, Comm. Algebra, 38(12) (2010), 4531-4544.
  • X. H. Fu, H. Y. Zhu and N. Q. Ding, On copure projective modules and copure projective dimensions, Comm. Algebra, 40(1) (2012), 343-359.
  • L. Fuchs and S. B. Lee, Weak-injectivity and almost perfect domains, J. Algebra, 321(1) (2009), 18-27.
  • Z. H. Gao and F. G. Wang, All Gorenstein hereditary rings are coherent, J. Algebra Appl., 13(4) (2014), 1350140 (5 pp).
  • S. Glaz, Commutative Coherent Rings, Lecture Notes in Math., 1371, Springer- Verlag, Berlin Heidelberg, 1989.
  • 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.
  • R. M. Hamsher, On the structure of a one dimensional quotient eld, J. Algebra, 19 (1971), 416-425.
  • H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004), 167-193.
  • E. G. Houston, On divisorial prime ideals in Prüfer v-multiplication domains, J. Pure Appl. Algebra, 42(1) (1986), 55-62.
  • K. Hu and F. G. Wang, Some results on Gorenstein Dedekind domains and their factor rings, Comm. Algebra, 41(1) (2013), 284-293.
  • C. U. Jensen, On the vanishing of lim (i), J. Algebra, 15 (1970), 151-166.
  • A. Jhilal and N. Mahdou, On strong n-perfect rings, Comm. Algebra, 38(3) (2010), 1057-1065.
  • I. Kaplansky, Commutative Rings (Revised edition), The University of Chicago Press, Chicago, Ill.-London, 1974.
  • S. B. Lee, h-Divisible modules, Comm. Algebra, 31(1) (2003), 513-525.
  • S. B. Lee, Weak-injective modules, Comm. Algebra, 34(1) (2006), 361-370.
  • B. H. Maddox, Absolutely pure modules, Proc. Amer. Math. Soc., 18 (1967), 155-158.
  • N. Mahdou and M. Tamekkante, On (strongly) Gorenstein (semi)hereditary rings, Arab. J. Sci. Eng., 36(3) (2011), 431-440.
  • L. X. Mao and N. Q. Ding, Relative copure injective modules and copure flat modules, J. Pure Appl. Algebra, 208(2) (2007), 635-646.
  • L. X. Mao and N. Q. Ding, Gorenstein FP-injective and Gorenstein at modules, J. Algebra Appl., 7(4) (2008), 491-506.
  • A. Mimouni, Integral domains in which each ideal is a w-ideal, Comm. Algebra, 33(5) (2005), 1345-1355.
  • L. Qiao and F. G. Wang, A Gorenstein analogue of a result of Bertin, J. Algebra Appl., 14(2) (2015), 1550019 (13 pp).
  • J. J. Rotman, An Introduction to Homological Algebra, Pure and Applied Mathematics, 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.
  • J. J. Rotman, An Introduction to Homological Algebra, 2nd ed. Universitext, Springer, New York, 2009.
  • L. Salce, Almost perfect domains and their modules, in Commutative algebra: Noetherian and non-Noetherian perspectives, Springer, New York, (2011), 363- 386.
  • W. V. Vasconcelos, The Rings of Dimension Two, Lect. Notes Pure Appl. Math., Vol. 22. Marcel Dekker, Inc., New York-Basel, 1976.
  • T. Xiong, Rings of copure projective dimension one, J. Korean Math. Soc., 54(2) (2017), 427-440.
  • T. Xiong, A characterization of Gorenstein Prüfer domains, submitted.
  • T. Xiong, F. G. Wang and K. Hu, Copure projective modules and CPH-rings (in Chinese), Journal of Sichuan Normal University (Natural Science), 36(2) (2013), 198-201.
  • T. Xiong, F. G. Wang, G. L. Xia and X. W. Sun, Change theorem of rings on copure flat dimensions (in Chinese), Journal of Natural Science of Heilongjiang University, 33(4) (2016), 435-437.
  • G. Yang, Z. K. Liu and L. Liang, Ding projective and Ding injective modules, Algebra Colloq., 20(4) (2013), 601-612.

Year 2017, Volume: 22 Issue: 22, 45 - 61, 11.07.2017

Tao Xiong

https://doi.org/10.24330/ieja.325922

Abstract

References

  • J. Abuhlail and M. Jarrar, Tilting modules over almost perfect domains, J. Pure Appl. Algebra, 215(8) (2011), 2024-2033.
  • H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc., 95 (1960), 466-488.
  • D. Bennis, Rings over which the class of Gorenstein at modules is closed under extensions, Comm. Algebra, 37(3) (2009), 855-868.
  • D. Bennis, A note on Gorenstein at dimension, Algebra Colloq., 18(1) (2011), 155-161.
  • J. L. Chen and X. X. Zhang, Coherent Rings and FP-injective Rings, Science Press, Beijing, 2014.
  • E. E. Enochs and O. M. G. Jenda, Copure injective resolutions, at resolvents and dimensions, Comment. Math. Univ. Carolin., 34(2) (1993), 203-211.
  • E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter & Co., Berlin, 2000.
  • X. H. Fu and N. Q. Ding, On strongly copure at modules and copure at dimensions, Comm. Algebra, 38(12) (2010), 4531-4544.
  • X. H. Fu, H. Y. Zhu and N. Q. Ding, On copure projective modules and copure projective dimensions, Comm. Algebra, 40(1) (2012), 343-359.
  • L. Fuchs and S. B. Lee, Weak-injectivity and almost perfect domains, J. Algebra, 321(1) (2009), 18-27.
  • Z. H. Gao and F. G. Wang, All Gorenstein hereditary rings are coherent, J. Algebra Appl., 13(4) (2014), 1350140 (5 pp).
  • S. Glaz, Commutative Coherent Rings, Lecture Notes in Math., 1371, Springer- Verlag, Berlin Heidelberg, 1989.
  • 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.
  • R. M. Hamsher, On the structure of a one dimensional quotient eld, J. Algebra, 19 (1971), 416-425.
  • H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004), 167-193.
  • E. G. Houston, On divisorial prime ideals in Prüfer v-multiplication domains, J. Pure Appl. Algebra, 42(1) (1986), 55-62.
  • K. Hu and F. G. Wang, Some results on Gorenstein Dedekind domains and their factor rings, Comm. Algebra, 41(1) (2013), 284-293.
  • C. U. Jensen, On the vanishing of lim (i), J. Algebra, 15 (1970), 151-166.
  • A. Jhilal and N. Mahdou, On strong n-perfect rings, Comm. Algebra, 38(3) (2010), 1057-1065.
  • I. Kaplansky, Commutative Rings (Revised edition), The University of Chicago Press, Chicago, Ill.-London, 1974.
  • S. B. Lee, h-Divisible modules, Comm. Algebra, 31(1) (2003), 513-525.
  • S. B. Lee, Weak-injective modules, Comm. Algebra, 34(1) (2006), 361-370.
  • B. H. Maddox, Absolutely pure modules, Proc. Amer. Math. Soc., 18 (1967), 155-158.
  • N. Mahdou and M. Tamekkante, On (strongly) Gorenstein (semi)hereditary rings, Arab. J. Sci. Eng., 36(3) (2011), 431-440.
  • L. X. Mao and N. Q. Ding, Relative copure injective modules and copure flat modules, J. Pure Appl. Algebra, 208(2) (2007), 635-646.
  • L. X. Mao and N. Q. Ding, Gorenstein FP-injective and Gorenstein at modules, J. Algebra Appl., 7(4) (2008), 491-506.
  • A. Mimouni, Integral domains in which each ideal is a w-ideal, Comm. Algebra, 33(5) (2005), 1345-1355.
  • L. Qiao and F. G. Wang, A Gorenstein analogue of a result of Bertin, J. Algebra Appl., 14(2) (2015), 1550019 (13 pp).
  • J. J. Rotman, An Introduction to Homological Algebra, Pure and Applied Mathematics, 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.
  • J. J. Rotman, An Introduction to Homological Algebra, 2nd ed. Universitext, Springer, New York, 2009.
  • L. Salce, Almost perfect domains and their modules, in Commutative algebra: Noetherian and non-Noetherian perspectives, Springer, New York, (2011), 363- 386.
  • W. V. Vasconcelos, The Rings of Dimension Two, Lect. Notes Pure Appl. Math., Vol. 22. Marcel Dekker, Inc., New York-Basel, 1976.
  • T. Xiong, Rings of copure projective dimension one, J. Korean Math. Soc., 54(2) (2017), 427-440.
  • T. Xiong, A characterization of Gorenstein Prüfer domains, submitted.
  • T. Xiong, F. G. Wang and K. Hu, Copure projective modules and CPH-rings (in Chinese), Journal of Sichuan Normal University (Natural Science), 36(2) (2013), 198-201.
  • T. Xiong, F. G. Wang, G. L. Xia and X. W. Sun, Change theorem of rings on copure flat dimensions (in Chinese), Journal of Natural Science of Heilongjiang University, 33(4) (2016), 435-437.
  • G. Yang, Z. K. Liu and L. Liang, Ding projective and Ding injective modules, Algebra Colloq., 20(4) (2013), 601-612.
There are 37 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Research Article
Authors

Tao Xiong This is me

Publication Date July 11, 2017
Published in Issue Year 2017 Volume: 22 Issue: 22

Cite

APA Xiong, T. (2017). Gorenstein semihereditary rings and Gorenstein Prüfer domains. International Electronic Journal of Algebra, 22(22), 45-61. https://doi.org/10.24330/ieja.325922
AMA 1.Xiong T. Gorenstein semihereditary rings and Gorenstein Prüfer domains. IEJA. 2017;22(22):45-61. doi:10.24330/ieja.325922
Chicago Xiong, Tao. 2017. “Gorenstein Semihereditary Rings and Gorenstein Prüfer Domains”. International Electronic Journal of Algebra 22 (22): 45-61. https://doi.org/10.24330/ieja.325922.
EndNote Xiong T (July 1, 2017) Gorenstein semihereditary rings and Gorenstein Prüfer domains. International Electronic Journal of Algebra 22 22 45–61.
IEEE [1]T. Xiong, “Gorenstein semihereditary rings and Gorenstein Prüfer domains”, IEJA, vol. 22, no. 22, pp. 45–61, July 2017, doi: 10.24330/ieja.325922.
ISNAD Xiong, Tao. “Gorenstein Semihereditary Rings and Gorenstein Prüfer Domains”. International Electronic Journal of Algebra 22/22 (July 1, 2017): 45-61. https://doi.org/10.24330/ieja.325922.
JAMA 1.Xiong T. Gorenstein semihereditary rings and Gorenstein Prüfer domains. IEJA. 2017;22:45–61.
MLA Xiong, Tao. “Gorenstein Semihereditary Rings and Gorenstein Prüfer Domains”. International Electronic Journal of Algebra, vol. 22, no. 22, July 2017, pp. 45-61, doi:10.24330/ieja.325922.
Vancouver 1.Xiong T. Gorenstein semihereditary rings and Gorenstein Prüfer domains. IEJA [Internet]. 2017 July 1;22(22):45-61. Available from: https://izlik.org/JA24WF75ZH