Gorenstein semihereditary rings and Gorenstein Prüfer domains

Year 2017, , 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.