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Year 2019, Volume: 26 Issue: 26, 53 - 75, 11.07.2019
https://doi.org/10.24330/ieja.586945

Abstract

References

  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974.
  • M. Auslander and M. Bridger, Stable Module Theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969.
  • H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc., 95 (1960), 466-488.
  • S. Bazzoni and D. Herbera, Cotorsion pairs generated by modules of bounded projective dimension, Israel J. Math., 174 (2009), 119-160.
  • S. Bazzoni and L. Salce, Almost perfect domains, Colloq. Math., 95(2) (2003), 285-301.
  • S. Bouchiba, Finiteness aspects of Gorenstein homological dimensions, Colloq. Math., 131(2) (2013), 171-193.
  • T. J. Cheatham and D. R. Stone, Flat and projective character modules, Proc. Amer. Math. Soc., 81(2) (1981), 175-177.
  • E. E. Enochs and O. M. G. Jenda, On Gorenstein injective modules, Comm. Algebra, 21(10) (1993), 3489-3501.
  • 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.
  • L. Fuchs and S. B. Lee, The functor Hom and cotorsion theories, Comm. Algebra, 37(3) (2009), 923-932.
  • L. Fuchs and S. B. Lee, Weak-injectivity and almost perfect domains, J. Algebra, 321(1) (2009), 18-27.
  • L. Fuchs and S. B. Lee, On weak-injective modules over integral domains, J. Algebra, 323(7) (2010), 1872-1878.
  • L. Fuchs and S. B. Lee, On modules over commutative rings, J. Aust. Math. Soc., 103(3) (2017), 341-356.
  • L. Fuchs and L. Salce, Modules Over Non-Noetherian Domains, Mathematical Surveys and Monographs, 84, American Mathematical Society, Providence, RI, 2001.
  • Z. H. Gao and F. G. Wang, Weak injective and weak at modules, Comm. Algebra, 43(9) (2015), 3857-3868.
  • H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004), 167-193.
  • S. B. Lee, Weak-injective modules, Comm. Algebra, 34(1) (2006), 361-370.
  • S. B. Lee, Weak-injective modules over commutative rings, Comm. Algebra, 46(6) (2018), 2500-2509.
  • L. Mao and N. Ding, FP-projective dimensions, Comm. Algebra, 33(4) (2005), 1153-1170.
  • C. Megibben, Absolutely pure modules, Proc. Amer. Math. Soc., 26 (1970), 561-566.
  • M. Raynaud and L. Gruson, Criteres de platitude et de projectivite: Techniques de \plati cation" d'un module, Invent. Math., 13 (1971), 1-89 (in French).
  • J. J. Rotman, An Introduction to Homological Algebra, Pure and Applied Mathematics, 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.
  • B. Stenstrom, Coherent rings and FP-injective modules, J. London Math. Soc., 2(2) (1970), 323-329.
  • J. Trlifaj, Covers, Envelopes, and Cotorsion Theories, Lecture notes, Cortona Workshop, 2000.
  • R. Wisbauer, Foundations of Module and Ring Theory, A handbook for study and research, Revised and translated from the 1988 German edition, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
  • J. Xu, Flat Covers of Modules, Lecture Notes in Mathematics, 1634, Springer- Verlag, Berlin, 1996.

INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE

Year 2019, Volume: 26 Issue: 26, 53 - 75, 11.07.2019
https://doi.org/10.24330/ieja.586945

Abstract

Several authors have been interested in cotorsion theories. Among these theories we figure the pairs $(\mathcal P_n,\mathcal P_n^{\perp})$, where $\mathcal P_n$ designates the set of modules of projective dimension at most a given integer $n\geq 1$  over a ring $R$. In this paper, we shall focus on homological properties of the class $\mathcal P_1^{\perp}$ that we term the class of $\mathcal P_1$-injective modules. Numerous nice characterizations of rings as well as of their homological dimensions arise from this study. In particular, it is shown that a ring $R$ is left hereditary if and only if any $\mathcal P_1$-injective module is injective and that $R$ is left semi-hereditary if and only if any $\mathcal P_1$-injective module is FP-injective. Moreover, we prove that the global dimensions of $R$ might be computed in terms of $\mathcal P_1$-injective modules, namely the formula for the global dimension and the weak global dimension turn out to be as follows $$\wdim(R)=\sup \{\fd_R(M): M\mbox { is a }\mathcal P_1\mbox {-injective left } R\mbox {-module} \}$$ and $$\gdim(R)=\sup \{\pd_R(M):M \mbox { is a }\mathcal P_1\mbox {-injective left }R\mbox {-module}\}.$$ We close the paper by proving that, given a Matlis domain $R$ and an $R$-module $M\in\mathcal P_1$, $\Hom_R(M,N)$ is $\mathcal P_1$-injective for each $\mathcal P_1$-injective module $N$ if and only if $M$ is strongly flat.

References

  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974.
  • M. Auslander and M. Bridger, Stable Module Theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969.
  • H. Bass, Finitistic dimension and a homological generalization of semi-primary rings, Trans. Amer. Math. Soc., 95 (1960), 466-488.
  • S. Bazzoni and D. Herbera, Cotorsion pairs generated by modules of bounded projective dimension, Israel J. Math., 174 (2009), 119-160.
  • S. Bazzoni and L. Salce, Almost perfect domains, Colloq. Math., 95(2) (2003), 285-301.
  • S. Bouchiba, Finiteness aspects of Gorenstein homological dimensions, Colloq. Math., 131(2) (2013), 171-193.
  • T. J. Cheatham and D. R. Stone, Flat and projective character modules, Proc. Amer. Math. Soc., 81(2) (1981), 175-177.
  • E. E. Enochs and O. M. G. Jenda, On Gorenstein injective modules, Comm. Algebra, 21(10) (1993), 3489-3501.
  • 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.
  • L. Fuchs and S. B. Lee, The functor Hom and cotorsion theories, Comm. Algebra, 37(3) (2009), 923-932.
  • L. Fuchs and S. B. Lee, Weak-injectivity and almost perfect domains, J. Algebra, 321(1) (2009), 18-27.
  • L. Fuchs and S. B. Lee, On weak-injective modules over integral domains, J. Algebra, 323(7) (2010), 1872-1878.
  • L. Fuchs and S. B. Lee, On modules over commutative rings, J. Aust. Math. Soc., 103(3) (2017), 341-356.
  • L. Fuchs and L. Salce, Modules Over Non-Noetherian Domains, Mathematical Surveys and Monographs, 84, American Mathematical Society, Providence, RI, 2001.
  • Z. H. Gao and F. G. Wang, Weak injective and weak at modules, Comm. Algebra, 43(9) (2015), 3857-3868.
  • H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189 (2004), 167-193.
  • S. B. Lee, Weak-injective modules, Comm. Algebra, 34(1) (2006), 361-370.
  • S. B. Lee, Weak-injective modules over commutative rings, Comm. Algebra, 46(6) (2018), 2500-2509.
  • L. Mao and N. Ding, FP-projective dimensions, Comm. Algebra, 33(4) (2005), 1153-1170.
  • C. Megibben, Absolutely pure modules, Proc. Amer. Math. Soc., 26 (1970), 561-566.
  • M. Raynaud and L. Gruson, Criteres de platitude et de projectivite: Techniques de \plati cation" d'un module, Invent. Math., 13 (1971), 1-89 (in French).
  • J. J. Rotman, An Introduction to Homological Algebra, Pure and Applied Mathematics, 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.
  • B. Stenstrom, Coherent rings and FP-injective modules, J. London Math. Soc., 2(2) (1970), 323-329.
  • J. Trlifaj, Covers, Envelopes, and Cotorsion Theories, Lecture notes, Cortona Workshop, 2000.
  • R. Wisbauer, Foundations of Module and Ring Theory, A handbook for study and research, Revised and translated from the 1988 German edition, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991.
  • J. Xu, Flat Covers of Modules, Lecture Notes in Mathematics, 1634, Springer- Verlag, Berlin, 1996.
There are 27 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Samir Bouchiba This is me

Mouhssine El-arabi This is me

Publication Date July 11, 2019
Published in Issue Year 2019 Volume: 26 Issue: 26

Cite

APA Bouchiba, S., & El-arabi, M. (2019). INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE. International Electronic Journal of Algebra, 26(26), 53-75. https://doi.org/10.24330/ieja.586945
AMA Bouchiba S, El-arabi M. INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE. IEJA. July 2019;26(26):53-75. doi:10.24330/ieja.586945
Chicago Bouchiba, Samir, and Mouhssine El-arabi. “INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE”. International Electronic Journal of Algebra 26, no. 26 (July 2019): 53-75. https://doi.org/10.24330/ieja.586945.
EndNote Bouchiba S, El-arabi M (July 1, 2019) INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE. International Electronic Journal of Algebra 26 26 53–75.
IEEE S. Bouchiba and M. El-arabi, “INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE”, IEJA, vol. 26, no. 26, pp. 53–75, 2019, doi: 10.24330/ieja.586945.
ISNAD Bouchiba, Samir - El-arabi, Mouhssine. “INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE”. International Electronic Journal of Algebra 26/26 (July 2019), 53-75. https://doi.org/10.24330/ieja.586945.
JAMA Bouchiba S, El-arabi M. INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE. IEJA. 2019;26:53–75.
MLA Bouchiba, Samir and Mouhssine El-arabi. “INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE”. International Electronic Journal of Algebra, vol. 26, no. 26, 2019, pp. 53-75, doi:10.24330/ieja.586945.
Vancouver Bouchiba S, El-arabi M. INJECTIVE MODULES WITH RESPECT TO MODULES OF PROJECTIVE DIMENSION AT MOST ONE. IEJA. 2019;26(26):53-75.