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NOTE ON THE DW RINGS

Year 2019, , 43 - 54, 08.01.2019
https://doi.org/10.24330/ieja.504108

Abstract

In this paper we are mainly concerned with DW rings, i.e., rings
in which every ideal is a w-ideal. We give some new classes of DW rings and we
show how the concept of DW domains is used to characterize Prufer domains
and Dedekind domains. Namely, we prove that a ring is a Prufer domain
(resp., Dedekind domain) if and only if it a coherent (resp., Noetherian) DW
domain with nite weak global dimension.

References

  • N. Q. Ding and J. L. Chen, The flat dimensions of injective modules, Manuscripta Math., 78(2) (1993), 165-177.
  • K. Fujita, In nite dimensional Noetherian Hilbert domains, Hiroshima Math. J., 5(2) (1975), 181-185.
  • S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics, 1371, Springer-Verlag, Berlin, 1989.
  • S. Glaz and W. V. Vasconcelos, Flat ideals. II, Manuscripta Math., 22(4) (1977), 325-341.
  • J. R. Hedstrom and E. G. Houston, Some remarks on star-operations, J. Pure Appl. Algebra, 18(1) (1980), 37-44.
  • W. Heinzer, Integral domains in which each non-zero ideal is divisorial, Mathematika, 15 (1968), 164-170.
  • H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189(1-3) (2004), 167-193.
  • E. G. Houston and M. Zafrullah, Integral domains in which each t-ideal is divisorial, Michigan Math. J., 35(2) (1988), 291-300.
  • E. G. Houston, S. Kabbaj and A. Mimouni, ?-Reductions of ideals and Prufer v-multiplication domains, J. Commut. Algebra, 9(4) (2017), 491-505.
  • B. G. Kang, Prufer v-multiplication domains and the ring R[X]Nv , J. Algebra, 123(1) (1989), 151-170.
  • I. Kaplansky, Commutative Rings, Revised Edition, Univ. Chicago Press, Chicago, 1974.
  • H. Kim and F. G. Wang, On LCM-stable modules, J. Algebra Appl., 13(4) (2014), 1350133 (18 pp).
  • N. Mahdou and M. Tamekkante, On (weak) Gorenstein global dimensions, Acta Math. Univ. Comenian. (N.S.), 82(2) (2013), 285-296.
  • N. Mahdou, M. Tamekkante and S. Yassemi, On (strongly) Gorenstein von Neumann regular rings, Comm. Algebra, 39(9) (2011), 3242-3252.
  • A. Mimouni, TW-domains and strong Mori domains, J. Pure Appl. Algebra, 177(1) (2003), 79-93.
  • A. Mimouni, Integral domains in which each ideal is a w-ideal, Comm. Algebra, 33(5) (2005), 1345-1355.
  • F. G. Wang, On w-projective modules and w- at modules, Algebra Colloq., 4(1) (1997), 111-120.
  • F. G. Wang, Finitely presented type modules and w-coherent rings, J. Sichuan Normal Univ., 33 (2010), 1-9.
  • F. G. Wang and H. Kim, w-Injective modules and w-semi-hereditary rings, J. Korean Math. Soc., 51(3) (2014), 509-525.
  • F. G. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications, 22, Springer, Singapore, 2016.
  • F. G. Wang and R. L. McCasland, On w-modules over strong Mori domains, Comm. Algebra, 25(4) (1997), 1285-1306.
  • F. G. Wang and R. L. McCasland, On strong Mori domains, J. Pure Appl. Algebra, 135(2) (1999), 155-165.
  • F. G. Wang and L. Qiao, The w-weak global dimension of commutative rings, Bull. Korean Math. Soc., 52(4) (2015), 1327-1338.
  • F. G. Wang and J. Zhang, Injective modules over w-Noetherian rings, Acta Math. Sinica (Chin. Ser.), 53(6) (2010), 1119-1130 (in Chinese).
  • H. Yin, F. G. Wang, X. Zhu and Y. Chen, w-Modules over commutative rings, J. Korean Math. Soc., 48(1) (2011), 207-222.
  • M. Zafrullah, The v-operation and intersection of quotient of integral domains, Comm. Algebra, 13(8) (1985), 1699-1712.
  • S. Q. Zhao, F. G. Wang and H. L. Chen, Flat modules over a commutative ring are w-modules, J. Sichuan Normal Univ., 35 (2012), 364-366 (in Chinese).
Year 2019, , 43 - 54, 08.01.2019
https://doi.org/10.24330/ieja.504108

Abstract

References

  • N. Q. Ding and J. L. Chen, The flat dimensions of injective modules, Manuscripta Math., 78(2) (1993), 165-177.
  • K. Fujita, In nite dimensional Noetherian Hilbert domains, Hiroshima Math. J., 5(2) (1975), 181-185.
  • S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics, 1371, Springer-Verlag, Berlin, 1989.
  • S. Glaz and W. V. Vasconcelos, Flat ideals. II, Manuscripta Math., 22(4) (1977), 325-341.
  • J. R. Hedstrom and E. G. Houston, Some remarks on star-operations, J. Pure Appl. Algebra, 18(1) (1980), 37-44.
  • W. Heinzer, Integral domains in which each non-zero ideal is divisorial, Mathematika, 15 (1968), 164-170.
  • H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189(1-3) (2004), 167-193.
  • E. G. Houston and M. Zafrullah, Integral domains in which each t-ideal is divisorial, Michigan Math. J., 35(2) (1988), 291-300.
  • E. G. Houston, S. Kabbaj and A. Mimouni, ?-Reductions of ideals and Prufer v-multiplication domains, J. Commut. Algebra, 9(4) (2017), 491-505.
  • B. G. Kang, Prufer v-multiplication domains and the ring R[X]Nv , J. Algebra, 123(1) (1989), 151-170.
  • I. Kaplansky, Commutative Rings, Revised Edition, Univ. Chicago Press, Chicago, 1974.
  • H. Kim and F. G. Wang, On LCM-stable modules, J. Algebra Appl., 13(4) (2014), 1350133 (18 pp).
  • N. Mahdou and M. Tamekkante, On (weak) Gorenstein global dimensions, Acta Math. Univ. Comenian. (N.S.), 82(2) (2013), 285-296.
  • N. Mahdou, M. Tamekkante and S. Yassemi, On (strongly) Gorenstein von Neumann regular rings, Comm. Algebra, 39(9) (2011), 3242-3252.
  • A. Mimouni, TW-domains and strong Mori domains, J. Pure Appl. Algebra, 177(1) (2003), 79-93.
  • A. Mimouni, Integral domains in which each ideal is a w-ideal, Comm. Algebra, 33(5) (2005), 1345-1355.
  • F. G. Wang, On w-projective modules and w- at modules, Algebra Colloq., 4(1) (1997), 111-120.
  • F. G. Wang, Finitely presented type modules and w-coherent rings, J. Sichuan Normal Univ., 33 (2010), 1-9.
  • F. G. Wang and H. Kim, w-Injective modules and w-semi-hereditary rings, J. Korean Math. Soc., 51(3) (2014), 509-525.
  • F. G. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications, 22, Springer, Singapore, 2016.
  • F. G. Wang and R. L. McCasland, On w-modules over strong Mori domains, Comm. Algebra, 25(4) (1997), 1285-1306.
  • F. G. Wang and R. L. McCasland, On strong Mori domains, J. Pure Appl. Algebra, 135(2) (1999), 155-165.
  • F. G. Wang and L. Qiao, The w-weak global dimension of commutative rings, Bull. Korean Math. Soc., 52(4) (2015), 1327-1338.
  • F. G. Wang and J. Zhang, Injective modules over w-Noetherian rings, Acta Math. Sinica (Chin. Ser.), 53(6) (2010), 1119-1130 (in Chinese).
  • H. Yin, F. G. Wang, X. Zhu and Y. Chen, w-Modules over commutative rings, J. Korean Math. Soc., 48(1) (2011), 207-222.
  • M. Zafrullah, The v-operation and intersection of quotient of integral domains, Comm. Algebra, 13(8) (1985), 1699-1712.
  • S. Q. Zhao, F. G. Wang and H. L. Chen, Flat modules over a commutative ring are w-modules, J. Sichuan Normal Univ., 35 (2012), 364-366 (in Chinese).
There are 27 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mohammed Tamekkante This is me

Refat Abdelmawla Khaled Assaad This is me

El Mehdi Bouba This is me

Publication Date January 8, 2019
Published in Issue Year 2019

Cite

APA Tamekkante, M., Assaad, R. A. K., & Bouba, E. M. (2019). NOTE ON THE DW RINGS. International Electronic Journal of Algebra, 25(25), 43-54. https://doi.org/10.24330/ieja.504108
AMA Tamekkante M, Assaad RAK, Bouba EM. NOTE ON THE DW RINGS. IEJA. January 2019;25(25):43-54. doi:10.24330/ieja.504108
Chicago Tamekkante, Mohammed, Refat Abdelmawla Khaled Assaad, and El Mehdi Bouba. “NOTE ON THE DW RINGS”. International Electronic Journal of Algebra 25, no. 25 (January 2019): 43-54. https://doi.org/10.24330/ieja.504108.
EndNote Tamekkante M, Assaad RAK, Bouba EM (January 1, 2019) NOTE ON THE DW RINGS. International Electronic Journal of Algebra 25 25 43–54.
IEEE M. Tamekkante, R. A. K. Assaad, and E. M. Bouba, “NOTE ON THE DW RINGS”, IEJA, vol. 25, no. 25, pp. 43–54, 2019, doi: 10.24330/ieja.504108.
ISNAD Tamekkante, Mohammed et al. “NOTE ON THE DW RINGS”. International Electronic Journal of Algebra 25/25 (January 2019), 43-54. https://doi.org/10.24330/ieja.504108.
JAMA Tamekkante M, Assaad RAK, Bouba EM. NOTE ON THE DW RINGS. IEJA. 2019;25:43–54.
MLA Tamekkante, Mohammed et al. “NOTE ON THE DW RINGS”. International Electronic Journal of Algebra, vol. 25, no. 25, 2019, pp. 43-54, doi:10.24330/ieja.504108.
Vancouver Tamekkante M, Assaad RAK, Bouba EM. NOTE ON THE DW RINGS. IEJA. 2019;25(25):43-54.